text
stringlengths 6
976k
| token_count
float64 677
677
| cluster_id
int64 1
1
|
|---|---|---|
Millbrae StatisticsTherefore, I have experience with many of the areas of discrete math typically encountered in introductory college coursework: set theory, combinatorics, probability theory, matrices and operations research. Actuarial Science is the field of study that utilizes mathematics and statistics in ass... | 677.169 | 1 |
As atudents progress in thier educational pathway, more knowledge and skills will be required. This course will foster a development and understanding of mathematics in the real world. Students will acquire skills in adding, subtracting, multiplyuing and diving signed numbers which will include integers. Students will solve multi-step equations involving the real number system and algebraic thinking. Problems solving in this course includes applications of ratios, proportion, fractions, and percents. It continues to develop other important mathematics topics including patterns, functions, gemoetry, measurement, probability, and statistics. It provides hands-on, visuals for students who are below grade level as well as renrichment for advanced students.
Algebra I is intended to build a foundation for all higher math classes. It is the brige from the concrete to the abstract study of mathematics. This course will review algebraic expressions, integers, and mathematical proporties that will lead into working with variables and linear equations. There will be an in-depth study of graphing, polynomials, quadratic equations, data analysis, and systems of equations through direct class instruction, group work, homework, and Fuse (I-pads). | 677.169 | 1 |
Discover Mathematics Through Investigation In Symmetry, Shape, and Space, geometry is the framework for an introduction to mathematics. The visual nature of geometry allows students to use their intuition and imagination while developing the ability to think critically. The beauty of the material lies in students discovering mathematics as mathematicians do through investigation. Many of the exercises require students to express their ideas clearly in writing, while others require drawings or physical models, making the mathematics a more hands-on experience. The book is written so that each chapter is essentially independent of the others to allow for flexibility. The text activities and exercises can serve as enrichment projects at elementary and secondary levels. Mathematics professionals and educators will enjoy its informal approach and will find the explorations of nontraditional geometric topics such as billiards, theoretical origami, tilings, mazes, and soap bubbles intriguing. A companion Sketchpad Student Lab Manual can be packaged with The Geometer Sketchpad or KaleidoMania at a special price.
Regularly:
£33.00 excl Vat
On Sale:
£17.00 excl Vat
Regularly:
£33.00 inc Vat
On Sale:
£17.00 inc Vat
Use the form below to email a friend about this product.
All required fields are marked with a star (*). Click the 'Submit' button at the bottom of this form to proceed. | 677.169 | 1 |
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more.
Course Hero has millions of course specific materials providing students with the best way to expand
their education.
Biology 111 Lecture 1 - Introduction to Content and Science 8/29/2007What is organismal biology? ! - Study of diversity of life on EarthI. Intro to ScienceA. What is Science? ! ! ! ! ! ! - A way of asking questions and gaining answers according t
Fall 2008Econ 3130Problem Set 1: Budget SetsThis problem set must be done on graph paper. If a problem calls for two or more graphs on the same axes, use dierent colors. 1. You have a wealth of $400 to spend on two commodities. Commodity 1 costs
Fall 2008Econ 3130Problem Set 2: Preferences and UtilityThis problem set must be done on graph paper. If a problem calls for two or more graphs on the same axes, use dierent colors. 1. Adam likes both pizza and wings. More,of course, is always b
Laboratory Exercise Number 1 Getting Started with Matlab Background: This laboratory exercise is intended to begin our introduction to programming in Matlab, its Integrated Development Environment (IDE), and its ability to draw graphs based on an equ
Laboratory Exercise Number 2 Assignment Statements and Functions Hot enough for you?orIt aint the heat, its the humidityorYes, I know its 122E here in Phoenix, but its a dry heat Assignment: This laboratory and its associated homework exercise
Math 115 First Midterm ExamFebruary 7, 2006Name: Instructor: Section Number:1. Do not open this exam until you are told to begin. 2. This exam has 8 pages including this cover. There are 8 questions. 3. Do not separate the pages of the exam. If 10, 2004 anMATH 115 FIRST MIDTERM EXAMOctober 8, 2003 any
MATH 115 FIRST MIDTERM EXAMOctober 8, 2003NAME: INSTRUCTOR: SOLUTION KEY SECTION NO:1. Do not open this exam until you are told to begin. 2. This exam has 9 pages including this cover. There are 10 questions. 3. Do not separate the pages of the
MATH 115 FIRST MIDTERM EXAM SOLUTIONS1. (2 points each) Circle True or False for each of the following problems. Circle True only is the statement is always true. No explanation is necessary.1 (a) log( A ) = log(A).TRUEFalse(b) If f (x) =
Practice Final Exam Each of the following species contains two bonds EXCEPT A) O2 B) HCCH C) CO2 D) CNE) N2 Which of the following species is(are) planar? (i) CO32(ii) XeF4 (iii) H2NNH21.2.A) (i) only B) (i) and (ii) only C) (i) and (iii) only
Math 115 First Midterm ExamSolutionsName: Instructor: Section Number:1. Do not open this exam until you are told to begin. 2. This exam has 8 pages including this cover. There are 8 questions. 3. Do not separate the pages of the exam. If any pagPractice Exam 3a 1. Figure 10.21 from the text shows HCl, with one orbital centered on hydrogen and one orbital centered on chlorine. What is the orbital centered on chlorine?(A) (B) (C) (D) (E)2p 3p 3d 3s 4s2. Which of the following ground sta
Thalassemia results in the under production of globin protein, often through mutations in regulatory genes, or structural abnormalities in the globin proteins themselves. The two conditions may overlap, however, since some conditions which cause abno | 677.169 | 1 |
Course Description
MTH105
Intro - Contemporary Mathematics
- F/W/Sp This course surveys the broad applicability of mathematics as a
problem-solving tool and the breadth of phenomena that mathematics can
model. A wide range of real world problems are examined using the tools of
mathematics. The course focuses on development of mathematical maturity, and
problem-solving. Course topics are selected from probability, statistics,
personal finance, population growth, symmetry, linear programming, fair
division and voting theory. A computer laboratory is required. | 677.169 | 1 |
Mathematics and Computer Science Undergraduate Courses
Mathematics Courses
MAT 40 Pre-College Mathematics
(4)
This course is designed to promote mathematical literacy among liberal arts students and to prepare students for GSR 104. The approach in this course helps students increase their knowledge of mathematics, sharpen their problem-solving skills, and raise their overall confidence in their ability to learn and communicate mathematics. Technology is integrated throughout to help students interpret real-life data algebraically, numerically, symbolically, and graphically. Topics include calculator skills, number sense, basic algebraic manipulation, solving linear equations, graphing of linear equations, and their applications. Access to mathematics instructional software is provided to support and enhance student learning. A graphing calculator is required.
MAT 45 Elementary Algebra
(4)
This course covers basic operations with algebraic expressions, solving equations in one variable, linear equations and their graphs, linear inequalities, exponents, multiplying and dividing polynomials, and factoring polynomials. Applications are included throughout. Access to mathematics instructional software is provided to support and enhance student learning. A graphing calculator is required.
MAT 55 Intermediate Algebra
(4)
This course covers rational expressions, systems of linear equations in two variables, radicals, and complex numbers, quadratic equations, graphs of quadratic functions, exponential and logarithmic functions. Applications are included throughout. Access to mathematics instructional software is provided to support and enhance student learning. A graphing calculator is required.
Prerequisite:
MAT 045 or equivalent, or a satisfactory score on appropriate placement exam.
MAT 101 Introductory Mathematical Applications
(3)
Linear, quadratic, exponential, and logarithmic functions. Ratios, percentages, matrices, and linear programming emphasizing applications to various branches of the sciences, social studies, and management. Credit will not be allowed if student has passed Math 130. This course will not be counted toward a major in the department.
Prerequisite:
MAT 055 or equivalent.
MAT 102 Introductory Probability and Statistics
(3)
Basic concepts of probability and statistics, and applications to the sciences, social sciences, and management. Probability, conditional probability, Bayes Formula, Bernoulli trials, expected value, frequency distributions, and measures of central tendency. Credit will not be allowed for MAT 102 if student has previously passed MAT 130; 102 will not be counted toward a major in the department.
Prerequisite:
MAT 055 or equivalent, or permission of the department chair.
MAT 125 College Algebra
(3)
This course provides a survey of the algebra topics necessary for Calculus. Topics covered include the analysis of graphs of basic functions, transformations of graphs, composition of functions, inverse functions, quadratic functions and their graphs, polynomial and rational inequalities, absolute value inequalities, radicals and fractional exponents, exponential and logarithmic functions and equations, exponential growth and decay problems, and the analysis of circles, parabolas, ellipses, and hyperbolas. MAT 125 consists of the first half of MAT 130. Passing both MAT 125 and 126 is equivalent to passing MAT 130.
Prerequisite:
MAT 055 or the equivalent, or a satisfactory score on appropriate placement exam.
MAT 130 Precalculus
(4)
This course emphasizes the meaning and application of the concepts of functions. It covers polynomial, rational, exponential, logarithmic and trigonometric functions and their graphs, trigonometric identities, and sequences and series. Passing both MAT 125 and 126 is equivalent to passing MAT 130.
Prerequisite:
A grade of C or above in MAT 055 or the equivalent, or a satisfactory score on appropriate placement exam.
MAT 145 Calculus for Business and Social Sciences
(3)
This course emphasizes the applications of the following topics in Business and Social Sciences: Functions and their graphs, exponential and logarithmic functions, limits and continuity, and differentiation's and integrations in one and several variables. Credit will not be allowed if student has passed MAT 150. This course will not be counted toward a major in the department.
Prerequisite:
MAT 130 or the equivalent.
MAT 150 Calculus I
(4)
Limit processes, including the concepts of limits, continuity, differentiation, and integration of functions. Applications to physical problems will be discussed.
Prerequisite:
A grade of C or better in either MAT 126 or MAT 130.
MAT 171 Basic Concepts of Mathematics for Early Childhood and Elementary School Teachers I
(3)
This course is the first part of a two-semester course sequence with MAT 172. This course is designed for prospective early childhood and elementary school teachers. The contents of this course include concepts and theories underlying early childhood and elementary school mathematics. The students will explore the "why" behind the mathematical concepts, ideas, and procedures. Topics include problem solving, whole numbers and numeration, whole numbers operations and properties, number theory, fractions, decimals, ratio and proportion, and integers.
Prerequisite:
GSR 104 or the equivalent, or permission of the department chair. This course is not open to mathematics majors.
MAT 172 Basic Concepts of Mathematics for Early Childhood and Elementary School Teachers II
(3)
This course is the second part of a two-semester course sequence with MAT 171. This course is designed for prospective early childhood and elementary school teachers. The contents of this course include concepts and theories underlying early childhood and elementary school mathematics. The students will explore the "why" behind the mathematical concepts, ideas and procedures. Topics include rational and real numbers, introduction to algebra, Euclidean and solid geometry, statistics, and probability.
Prerequisite:
MAT 171. This course is not open to mathematics majors.
MAT 195 Special Topics
(1-5)
Special topics in the discipline, designed primarily for freshmen. Students may enroll in 195 Special Topics multiple times, as long as the topics differ.
MATMAT 307 Linear Algebra
(3)
This course covers the fundamental concepts of vector spaces, linear transformations, systems of linear equations, and matrix algebra from a theoretical and a practical point of view. Results will be illustrated by mathematical and physical examples. Important algebraic (e.g., determinants and eigenvalues), geometric (e.g., orthogonality and the Spectral Theorem), and computational (e.g., Gauss elimination and matrix factorization) aspects will be studied.
Prerequisite:
MAT 205 or permission of department chair.
MAT 313 Introduction to Probability
(3)
This course is the first part of a two-semester sequence with MAT 314, with a focus on basic probability. It covers descriptive statistics, sample spaces and events, axioms of probability, counting techniques, conditional probability and independence, distribution of discrete and continuous random variables, joint distributions, and the central limit theorem.
Prerequisite:
MAT 205.
MAT 314 Applied Statistics I
(3)
This course is the second part of a two-semester course sequence with MAT 313, with a focus on applied statistics. It covers basic statistical concepts, graphical displays of data, sampling distribution models, hypothesis testing, and confidence intervals. A statistical software package is used.
Prerequisite:
MAT 313.
MAT 320 History of Mathematics
(3)
A survey of the history of mathematics from antiquity through modern times.
MAT 451 Internship
(3)
This is a one-semester internship in which the student works for at least 60 hours in an applied mathematical or statistical setting under the supervision and guidance of the course instructor and on-site professionals in the field.
Prerequisite:
Mathematics major and permission of the instructor.
MAT 455 Advanced Calculus I
(3)
This course is the first part of a two-semester course sequence with MAT 456. This course covers a theoretical approach to calculus of functions of one and several variables. Limits, continuity, differentiability, Reimann integrability, sequences, series, and contour integration.
Prerequisite:
MAT 206, 210, 307.
MAT 456 Advanced Calculus II
(3)
This course is the second part of a two-semester course sequence with MAT 455. This course covers a theoretical approach to calculus of functions of one and several variables. Limits, continuity, differentiability, Reimann integrability, sequences, series, and contour integration.
Prerequisite:
MAT 455.
MATMAT 499 Independent Study
(1-3)
Intensive supervised study and research on topics of the student's selection.
CSC 150 Computer Programming II
(3)
This course will continue the development of discipline in program design, in style and expression, and in debugging and testing, especially for larger programs. It will also introduce algorithms analysis and basic aspects of string processing, recursion, internal search/sort methods, and simple data structures.
Prerequisite:
A grade of C or better in CSC 130.
CSC 195 Special Topics
(1-5)
Special topics in the discipline, designed primarily for freshmen. Students may enroll in 195 Special Topics multiple times, as long as the topics differ.
Prerequisite:
Permission of the instructor.
CSC 201 Introduction to Computer Organization
(3)
This course gives the organization and structuring of the major hardware components of computers. It provides the fundamentals of logic design and the mechanics of information transfer and control within a digital computer system.
CSCCSC 305 Introduction to File Processing
(3)
This course will introduce concepts and techniques of structuring data on bulk storage devices, provide experience in the use of bulk storage devices, and provide the foundation for applications of data structures and file processing techniques.
Prerequisite:
CSC 150; MAT 140.
CSC 315 Data Structure and Algorithm Analysis
(3)
This course will apply analysis and design techniques to nonnumeric algorithms that act on data structures. It will also use algorithmic analysis and design criteria in the selection of methods for data manipulation in the environment of a database management system.
Prerequisite:
CSC 150; MAT 140; CSC 305 recommended.
CSC 326 Operating Systems and Computer Architecture
(3)
The course will introduce the major concept areas of operating systems principles, develop an understanding of both the organization and architecture of computer systems at the register-transfer and programming levels of system description, and study interrelationships between the operating system and the architecture of computer systems.
Prerequisite:
CSC 150, CSC 315; MAT 140; CSC 202 recommended.
CSC 336 Organization of Programming Languages
(3)
This course will develop an understanding of the organization of programming languages, especially the run time behavior of programs. It will also introduce the formal study of programming language specification and analysis and will continue the development of problem solution and programming skills introduced in the elementary level material.
CSC 341 Software Engineering
(3)
This course will present a formal approach to state-of-the-art techniques in software design and development. It will expose students to the entire software life cycle, which includes feasibility studies, the problem specification, the software requirements, the program design, the coding phase, debugging, testing and verification, benchmarking, documentation, and maintenance. An integral part of the course will be involvement of students working in teams in the development of a large scale software project.
CSC 403 Computer Networking
(3)
The fundamental principles of computer communications. The Open Systems Interconnection Model is used to provide a framework for organizing computer communications. Local area and wide area networks are discussed. The principles of Internetworking are introduced. Communications software is used to illustrate the principles of the course.
Prerequisite:
CSC 150.
CSC 406 Object Oriented Programming
(3)
This course will cover all of the major features of a selected Object Oriented programming language as well as Object Oriented design principles such as: reusability of code, data abstraction, encapsulation, and inheritance.
CSCCSC 499 Independent Study
(1-3)
Intensive supervised study and research on topics of the student's selection. | 677.169 | 1 |
resource book which looks at the design of mechanisms, for example gears and linkages, through the eyes of a mathematician. There are a wide variety of examples including car steering, anglepoise lamps, bicycles, cine cameras, folding push chairs and the design of robots. Readers are encouraged to make models throughout and to look for further examples in everyday life. Suitable for GCSE, A level, and mathematics/technology/engineering courses in Further Education. | 677.169 | 1 |
Course Overview
Most students who enter this course are used to calculation-based mathematics, such as algebra, trigonometry, and calculus. The purpose of this course is to help you make the transition to the later math courses in which proofs, logic, language, and notation play an integral role. The prerequisites are a logical mind, enjoyment of patterns, and a willingness to work. Usually completion of Calculus II and your interest are enough.
Content Goals:
facility in interpreting and using mathematical language and notation;
a firm background in elementary logic and practice in reasoning;
experience dealing with sets, functions, relations;
improvement of computer skills, particularly Mathematica;
Skills:
asking good questions;
discovering and writing proofs;
evaluating the proofs of others;
knowing when you are correct, when you are on a useful path, and when you're lost;
This year the course becomes 4 credit hours. The content includes material from my own text, Essentials of Mathematics. Most of your time will be spent on digesting the definitions, axioms, and theorems in the book and proving the theorems. In addition, there will be lab problems from various areas of math. These will help you to expand your mathematical horizons, learn to explore, and develop computer skills. I also encourage you to read the recommended book by Ian Stewart, Letters to a Young Mathematician. Other interesting books are found in additional resources.
Grading
Your grade will be based on the following:
homework
15%
labs
10%
3 tests
10% 15% 15%
paper and presentation
10%
final exam
25%
Homework should consume about 8 hours per week outside of class. The first test will occur at the end of Chapter 1, approximately the week of 9/17. Subsequent tests will be announced a week ahead of time. After each test, you will receive an update on your cumulative grade. Shortly after the first test, you will begin work on the paper. Due toward the end of the semester, it will report on an article you have read. Please see the paper and talk guidelines. Thursdays are lab days, meeting in 205E. Lab due dates are announced with each lab. The final exam is Tuesday 12/11, 4-6 pm.
Policies
Becoming a mathematician involves learning new skills and adopting new habits of mind. The purpose of this course is to gently push you in the right directions. Two of the goals, asking your own questions and dealing with the frustration of not knowing answers immediately, are achieved only when the professor takes a step away. Therefore, my teaching style is to let you explore on your own until you're ready for help. I have not abandoned you, I'm simply transferring to you the responsibility of forming the question.
Class work is subject to some rules that both further our goals and assure that the class runs smoothly.
Read your email regularly, as I sometimes send class announcements that way. Upon request, I can also send confidential grade information to you at your Stetson email address or through Blackboard. You can forward your Stetson email or configure Blackboard for another address. IT can provide help (ext. 7217). If you have special needs, don't hesitate to discuss them, either with me or with the Academic Resources Center. I hope you're looking forward to the semester. | 677.169 | 1 |
Book Description: Certain contemporary mathematical problems are of particular interest to teachers and students because their origin lies in mathematics covered in the elementary school curriculum and their development can be traced through high school, college, and university level mathematics. This book is intended to provide a source for the mathematics (from beginning to advanced) needed to understand the emergence and evolution of five of these problems: The Four Numbers Problem, Rational Right Triangles, Lattice Point Geometry, Rational Approximation, and Dissection. Each chapter begins with the elementary geometry and number theory at the source of the problem, and proceeds (with the exception of the first problem) to a discussion of important results in current research. The introduction to each chapter summarizes the contents of its various sections, as well as the background required. The book is intended for students and teachers of mathematics from high school through graduate school. It should also be of interest to working mathematicians who are curious about mathematical results in fields other than their own. It can be used by teachers at all of the above mentioned levels for the enhancement of standard curriculum materials or extra-curricular projects. | 677.169 | 1 |
ne... read more
Customers who bought this book also bought:
Our Editors also recommend:
The Beauty of Geometry: Twelve Essays by H. S. M. Coxeter Absorbing essays demonstrate the charms of mathematics. Stimulating and thought-provoking treatment of geometry's crucial role in a wide range of mathematical applications, for students and mathematicians.
Geometry: A Comprehensive Course by Dan Pedoe Introduction to vector algebra in the plane; circles and coaxial systems; mappings of the Euclidean plane; similitudes, isometries, Moebius transformations, much more. Includes over 500 exercises.
Euclidean Geometry and Transformations by Clayton W. Dodge This introduction to Euclidean geometry emphasizes transformations, particularly isometries and similarities. Suitable for undergraduate courses, it includes numerous examples, many with detailed answers. 1972 edition.
Product Description:
needed to develop a feeling for the subject or when they illustrate a general method. On the other hand, an unusual amount of space is devoted to the discussion of the fundamental concepts of distance, motion, area, and perpendicularity. Topics include the projective plane, polarities and conic sections, affine geometry, projective metrics, and non-Euclidean and spatial geometry. Numerous figures appear throughout the text, which concludes with a bibliography and index | 677.169 | 1 |
Application for Math Lab Employment
If
interested in working in the math lab, please fill out the
form below. The math lab director will contact you shortly
thereafter for an initial interview. For a printable form,
please click here: Printable
Math Lab Application.
Contact Information
Name:
E -mail:
Local Phone:
Local Address:
Educational Information
Major:
Minor:
Cumulative GPA
Major GPA:
# Semesters Here:
Expected
Graduation Date:
Course information for tutors:
Math Lab tutors are required to have completed Math 114 or
equivalent.
Indicate courses you have
taken at UWEC
(check all that apply)
Indicate courses you worked
as a grader
(check all that apply)
Math
010
Math
010
Math
020
Math
020
Math
104
Math
104
Math
108
Math
108
Math
109
Math
109
Math
111
Math
111
Math
112
Math
112
Math
113
Math
113
Math
114
Math
114
Math
215
Math
215
Math
216
Math
216
Math
246
Math
246
Math
314
Math
314
Math
324
Math
324
Math
345
Math
345
Math
346
Math
346
Math
347
Math
347
Math
425
Math
425
References
List at least two faculty who can serve as references. Please
include how they know you (Which class(es) did you have with
them? Did you grade for them?).
Reference 1:
Name:
Reference 2:
Name:
Please briefly answer the following
1. Why do you want to be a Math Lab tutor?
2. What skills and qualities do you posses that will enhance
your ability to tutor?
3. Have you had any experience tutoring (formally or
informally)? If yes, please describe. | 677.169 | 1 |
Algebra 2 and lists the items related to each objective that appear in the. Pre-Course Diagnostic Test, Post-Course Test, and End-of-Course. Practice Tests in this ... Chapter Standardized Tests A and B Two parallel versions of a standardized ...
for STAR District and Test Site Coordinators and Research ... Table of Contents. Acronyms and Initialisms in the Post-Test Guide. ... Chapter I.2 Introduction .
CHAPTER 14 The Core-Plus Mathematics Project Perspectives and ...
This chapter provides a brief overview of the CPMP curriculum in terms of its ..... For the Course 2 study, we asked these 11 schools to posttest as many of the .... 2 Posttest (Part 1) also contains two contextual subtests, one algebraic and the ... | 677.169 | 1 |
My philosophy is simple: Quality learning takes TIME, PATIENCE AND COMMITMENT. I will provide you time and patience each day to be successful, please make sure you are committed and you will have great success this year in math. Learning isn't a race-so relax and focus on the task at hand.
A list of materials you will need are: TI-30 calculator (solar is the best) Two notebooks or loose leaf paper, pencil and highlighters. NOTE: All notes and homework assignments for each day will be updated on a regular basis. If you are ill or miss school please download and print out this information so you stay current with the class. Refer to your specific class for more information.
Algebra I This course develops students' ability to recognize, represent, and solve problems involving relations among quantitative variables. Key functions studied are linear, exponential, power and periodic functions using graphic, numeric and symbolic representations. Students will also develop the ability to to analyze data, to recognize and measure variation and to understand the patterns that underlie probabilistic situations.
The Online Holt Algebraintalg Password: erhsintalg
Please refer to the calendar for homework assignments and class downloads for section notes (where applicable). NOTE: Use the link below to view these downloads.
Geometry A/B (2 terms,2 credits) This course is designed to develop formal and informal reasoning. Geometry develops visual thinking and students' ability to construct, reason with, interpet, and apply mathematical models of patterns in two and three dimensions.
The Online Holt Geometrygeom1 Password: erhsgeom
Please refer to the calendar for homework assignments and class downloads for section notes (where applicable). NOTE: Use the link below to view these downloads. | 677.169 | 1 |
0321385179
9780321385178
0321830881
9780321830883 Since they are fundamental to the study of linear algebra, students' understanding of these concepts is vital to their mastery of the subject. David Lay introduces these concepts early in a familiar, concrete Rn setting, develops them gradually, and returns to them again and again throughout the text so that when discussed in the abstract, these concepts are more accessible. «Show less... Show more»
Rent Linear Algebra and Its Applications 4th Edition today, or search our site for other Lay | 677.169 | 1 |
Grade 9, Principles of Mathematics, Academic (Enriched)
Course Code:
MPM1D3
Same as MPM1D but it is enriched. In addition, many interesting math topics will be explored. Graphing Calculators will used extensive in exploring mathematics properies and mathematics modelling. All students will prepare and participate in Garde 9 Pascal mathematics competition sponsored by University of Waterloo. Grade 9 Principles of Mathematics, Academic: This course covers four strands: 1) Number Sense and Algebra; 2) Relationships; 3) Analytic Geometry; 4) Measurement and Geometry. The course enables students to develop generalizations of mathematical ideas and methods through the exploration of applications, the effective use of technology, and abstract reasoning. Students will investigate relationships to develop equations of straight lines in analytic geometry, explore relationships between volume and surface area of objects in measurement, and apply extended algebraic skills in problem solving. Students will engage in abstract extensions of core learning th ore skills and deepen their understanding of key mathematical concepts. | 677.169 | 1 |
These
Interactive Diagrams are visual tools to help students explore the
dynamic aspects of mathematics and to help them connect physical
and dynamic representations of real-world situations to algebraic
symbolism. | 677.169 | 1 |
understanding an assignment
prepping for a quiz or exam
analyzing data
writing computer code
solving a problem
calculating and interpreting statistics
proving or applying a theorem
using quantitative software
designing an experiment
Credit for
photo: Albert Einstein Institute at the Max Planck Institute for
Gravitational Physics and Konrad-Zuse-Zentrum, Berlin. Visualization by
Werner Benger and Edward Seidel, director of Louisiana State University's
Center for Computation and Technology | 677.169 | 1 |
Trigonometry and Calculus Problem of the Week - Math Forum
Trigonometry and calculus problems from a variety of sources, including textbooks, math contests, NCTM books, puzzle books, and real-life situations, designed to reflect different levels of difficulty. From 1998 until 2002, the goal was to challenge students with non-routine problems and encourage them to put their solutions into words. Different types of problems were used to reach a diverse group of calculus students.
more>>
Carousel Math - Web Feats Workshop II
Carousels are as reliant on the laws of motion as roller coasters. Let's take a ride on the new Bear Mountain Carousel at Bear Mountain State Park in New York. After enjoying the ride, take the data that has been collected and use the applet to explore
...more>>
CASTLE Software
Computer Assisted Student Tutorial Learning Environment (CASTLE) software for Windows, for high school science, social studies, and mathematics - algebra, geometry (including probability), and trigonometry. Includes a tutorial program for student review,
...more>>
ClassZone - McDougal Littell, Publishers
In ClassZone, McDougal Littell has gathered together all of its activities, interactive online resources, research links, chapter quizzes, and other materials that support its popular middle and high school mathematics textbooks, such as Integrated Math
...more>>
CLK-Calculator - Lars Kobarg
Reverse Polish notation Calcualtor is a calculator program for Windows with unit conversion and matrix calculator, functions for vectors and complex numbers, and a
function plotter.
...more>>
Complex Numbers & Trig - Alan Selby
Complex Numbers and the Distributive Law for Complex Numbers, offering a short way to reach and explain trigonometry, the Pythagorean theorem, trig formulas for dot- and cross-products, the cosine law, and a converse to the Pythagorean theorem. A geometric
...more>>
The Constants and Equations Pages - Jonathan Stott
A growing reference resource providing alphabetically listed categories of some of the more important and useful aspects of maths and special sections on numbers, algebra, trigonometry, integration, differentiation, and SI units and symbols, with in addition
...more>>
CPO Online - Cambridge Physics Outlet
A company founded by teachers and scientists that creates hands-on equipment and curriculum for teaching science, math, and technology from grades 4-12 and beyond, and provides effective professional development in science and math that is both content
...more>>
Creative Geometry - Cathleen V. Sanders
Teachers and students will find creative and interesting "hands-on" projects for most topics in the geometry curriculum. Each project is designed to help students understand, remember, and find value in the concepts of geometry. The pages are organized
...more>>
DeadLine OnLine - Ionut Alex. Chitu
Freeware that graphs equations and precisely estimates their roots. It includes the option to evaluate the function and the first two derivatives, find extrema of the function and integrate numerically. For Visual Basic-enabled computers running Windows
...more>>
Emaths.Info - Vinod Sebastian
Math tools, formulas, tutorials, videos, tables, and "curios," such as the unusual properties of 153, 1729, and 2519. See in particular Emaths' interactive games, which include the N queens problem, Towers of Hanoi, and partition magic, which finds a
...more>> | 677.169 | 1 |
Instructor Class Description
Functions, Models, and Quantitative Reasoning
Explores the concept of a mathematical function and its applications. Explores real world examples and problems to enable students to create mathematical models that help them understand the world in which they live. Each idea will be represented symbolically, numerically, graphically, and verbally. Prerequisite: minimum grade of 2.0 in B CUSP 122, a score of 145-153 on the MPT-AS assessment test, or a score of 147-165 on the MPT-GS assessment test. Offered: AWSp.
Class description
Functions are the key to how mathematical models are built. Various mathematical models will be created through the usage of real world examples. This course is designed to prepare students for calculus I and serves as a prerequisite for B CUSP124. Upon successful completion of the course, students are expected to build solid skills in algebra, trigonometry, logarithms, exponentials, composition of functions, and graphing.
The class will be taught with a mixture of group activities and projects, as well as interactive lectures.
Recommended preparation
Appropriate score on the UWB math placement test.
To prepare for this class, please review your high school algebra and trigonometry. Everything we do will build on these skills.
Class assignments and grading
Homework will be assigned regularly. It will be completed using the Wiley Plus online system. You will need to purchase a registration code for this. Since that registration code includes a complete electronic version of the textbook, there is no need to buy hard copy of the text unless you prefer that and do not wish to print pages from the electronic version yourself. I will also distribute additional materials in class.
In addition to online homeworks, there will be week-long laboratory assignments, in-class worksheets, exams.
Grades will be based on student's performance on assigned work, projects, in-class worksheets and Bilin Z Stiber | 677.169 | 1 |
Algebra I (PAP/Honors)
One Credit Course
Instructor: Ms. Brisa H. Bermea
Algebra I covers the basic structure of real numbers, algebraic equations, and functions. The topics studied are linear equations, inequalities, functions and systems, quadratic equations and functions, polynomial expressions, data analysis, probability and properties of functions. The honors courses provide higher expectations for students. Students have the opportunity to work at an accelerated pace and develop their higher-order thinking skills.
Geometry (Regular)
One Credit Cours
Geometry (PAP/Honors)
One Credit Course The honors courses provide higher expectations for students. Students have the opportunity to work at an accelerated pace and develop their higher-order thinking skills.
Algebra II (Regular)
One Credit Course
Instructor: Ms. Brisa H. Bermea
Algebra II (PAP/Honors)
One Credit Course
Instructor: Mr. Samuel Ayala. The honors courses provide higher expectations for students. Students have the opportunity to work at an accelerated pace and develop their higher-order thinking skills.
Advanced Math
One Credit Course
Instructor: Mrs. Yirah Valverde
This course is designed for students who have completed Algebra 1, Geometry and Algebra II and are in need of additional support after taking Algebra 2. Students in this course will apply concepts from Algebra 1, Geometry and Algebra II to solve meaningful real-world problems. In doing so, they will reinforce their algebra and geometry skills before entering the Pre-Calculus and Calculus class
Pre-Calculus
One Credit Course
Instructor: Mr. Samuel Ayala
Precalculus covers many of the topics covered on previous classes. The topics studied are polynomial, exponential, logarithmic, rational, radical, piece-wise, and trigonometric and circular functions and their inverses. Parametric equations, vectors, and infinite sequences and series are also studied.
Pre-Calc (EA)
One College Credit Course
Instructor: Mr. Reichenbach
This course covers the basic concepts of trigonometry and analytic geometry, including trigonometric functions and their graphs, relationships, and applications. Basic analytical geometry topics include the conics, translations, rotations, and basic vector geometry. At the end of the course the students receiving a passing grade will also receive college credit for this class.
Calculus (EA)
One College Credit Course
Instructor: Dr. Liguori
The Calculus topics are those traditionally offered in the first year of calculus in college, and are designed for students who wish to obtain a semester of advanced placement in college. The topics studied include limits, continuity, derivatives and integrals of algebraic and transcendental functions and their applications, and elementary differential equations. At the end of the course the students receiving a passing grade will also receive college credit for this class. | 677.169 | 1 |
Books
What are some good physical books that actually break down the formulas and algorithms for geometries and processes of algorithmic and generative design, as opposed to books based on their history and examples in architecture. thanks
If i remember right i think its just the cost. For the academic one i think i sent my student ID as proof. As bought mine a few years ago. Its an expensive book but as it will never really date its well worth the investment. | 677.169 | 1 |
grasp concepts and cement your comprehension. You'll also find coverage of the graphing calculator as a problem-solving tool, plus hands-on activities in each chapter that allow you to practice statistics firsthand.
DJVUA Theoretical Introduction to Numerical Analysis presents the general methodology and principles of numerical analysis, illustrating these concepts using numerical methods from realanalysis, linear algebra, and differential equations. The book focuses on how to efficiently represent mathematical models for computer-based study. ...
"[Sedgewick and Flajolet] are not only worldwide leaders of the field, they also are masters of exposition. I am sure that every serious computer scientist will find this book rewarding in many ways." —From the Foreword by Donald E. Knuth Despite growing interest, basic information on methods and models for mathematically analyzing algorithms has rarely been directly accessible to practitioners, researchers, or students. An Introduction to the Analysis of Algorithms, Second Edition, organizes and presents that knowledge, fully introducing primary techniques and results in the field.
This best-selling introduction to language studies includes a huge range of activities and projects, introducing core areas of language structure and grammar through analysis of real texts. Ideal for both A level and beginning undergraduate students, this second edition includes:[list][*]an introductory section on how to use the book ideas over excessively formal treatment while thoroughly covering the material required in an introductory algorithms course. Popular puzzles are used to motivate students' interest and strengthen their skills in algorithmic problem solving. Other learning-enhancement features include chapter summaries, hints to the exercises, and a detailed solution manual.
This book is intended to be a thorough overview of the primary techniques used in the mathematical analysis of algorithms. The material covered draws from classical mathematical topics, including discrete mathematics, elementary realanalysis, and combinatorics; as well as from classical computer science topics, including algorithms and data structures. The focus is on "average-case'' or "probabilistic'' analysis, though the basic mathematical tools required for "worst-case" or "complexity" analysis are covered, as well.... | 677.169 | 1 |
M115A-MidtermOneAdvice
Course: MATH 172a, Fall 2012 School: UCLA Rating:
Word Count: 405
Document Preview solve a problem if you do not have the proper tools.
3. You should understand all of the proofs given in class and be able to recreate them on the midterm.
This does not mean you should go out and memorize all of the proofs. You should go and understand
the main idea, trick, and technique of each proof. Most techniques can be repeated or are useful in
other problems.
4. If you are asked to prove something from lecture, you should try to give the proof done in lecture and
cannot use material after the proof given in lecture. If you are asked to prove something new, anything
from class is fair game.
5. Manage your time during the test. When proving result a on the midterm, you should include as much
detail as you deem necessary. If you are unsure whether to go into more detail, leave it and come back
to the problem if you have time.
6. You should try to have an intuition about how to approach problems. This is the most dicult thing
to do in this course and, if you can do this, you should most denitely succeed. When trying to solve
a problem, you should think about what you are given and what the givens imply, think about what
you are trying to do and how you could do it, think about what techniques we have and which might
be useful, and think about which tools (i.e. results and theorems) you have and how they might be
applied.
7. Do not give up on a problem. Most problems in this course do not involve an absurdly dicult trick
and can be solved by reasoning using the denitions and results done in class.
8. Get a good night sleep the night before the midterm. If you are too tired to think because you crammed
all night, the cramming with not be eective and you will not be able to think criticallyCommon Notation and Symbols in Linear AlgebraPaul SkoufranisSeptember 20, 2011The following is an incomplete list of mathematical notation and symbols that may be used MATH 115A.Shorthand Notation:for allthere existstherefores.t.such that=impli
MATH 115A - Practice Final ExamPaul SkoufranisNovember 19, 2011Instructions:This is a practice exam for the nal examination of MATH 115A that would be similar to the nalexamination I would give if I were teaching the course. This test may or may not
MATH 115A - Practice Midterm OnePaul SkoufranisOctober 9, 2011Instructions:This is a practice exam for the rst midterm of MATH 115A that would be similar to a midterm I wouldgive if I were teaching the course. This test may or may not be an accurate
MATH 115A - MATH 33A Review QuestionsPaul SkoufranisSeptember 13, 2011Instructions:This documents contains a series of questions designed to remind students of the material discussed inMATH 33A. It is recommended that students work through these ques
University of California, Los AngelesMidterm Examination 2November 9, 2011Mathematics 115A Section 5SOLUTIONS1. (6 points) Each part is worth 3 points. For each of the following statements, prove or nd a counterexample.(a) Let V and W be nite dimens
Problem Set 1Math 115A/5 Fall 2011Due: Friday, September 30Please note a typo was corrected in problem 3.Read Chapter 2 of the supplemental material in the text.Problem 1Consider the set Mnn (R) of all n n matrices with real entries with the operati
Problem Set 6Math 115A/5 Fall 2011Due: Friday, November 18Problem 1 (4.4.6)Prove that if M Mnn (F ) can be written in the formM=AB0C,where A and C are square matrices, then det(M ) = det(A) det(C ).Problem 2 (5.1.3)For each of the following mat
Problem Set 7Math 115A/5 Fall 2011Due: Monday, November 28Please note a typo was corrected in problem 2.Problem 1 (5.2.3)For each of the following linear operators T on a vector space V , test T for diagonalizability,and if T is diagonalizable, nd a
Problem Set 8Math 115A/5 Fall 2011Due: Friday, December 2Problem 1 (6.2.2)In each part, apply the Gram-Schmidt process to the given subset S of the inner productspace V to obtain an orthogonal basis for span(S ). Then normalize the vectors in the bas
Section 1.3, exercise 12Prove that the upper triangular matrices form a subspace of Mmn (F).Proof. Let W be the set of upper triangular matrices in the vector space Mmn (F). SinceMmn (F) is a vector space, it contains a zero vector and this vector is t
Life Science 15: Concepts and IssuesLecture 2: Intro to Life Science/Science as a Religion1/12/12I. Age of ScienceII. Who am I?III. Scientific ThinkingScientific MethodOrganizedEmpiricalMethodicalStructured way of finding info about observable e
Life Science 15: Concepts and IssuesLecture 3: Scientific thinking and decision making1/17/12I.II.III.IV.Scientific thinking- an efficient way to learn about and understand the worldHypotheses must be tested with critical experimentsControlling v
Life Science 15: Concepts and IssuesLecture 4: Darwins dangerous idea1/19/12I.II.III.The evolution of starvation resistanceWhat is evolution?What is natural selection?Q: How long can a fly live without food? Can we increase the average time to st
Life Science 15: Concepts and IssuesLecture 5: Nurturing nature: the power of culture1/24/12I.II.III.The four ways that evolution can occurSexual selection: NS can create sex differencesThe norm of reaction illustrates the relationship between nat
Life Science 15: Concepts and IssuesLecture 6: What did Mendel discover?1/26/12I.II.III.IV.Who was Mendel?Physical structure of the genomeWhat did Mendel discover?Sex DeterminationMendel:- why do offspring look like their parents?- 1859: Orig
Life Science 15: Concepts and IssuesLecture 8: Friend and foe are fluid categories2/2/12I.II.III.IV.evidence of kin selectioncooperation is rare in the animal worldcertain conditions are conducive to altruism among non-kinreciprocal altruism in
Life Science 15: Concepts and IssuesLecture 9: Unexpected conflict, unexpected cooperation2/7/12I.II.III.inbreeding and unexpected cooperationmothers love and unexpected conflictreciprocity instills us with a sense of fairnessReduce the perceived
Life Science 15: Concepts and IssuesLecture 12: Proteins, carbs, and fats: nutrition and health2/16/12Macromolecule 1: LipidsFeatures:o not water solubleo major storehouses of energyo good insulatorsMajor Types:o fats/triglycerideso phospholipid
Life Science 15: Concepts and IssuesLecture 13: The trouble with testosterone: hormones and sexdifferences2/21/12Hormones: Chemical signals, secreted into body fluids May reach many cells, but only target cells respond Elicit specific responses in
Life Science 15: Concepts and IssuesLecture 14: Reproduction: eggs are big, sperm are small, and menare dogs2/23/12I.II.III.Were built differently1. Early nurturing is necessarily female, 2. Males have greater reproductivecapacity but no paternit
Life Science 15: Concepts and IssuesLecture 15: Reproduction and mating systems2/28/12I.II.III.IV.What is a mating system?How does an embryo become male or female?Symmetry, heterozygosity, and beautyAnother mysterious motivator: the waist-to-hip
Life Science 15: Concepts and IssuesLecture 18: Why are drugs so good? Caffeine and alcohol: casestudies3/8/12I.II.III.The synapseDo-it-again centers in the brainDrugs can hijack pleasure pathwaysAction potential comes down axon in pre-synaptic
Life Science 15: Concepts and IssuesLecture 19: Flourishing in our alien, industrial environment3/13/12I.II.III.Industrial societies: life in an alien environmentWhat is culture?Culture breaks down the fundamental reproductive equationQ: Why is o
Life Science Final Review:1. Happiness: why is rate/direction of change more important than absolutelevel? How does this relate to material acquisitions? The emotion of happiness is a tool our genes use to cause u to behavein ways that will benefit th
Modern Art: Lecture 1 (1/10/12)To be modern is to know what is not possible anymore. Roland Barthes Typical idea of modernism: revolution, liberation, new possibilities Modern = to be self aware of limitationsQuickTime and adecompressorare needed to
Modern Art: Lecture 3 (1/17/12)Gustave Courbet Burial at Omans, 1850Political reactionHistorical painting- an event from his hometownGenre painting- everyday life, subject turns to the peopleEqualizing of attention across canvasQuickTime and adecom
Modern Art: Lecture 4 (2/19/2012)Manet Nymph Surprised, 1861Allusion to bibles Susanna: surprised while bathing by eldersManet takes out the elders, so the audience = the perpetratorsRole of vision in paintingGaze looks back at the viewer- breaks sto
Modern Art: Lecture 5 (1/24/2012)Opposition to culture of the time and monumental works1874- young painters showed work in opposition to any juried public Salon:Impressionists Exhibition.Called themselves anonymous society/corporationQuickTime and a
Modern Art: Lecture 7 (1/31/2012)QuickTime and adecompressorare needed to see this picture.Georges Seurat Bathers Asnires 1883-84Neo-Impressionists subgroup- rejected by Academy/Salon Salon of the IndependentsMovement toward landscape in Impressioni
Modern Art: Lecture 8 (2/2/2012)QuickTime and adecompressorare needed to see this picture.Paul Gauguin Spirit of the Dead Watching, 1892Liberated, arbitrary color not based upon copying the realLeaves France Tahiti (S. Pacific)MythologizeSearching
Modern Art: Lecture 10 (2/9/2012)QuickTime and adecompressorare needed to see this picture.Cezanne Large Bathers, 1898-1906Painting of bodily experienceRead as swimmers immersed in paintingParts left black, canvas showingGathering of bodies- tendi
Modern Art: Lecture 14 (3/1/2012)Franois Rude Departure of the Volunteers of 1792, 1833 Public monument in reliefo 3D with flat background, like picture planeo hybrid of 2D-3D Equivalent to history painting = narrative sculpture Series of bodies ord
Manet Review no clear narrative- fragmentation, citation of old masters concept of looking recognizable at one instant scenes from everyday life urbanism and its effects- some scenes of bourgeois leisure no event, no pregnant moment idea of uncerta
QuickTime and adecompressorare needed to see this picture.Gustav Klimt The Kiss, 1907-08Unification, love, reconciliationQuickTime and adecompressorare needed to see this picture.Gustav Klimt Expectation, 1905-09Excessive focus on lineStart from
Modern Art: Final Review SessionMatisse, Music: Use of color, pure planes Distortions of body, folding in on itself Has Fauvist concernsDistinction between modernism and avant-garde: Crow* Modernism: class of everything weve seen Avant-garde: at t
Management and Organization TheoryWhy do I need this class?What is management?What skills are needed to be successful?What mistakes do managers make?Can management skills be taught?How do you build a tech powerhouse without offering anystock option
Management and Organization TheoryEnvironment and Competitive AdvantageWhy Do Good Companies Go Bad? 2005 Robert H. Smith School of BusinessUniversity of MarylandHow Good Companies Emerge inthe First Place Good companies successfully emerge in the
SOUTHWEST AIRLINES The essay question stated: Southwest Airlines wasnamed to Business Weeks list of Customer ServiceChamps in 2007 and 2008. The Business Week listranks the best providers of Customer Service, and digsinto the techniques, and tools t
Lessons Business Graduates Apply to theReal World May Include Cheating Business students cheat more than students from any otheracademic discipline. Students have become more cavalier about cheating overthe years. They say theyre only acquiring skill
Ethics/Corporate SocialResponsibilityBMGT364Management and OrganizationalTheory 2005 Robert H. Smith School of BusinessUniversity of MarylandPublic recognitionThe banana giant that found its gentle side Financial Times (UK), December 2002Chiquit
Strategic ManagementDoes Strategy Matter? 2005 Robert H. Smith School of BusinessUniversity of MarylandDifferences in Industry ProfitabilityThe average return on invested capital varies markedly fromindustry to industry.Between 1992 and 2006, for e | 677.169 | 1 |
The Advanced Algebra Tutor: Learning By Example DVD Series teaches students through step-by-step example problems that progressively become more difficult. This DVD covers graphing rational functions in Algebra, as well as a discussion of what rational functions are and why they are important in algebra. Grades 9-College. 36 minutes on DVD. | 677.169 | 1 |
Welcome to the Algebra2go™ study tips page. Mastering math is very much like learning to play a musical instrument or play a sport. These are all heavily skill-based activities in which new techniques are built out of previously learned individual skills. All of these pursuits require time, patience, practice, practice, and more practice. I've collected some of the study tips that I've learned in Professor Perez' classes and listed them here for you. Start using them and soon you might be a math expert like me, Charlie!
(Well maybe not exactly like me.)
Using Your Textbook
Your textbook is one of your most valuable resources, so you should make sure that you use it to its fullest. Before the semester begins, familiarize yourself with the contents. Look for any useful features such as a glossary, mathematical tables, formula lists, an index, practice exams, or homework solutions. Make certain that you know where these resources are located and how to use them.
Reading your textbook can make a huge difference in your performance. When reading the text you need to keep in mind that math textbooks differ from most other literature in three main ways.
1. Math texts present information in a very condensed form. Every sentence is important.
2. Math textbooks use a very precise technical language.
3. Math texts attempt to teach skills as well as to convey information.
Because of these differences, math texts must be approached in a different fashion than most other texts. The approach below uses several techniques to improve understanding.
1. The text is read multiple times improve retention.
2. The text is read in a different manner at each reading to maximize understanding.
3. The text is read actively so as to develop your problem solving skills.
One reading schedule that incorporates these techniques is given below. It can easily be customized to fit your personal preferences. Find an approach that works for you and stick with it.
First Reading: Preview
Preview the text before the lecture. At of the applications of these techniques.
Interlude: Lecture
If
Second Reading: Outlining
Your second reading will be the most comprehensive. Here you will carefully read every sentence. This reading should be an active process and will require that you have pencil and paper (and maybe a calculator). You should record every new vocabulary term and its definition as you encounter them. You should also work through every example problem.
Interlude: Homework
Homework is the first place to apply what you have learned. While working through the problems you will probably use your book as a reference. Try to more than just copy the techniques illustrated by the example problems. The goal of homework is to help you learn how to apply general principles to specific situations. Try to keep your eye on the big picture.
Third Reading: Highlighting
During your third reading the idea is to review topics and examples which are still giving you trouble. When you complete this reading you should be able to answer every problem in the homework and the lecture.
Later Readings: Review
It is a good idea to read the text again before each quiz or test. At each reading the concepts will become more clear and memorable.
As you read through the rest of these study tips, you will notice these readings have already been included.
Lectures
During the lecture your instructor will attempt to give you an overview of the current topic. Being adequately prepared for the lecture and taking full advantage of the opportunity to interact with your instructor will make later parts of the learning process go more smoothly.
Preview the text before the lecture.
At applications of these techniques. If Come to class prepared to ask questions about any new concepts which are not clear after reading the text.
Attendance.
Lectures are an important component of the learning process. Make sure you attend class. Be on time and remain until the end of each session. Eliminate any distractions during class (no texting, etc.). Taking notes can help you organize the material, but it can also prevent you from listening to all parts of the lecture. You need to find a balance between these two outcomes.
Ask questions.
If you have questions about the homework problems, get your questions answered as they arise, either in class or in your instructor's office. Many colleges offer tutoring, and you should take advantage of such services. Don't save up your questions for the day before the exam.
Read the text after the lecture.
This reading should be more comprehensive than the preview was. This is an active process and will require that you have pencil and paper (and maybe a calculator) handy. Read every sentence and record every new vocabulary term and definition as you encounter them. Work through every example problem.
Doing Your Homework
Homework is the first place to apply what you have learned. While working through the problems you will probably use your book as a reference. Try to do more than just copy the techniques illustrated by the example problems. The goal of homework is to help you learn how to apply general principles to specific situations. Try to keep your eye on the big picture.
Do your homework.
Do all the assigned homework problems immediately after the section has been discussed in class. When you work the homework, you should work a group of problems at a time before checking your answers with those in the back of the text. Be sure you make an honest attempt at a problem before looking up the answer. Many students become very good at working backwards from the answers to obtain the solutions to problems. Unfortunately, the answers are not provided on exam and quiz problems.
Manage your time wisely.
Spend some time every day on the course. Spending comparatively little time each day will be more productive than saving up all your work for the weekend or for the week or day before the exam. You should expect to spend at least three hours outside of class for each hour of class time.
Review your homework.
Athletes and musicians often record their performances so they can study them later. Your homework can serve the same purpose. After you've finished the problems, examine the results. You will notice that the problems tend to be grouped together. Ask yourself why the text grouped them that way. What was similar about the problems in one group? What was different about the problems in other groups? What clues would tell you which type you were solving on an exam? Are there faster ways to solve these problems? Learning to analyze problems this way will help you see how to approach problems in the future.
Try using spiral review.
Each time you finish a homework assignment, go back and work a few problems from previous assignments. This technique is known as spiral review. It helps older material remain fresh in your mind, and can also help you see connections between different topics. Repeatedly returning to a topic over an extended period of time is one of the best ways to fully assimilate knowledge.
Review the text.
After you have finished the homework, it is a good idea to review topics and examples which are still giving you trouble. When you complete this reading you should be able to answer every problem in the homework and the lecture.
Focus on the big picture.
Concentrate on learning the concepts behind the solutions to the problems rather than the solutions to individual problems. The point of the homework is to help you master these concepts, not to obtain answers to every problem in the text. After working a series of problems, ask yourself what concepts were illustrated in the problems. Make sure that you understand not only how to apply a certain procedure to a given problem but also why the procedure can be applied and why it works.
Preparing for Exams
Preparing for a math exam can be as important as preparing for a track meet or a musical performance. Attending class, reading the text and working on homework problems will all help you succeed, but there are additional things you can do to improve your performance.
Review the text.
It is a good idea to read the text again before each quiz or test. At each reading the concepts will become more clear and memorable. After you have mastered the details of each problem type, reviewing the text can help you see them in their larger context.
Study with others.
It is often said that two heads are better than one. When studying for a math test, it is certainly true. Working in a group provides you with multiple points of view that you can draw on during an exam. Too large a group can become unwieldy, so you might want to limit the size to around four people.
Randomize chapter tests/reviews.
On most homework assignments similar problems are grouped together. A student will read the instructions for the first problem in a set, and then work out the rest of the problems using the same instructions. The result is that the student only read the instructions one time. This can make it difficult to recognize a problem when it appears out of context on an exam. Randomly select problems from the chapter review or chapter test, and try to identify the type of problem from the instructions. It can be difficult at first, but this skill will save you time and stress on an exam.
Positive attitude.
Thinking positively won't make you a math whiz by itself, but it can help you overcome the nervousness which might prevent you from showing off your math talents. Try thinking of the exam as a chance to demonstrate how much you know, rather than a judgement of your person. It can help if you practice making positive statements out loud. For example, "I can't wait for this test!" and "I am going to ace this exam!" You may feel silly at first, but you'll find it hard to be afraid of a test when you keep hearing yourself say positive things about it.
Get a good night's sleep.
If your body is not well rested then neither is your mind. It is often tempting to stay up all night cramming before a test. The fact is that this approach is rarely useful. The relationships between mathematical concepts are very complicated and it takes time for your mind to assimilate them. If you have not grasped them yet then one more night is probably not going to make much of a difference. On the other hand being so tired that you can not think clearly can significantly lower you test score.
Eat right.
Taking a test is work and like all work it requires energy. You need to keep your blood sugar up so that your brain has ready access to fuel when it needs it. Do not skip meals out of test anxiety. You might try having a healthy snack like nuts or fresh fruit shortly before the test.
Exercise.
A little nervousness isn't necessarily bad (it can keep you alert), but too much fear overwhelms your ability to think clearly. Exam anxiety has physical aspects as well as mental. Physical activity can help you focus that nervous energy. Some people like to jog or swim a few hours before an exam. If you already use some technique to help you relax before a sports contest or artistic performance, then try using the same method before an exam. Otherwise, experiment with different activities until you find one that works for you.
Taking Exams
For many students, exams are the most stressful and frightening part of a math course. But exams are merely a chance for you to demonstrate the skills that you have put so much time into mastering. Here are some simple practices which can make your testing experience less threatening and more successful.
Read the entire exam first.
Give yourself a few minutes to read the entire exam. It will enable you to strategize about which problems to do first, help you manage your time, and let your subconscious process the questions while you work on other problems. Investing these few minutes at the start of the exam can save you many more minutes later.
Skip around.
Everyone has strengths and weaknesses when it comes to math. On a test you need to capitalize on your strengths and minimize your weaknesses. You can do this by identifying the problems that are easiest for you while you read through the entire test. Early success on these problems will help you build the confidence to tackle the tougher questions.
Read each question carefully.
Students often lose points on exams simply because they do not read the question carefully. When solving a problem the first thing that you should do is to identify the quantity for which you are looking. Do not stop working until you have found that quantity. Since math uses a very precise language, it is also easy to misread a question. If you rush through a question, you could easily misread the square of x as the square root of x, but these will produce very different results. Remember that you won't get many points if you do not answer the question that was asked.
Move your pencil.
Many math problems can't be solved in your head. You have to reorganize the information before you will see the solution. Try writing the given information in a different form (a list, a table, a picture, etc.). Often you will find that when your pencil starts to move your mind will follow.
Manage your time.
Time management is critical. If you have one hour to complete a one-hundred point exam, then a pace of two points per minute will enable you to complete the exam and still leave ten minutes to check your answers. So you want to spend about ten minutes on a twenty point question, five minutes on a ten point question, etc. This is only a rough guide - many problems will require eother more or less time than this rule suggests - but it gives you a framework for assessing your progress.
Breathe.
Don't forget to keep breathing deeply and slowly. Relaxation is important for your concentration. Be aware of your physical reactions to the experience of taking an exam, and try not to let them interfere with your work.
Check your answers.
Whenever possible, verify your conclusions. Sometimes this only requires substituting your solution into the original question, and other times it requires checking every step. Knowing that you have successfully completed a problem will free your mind to move onto new challenges.
Be careful changing answers.
It is common for students to second-guess themselves, erase their work, and either replace it with incorrect work or leave the problem blank. Having any answer is at least as good as leaving it blank, and if your mistake was small and near the end of the process it might be worth nearly all of the points. So even if you have determined that your answer is incorrect, follow this simple rule - don't change an answer until you are sure you have something better to replace it with.
Use the full time.
Just because you finish the exam early doesn't mean you have to turn it in. Time is a precious resource when taking exams, so you should use it all. You'll never have those particular minutes back. Re-read the exam and make sure that you answered the questions which were asked. Re-check any answers that can be checked, and re-work those that can't. Remember that an exam isn't a race - there aren't any prizes for finishing early. | 677.169 | 1 |
Homework and Problem Solving
When doing homework, you are actively using the knowledge you learned to solve problems. Reading textbook and attending lectures are somewhat "passive." You get a much better understanding after doing some problems.
In some sense, material is learned by doing many, many problems, especially the harder problems. A fatal mistake many students make is, after arriving at the solution once, they assume they have mastered the material. This is not enough practice to commit the material to long-term memory, so when the material is presented on the exam, students are unable to recognize the solution because they had not practiced enough.
It is extremely important that you do enough problems after you have learned the material, i.e., after your reading, lecture and review. Getting the answers right is not your goal (we already know the answer). It is the path of how you get the answers that is important. Only after you used the theorems, formulas many times, can you have a solid mastery of them and now you can say that you really "got it''.
The methods:
The most important secret to being a good problem solver is simply paying attention to the techniques, methods, or "tricks'' found from examples, proofs and some of the homework problems. You should study each method thoroughly, keep a list of them, and know when and where to apply them.
The problems:
Normally, each chapter has several typical problems. A problem becomes "typical'' either because of its relation to a theorem, a formula, an application, or because of the solution method. You should be able to recognize the problems, make a thorough study of them, know their possible variations and keep a list of them.
More vs. less, forest vs. tree:
You should do some well-selected problems very carefully to get the depth, to know all the details. After that, you should do more problems but less carefully to get general ideas and to become "wide.'' You can even just read the problems and do them in your mind without writing the solutions down.
One vs. several:
It is very beneficial to try to use one method to solve as many problems as you can, or to try to use several methods to solve one problem.
Getting something out of each problem:
You should always try to get something out of each problem you solved. After you have done your homework problems, you should think about them again briefly to see what you have learned and what methods are worthy of keeping. This step is important for you to retain what you learned. Without this step, most of the effort you made doing the homework will simply be wasted.
Summarize the material for each chapter:
Once you have practiced and mastered the material in the chapter, you will have confidence that you have learned the chapter's material. Now is the time to summarize. For example, calculus problems can be classified as follows: | 677.169 | 1 |
TalkAndWrite is a new kind of whiteboard that allows total interaction between two people using handwriting, drawing, typing, pointing out and erasing, all in real time, while you use the regular Skyp... More: lessons, discussions, ratings, reviews,...
With a scope that spans the mathematics curriculum from middle school to college, The Geometer's Sketchpad brings a powerful dimension to the study of mathematics. Sketchpad is a dynamic geometry cons... More: lessons, discussions, ratings, reviews,...
Interactive graphs and formulas make things move, adding a new depth of meaning for key ideas. MathWorlds uses the familiar and flexible structure of computer documents and menus connected to the curr... More: lessons, discussions, ratings, reviews,...
TI InterActive! is an integrated learning environment in which you can create interactive math and science documents. Documents may include formatted text, graphics, movies, and live integrated mathem...The multitouch interface of the Sketchpad Explorer app lets you interact with, and investigate, any document created with The Geometer's Sketchpad. Drag, manipulate and animate visual mathematics to dFathom user Kathy Powenski wrote Bill Finzer, asking how to hide equations in a function plot, so her pre-calc students can use sliders to match some basic functions. Although you can't "hide" func... More: lessons, discussions, ratings, reviews,...
An interactive applet that allows the user to graphically explore the properties of a cubic function. Specifically,
it is designed to foster an intuitive understanding of the effects of changinAn applet essentially mimicking a graphing calculator, this is used in a number of activities from the same author. Graph functions, experiment with parameters, distinguish between functions by graph | 677.169 | 1 |
Basic Business Math Skills
Fee: $195.00 Program #: SS03734
Cart is empty.
The Basic Business Math Skills course is designed for anyone who needs to apply basic math skills to business. Learners will review crucial math terms, basic mathematical concepts, and how to apply math concepts to the business environment. This course also instructs the learner in the following: how to use decimals, including addition, subtraction, multiplication, and division; how to solve problems involving percentages to determine portions, a rate, a whole unit, and increases and decreases—and how to apply these operations in business settings. Finally, using real-world scenarios, this course explains the concepts of ratio, proportion, and how to compare different kinds of numbers; and discusses simple, weighted, and moving averages. | 677.169 | 1 |
Specification
Aims
This course unit aims to introduce students to ordinary differential equations, primarily covering methods of solution and applications to physical situations.
Brief Description of the unit
The unit will cover first and second order ordinary differential equations including classification and standard solution methods. Applications will be drawn from the field of classical mechanics, but no prior experience in mechanics is expected or required. Matlab will be used to illustrate some of the ideas and methods.
Learning Outcomes
On completion of this unit successful students will be able to solve first order and second order linear problems and first order separable equations analytically. Use substitution methods and power series methods to find solutions. Be able to investigate solutions using direction fields and Euler's method. Have used Matlab as a mathematical tool and used differential equations to solve problems in mechanics and other applications.
Future topics requiring this course unit
Almost every applied mathematics course unit and many in pure mathematics and statistics. | 677.169 | 1 |
Reviews
I believe that the math study skills link will help a lot of people learn to study better the math studylink not only tells you how you can learn to be a better student, but it also gives you a little encouragement to ask questions when you need the help. So don't be afraid to ask.
Good Advice!
Spartanburg Community College, Spartanburg, SC
"Math Study Skills" provides all the suggestions I give to my math students, but in a much more concise form. Great reading for any student who wishes to succeed in college-level math.
This article talks about the skills of studying math. As it mentioned "Math is learned by doing problems." The more you practice, the better you learn. Moreover, never be shy or afraid to ask your classmates and instructors for help. Everyone is nice. They would like to help you out.
Also, it's a good way to make you absorb the knowledge you learn in class by reading the textbook before the classes begin. Learning math is a wonderful trip, if you put your heart into it. It will give you a fantastic experience.
This resources is talking about math skills. To be active, students should take responsibility to themselves, they should attendclass and participate in the course. Besides, college math is different than high school. Class always is less time per week than in high school but you have more information. Thus if students missed the class, they have to take more time to catch up.
Moreover, asking questions is another important skill. Math is different than other courses. It focuses on "doing" rather than writing. If students don't know how to do, they can't get the right answer, but in the writing course, students can get some grades if they written down something that is related. Therefore, math skill is very important to learn. The best way to do it,is to just spend time and do problems over and over again until you totally understand | 677.169 | 1 |
...Students learn the rules how to work with fractional variables, graphing and solving two or more variable equations,inequalities, and inequalities. Students learn to construct, solve and check real world problems. They also learn how to manipulate an equation in terms of another variable Emphasis on logic proofs | 677.169 | 1 |
Description and Objectives
Introduction to
use of computers to solve scientific and
engineering problems. Application
of mathematical judgment in
selecting tools to solve problems and to
communicate results.
MATLAB is the primary tool used for numerical
computation.
Although the subject matter of Beginning Scientific
Computing can be made rather difficult, I
will attempt to present the course
material in as simple a manner as
possible. More theoretical aspects, such
as proofs, will not be presented.
Applications will be emphasized.
Schedule and Homework
Follow links in
the table below to obtain a copy of the homework in Adobe
Acrobat (.pdf) format. You may also obtain here solutions to some of
the
homework and exam problems. An item shown below in plain text is not
yet
available. For additional information regarding viewing and printing
the
homework and solution sets,
click
here.
Grading
Your course grade will be calculated by weighing the homework,
the Midterm, and the Final in the proportions 50%, 20%, and 30%,
respectively. Homework problem
sets will be assigned bi-weekly. Homework
constitutes 50% of your final grade.
There will also be a one-hour-long
midterm and comprehensive final
for 20% and 30% or your grade respectively.
LATE HOMEWORK WILL NOT BE ACCEPTED.
Homework will be submitted and graded on-line. You have up to three
attempts per homework to get everything correct. If everything is correct
the first time a homework is submitted, you will receive a 100% for that homework.
If something is not correct, then you must fix it and re-submit the homework.
Your highest submitted homework grade will be your final grade for that particular
homework.
Matlab Resources
In this course, we will
make extensive use of Matlab, a technical computing
environment for
numerical computation and visualization produced by The MathWorks, Inc. A Matlab manual is
available in the MSCC Lab. If
you
are working in the Windows environment, be sure to check out the
Matlab
notebook feature that integrates Matlab with Microsoft Word. | 677.169 | 1 |
The entries in the Encyclopaedia are extracts from our books on the website or were written specifically for the Encyclopaedia by our author.
If you would like more detailed information on a topic over and above that which is contained in this Encyclopaedia – then you should go to the related material on covering ages from 9 up to 18 - Key Stage 2, Key Stage 3, GCSE (Ordinary), GCSE (Additional), AS and A Level.
In addition, we have special books on algebra, trigonometry and the calculus; 'Algebra – the way to do it' covers algebra from foundations upwards, 'Trigonometry – the
way to do it' and 'Calculus - the way to do it' Book 1 are appropriate for students of GCSE Additional or GCE Advanced Subsidiary level of study (age 17 approximately) and the material in 'Calculus - the way to do it' Book 2 is designed for GCE Advanced level courses of study (age 18 approximately).
All of the material is available in pdf from for instant download to your computer.
Of necessity, each entry in the Encyclopaedia is a condensed version of the material contained on The material in the Encyclopaedia covers age groups from 9 (Key Stage 2) to around 17 years (Advanced Subsidiary) and should prove extremely useful for examination revision. | 677.169 | 1 |
"We are what we do … Excellence therefore is not an act, but a habit!" Aristotle The Mathematics Department of Harold M. Brathwaite Secondary School is here to help you aim for excellence. We are dedicated to student learning, assisting through Counting on You, extra help sessions and contests. We use technology in the classroom for explorations and to reinforce concepts. The department office is located on the second floor at the front of the school building. Watch for news in the display case across the hall from the office.
Grade 11 MEL 3E0 Mathematics for Work and Everyday Life MBF 3C0 Foundations for College Mathematics MCF 3M0 Functions and Applications College/University MCR 3U0 Functions University
Grade 12 MEL 4E0 Mathematics for Work and Everyday Life MAP 4C0 Foundations for College Mathematics MCT 4C0 Mathematics for College Technology MHF 4U0 Advanced Functions University MDM 4U0 Mathematics of Data management University MCV 4U0 Calculus and Vectors University | 677.169 | 1 |
The present book aims at providing a detailed account of the basic concepts of vectors that are needed to build a strong foundation for a student pursuing career in mathematics. These concepts include Addition and Multiplication of vectors by Scalars, Cen | 677.169 | 1 |
This course is intended for those who want to brush up on their math skills and need guidance and practice in solving the types of problems encountered on the GRE or GMAT. Topics will include arithmetic, algebra, geometry and problem-solving. Textbooks can be purchased from the college bookstore and are required at the first day of class.
Prerequisite: None
Follow-up courses: GRE (GREPREP) or GMAT (GMTPREP) | 677.169 | 1 |
The second course in a three part calculus sequence. Topics include: the Riemann integral, applications of integration, techniques of integration, and transcendental functions. Prerequisite: MATH 215 with grade of C or higher.
Prerequisite(s) / Corequisite(s):
MATH 215 with a grade of C or higher.
Text(s):
Most current editions of the following:
Most current editions of the following:
Calculus
By Finney, R. & G. Thomas. (Addison-Wesley) Recommended
Calculus
By Stewart (Brookes-Cole) Recommended
Course Objectives
To use calculus to formulate and solve problems and communicate solutions to others.
To use technology as an integral part of the process of formulation, solution and communication.
To understand and appreciate the connections between mathematics and other disciplines.
Measurable Learning Outcomes:
• Compute definite integrals as the limit of Riemann sums and approximate integrals using finite Riemann sums. • Evaluate definite and indefinite integrals using the Fundamental Theorem of Calculus and the method of substitution. • Compute areas and volumes using definite integrals. • Identify the natural exponential and logarithmic functions as inverses of each other and find their derivatives and integrals. • Solve exponential growth and decay problems arising from biology, physics, chemistry, and other sciences. • Compute derivatives and integrals of functions containing inverse trigonometric functions. • Analyze various indeterminate forms and apply L'Hospital's rule to evaluate limits of such forms. • Use the Substitution Rule and the Integration by Parts formula to evaluate indefinite and definite integrals. *Describe and explain special methods required to integrate trigonometric and rational functions. * Apply numerical methods of integration such as Simpson's Rule to approximate definite integrals.
Topical Outline:
Integrals
Applications of integrals
Transcendental functions
Techniques of integration | 677.169 | 1 |
First-order differential equations
This unit introduces the topic of differential equations. The subject is developed...During this unit you will:
learn some basic definitions and terminology associated with differential equations and their solutions;
be able to visualize the direction field associated with a first-order differential equation and be able to use a numerical method of solution known as Euler's method;
be able to use analytical methods of solution by direct integration; separation of variables; and the integrating factor method.
Contents
First-order differential equations | 677.169 | 1 |
Tutorial fee-based software for PCs that must be downloaded to the user's computer. It covers topics from pre-algebra through pre-calculus, including trigonometry and some statistics. The software pos... More: lessons, discussions, ratings, reviews,...
Zoom Algebra is a Computer Algebra System App for TI-83 Plus and TI-84 Plus graphing calculators. Its patent-pending interface is visual and easy to use, with many little shortcuts. For example, ...An algebra practice program for anyone working on simplifying expressions and solving equations. Create your own sets of problems to work through in the equation editor, and have them appear on all of... More: lessons, discussions, ratings, reviews,...
Students answer the question: Is the ratio of our arm span to our height really equal to 1? If an emphasis is given to a/h = 1, this can be an engaging activity using variables. Use a spreadsheet to w... More: lessons, discussions, ratings, reviews,...
This eModule presents sequences of geometric patterns and encourages students to generate rules and functions describing relationships between the pattern number and characteristics of the pattern. S... More: lessons, discussions, ratings, reviews,...
A classroom activity, to be explored through large movement experience, manipulatives, and an interactive Java applet. Students then revisit the activity, look for patterns, and write the answer algeb... More: lessons, discussions, ratings, reviews,...
This collection of activities is intended to provide middle and high school Algebra I students with a set of data collection investigations that integrate mathematics and science and promote mathemati problem could be used in varying degrees with 6th graders through high school. It encourages students to use good problem-solving heuristics. Logo is used to extend this problem and to encourageQuestion: Is the ratio of our arm span to our height really equal to 1? If an emphasis is given to a/h = 1, this can be an engaging activity using variables. Use a spreadsheet to work with the data. More: lessons, discussions, ratings, reviews,...
These guided, interactive activities present sequences of geometric patterns and encourage students to generate rules and functions describing relationships between the pattern number and characterist... More: lessons, discussions, ratings, reviews,...
A classroom activity (also called Hop, Skip, Jump) aligned to the NCTM and California Standards, to be explored through large movement experience, manipulatives, and an interactive Java applet. Studen... More: lessons, discussions, ratings, reviews,...
Understanding factoring through geometry: students work cooperatively to display a numeral as the area of a rectangle, and make as many rectangular arrangements as possible for each numeral given. | 677.169 | 1 |
Vector Spaces
In this lesson our instructor talks about vector spaces. First, he talks about vector spaces and complex vector spaces. Then he does some example problems. He ends the lesson with a discussion on properties of vector spaces.
This content requires Javascript to be available and enabled in your browser.
Vector Spaces | 677.169 | 1 |
Elementary using Elementary Algebra, Second Edition, you will find that the text focuses on building competence and confidence. The authors present the concepts, show how to do the math, and explain the reasoning behind it in a language you can understand. The text ties concepts together using the Algebra Pyramid, which will help you see the big picture of algebra. The skills Carson presents through both the Learning Strategy boxes and the Study System, introduced in the Preface and incorporated throughout the text, will not only enhance your elemen... MOREtary algebra experience but will also help you succeed in future college courses. Book jacket. Elementary Algebra is a book for the student. the authors' goal is to help build students' confidence, their understanding and appreciation of math, and their basic skills by presenting an extremely user-friendly text that models a framework in which students can succeed. Unfortunately, students who place into developmental math courses often struggle with math anxiety due to bad experiences in past math courses. Developmental students often have never developed nor applied a study system in mathematics. to address these needs, the authors have framed three goals for Elementary Algebra: 1) reduce math anxiety, 2) teach for understanding, and 3) foster critical thinking and enthusiasm. The authors' writing style is extremely student-friendly. They talk to students in their own language and walk them through the concepts, explaining not only how to do the math, but also why it works and where it comes from, rather than using the "monkey-see, monkey-do" approach that some books take. | 677.169 | 1 |
This program is about every thing you want to know about real numbers. It discusses introduction to real numbers in a very simple understandable mathematical language because the aim of this program is to understand and not memorize mathematics. It also include fractions, addition and subtraction, multiplication and division of real numbers. Exponential and order of operations, algebraic expressions, properties of real numbers, and how to use these properties in algebra. All discussions are self-learning so that you have your own tutor at home, and you can study it at your own pace. All of these are present in one package | 677.169 | 1 |
Sixth Form: Mathematics
Subject Overview
Mathematics A Level is a chance to extend you skils in Mathematical techniques and problem solving. It covers four modules of Pure Maths, using a lot of your previous knowledge of algebra to build on your previous work with topics such as trigonometry and vectors. As well as learning about new Mathematical concepts such as differentiation and integration, you will also do two applied modules, where you will have the choice of studying Mechanics or Statistics.
Syllabuses (Course Outline and Structure)
At Heart of England Sixth Form we follow the AQA Syllabus. Both the AS and A2 sections of the course are marked out of 300 UMS points. The two sets of marks together constitute the entire A level, out of a total of 600 UMS points. The course is split into AS and A2 Mathematics as follows:
AS Mathematics
Pure Core 1 - Algebra methods, extended from GCSE, adding, subtracting, multiplying, dividing, sketching and translating polynomials, some work with coordinate geometry extending GCSE methods to work with circles and an introduction to calculus both differentiation and integration.
Pure Core 2 - Transforming functions and their graphs, introducing the concept of series—summing sequences, extending your GCSE work on trigonometry and introducing the measure of radians, some work with indices, introduction to the topic of logarithms and an extension of the work with differentiation and integration.
Choice of:
Mechanics 1 - Mathematical modelling of real life situations, displacement, velocity and acceleration in one and two dimensions, forces—including friction and tension, momentum, Newton's laws of motion, problems involving connected particles and projectiles or
A2 Mathematics
Pure Core 3 - Work with functions including inverse, compositions and combinations of transformations, extension of trigonometry including inverse and reciprocal trig functions, exponentials and logarithms, differentiation using the product, quotient and chain rules, integration by substitution and by parts, use of integration to find the volume of a revolution, iterative methods for solving equations and numerical methods of integration.
Pure Core 4 - Rational functions, algebraic division, partial fractions, conversion between Cartesian and parametric equations, extension of work with binomial series, further work with trigonometry including use of harmonic form and double angle formulae, exponential growth and decay, solving differential equations, differentiating parametric equations, integrating partial fractions and work on vectors including vector equation of lines and scalar product.
Choice of:
Mechanics 2 - Moments, finding the centre of mass, further work with displacement, velocity and acceleration including in three dimensions, Newton's Laws in up to three dimensions, application of differential equations, uniform circular motion and vertical circular motion as well as looking at work and energy, including GPE, KE and Hooke's Law or
The official AQA specifcations for all of the Maths modules are available in pdf form at:
AQA Mathematics Specification 2013
Entry Requirements
To study A Level Mathematics you must have at least five GCSEs at grade C or above, and we would recommend at least a grade B in your GCSE Maths. Since the course is very algebra based you must also have good skills in manipulating algebra and you will be tested on this during the first week of the course.
The 'step up' from GCSE Maths to A Level is quite significant and for those students who would like to get a good start on it, particularly if their algebra skills need a little brushing up, we recommend the CGP text 'Head Start to AS Maths'
Activities and Trips
We have no mandatory trips or activities in Maths, although throughout the two years there may be the opportunity to take part in activities, such as revision days and team challenges, run by the 'Further Maths Network' based at Warwick University, with whom we have been cultivating links over the last few years.
This would incur a small cost, usually of between £10 and £30 dependant on the type of activity, and may involve students arranging their own transport to and from the University.
Expected Costs
Other than the cost of the activities that we may run with the 'Further Maths Network' there are no expected costs associated with the Maths A Level. All the text books are lent to students for the duration of the course and they will only need to pay for them if they fail to return or badly damage them. There are no mandatory excursions and the only equipment they are required to have (other than the usual contents of a pencil case) is a scientific calculator, which they should have anyway from GCSE. Complementary Subject Combinations and Enrichment Activities
The main links between other subjects and Maths come from the choice of applied topic:
Mechanics – fits well with Physics as there is a lot of overlap in the content of the courses
Statistics – fits well with Pyschology and Biology as they use statistical analysis in some of their coursework.
Subject Resources
Schemes of Work
In Maths the Scheme of Work is based on the text books. For each module we have a text book produced by AQA which covers all the topics needed for that course. Students will be loaned these text books for the duration of their study
Past Papers
Past papers are an essential part of the revision process for Mathematics, it is important to get plenty of practice of the type of questions you will be asked in exams. At the end of each chapter in the text book there is a revision exercise made up of past exam questions and we always leave plenty of time after completing the learning for the module to do past paper practice, both under exam conditions and as an open book revision tool.
The AQA Maths past papers (and several other useful documents) can be found at:
AQA A Level Maths Materials
Useful Links
The AQA link above is very useful and provides access to past papers, mark schemes, examiners reports, specifications, practice papers for new specifications, the formula booklet and many other useful documents.
Also the school has paid for access to the website My Maths which students may have used in Key Stage 3 and 4 but which also has a wealth of resources for A Level revision. This can be accessed by asking your teacher for the school's login and password information.
Other Information
Maths A Level will support students who go on to study a wide range of different subjects at University or in other forms of Higher Education, the more obvious ones being Maths, Science and Engineering. It's logical thinking and problem solving based structure make it a qualification that can pick students 'out of the crowd' in the eyes of many universities and employers, even in non-Maths based courses or industries. | 677.169 | 1 |
Elementary Linear Algebra with Applications - 3rd edition
This book is intended for the first course in linear algebra, taken by mathematics, science, engineering and economics majors. The new edition presents a stronger geometric intuition for the ensuing concepts of span and linear independence. Applications are integrated throughout to illustrate the mathematics and to motivate the student.Edition/Copyright: 3RD 96 Cover: Hardback Publisher: Saunders College Division Published: 09/08/1995 International: No
View Table of Contents
Preface. List of Applications.
1. Introduction to Linear Equations and Matrices.
Introduction to Linear Systems and Matrices. Gaussian Elimination. The Algebra of Matrices: Four Descriptions of the Product. Inverses and Elementary Matrices. Gaussian Elimination as a Matrix Factorization. Transposes, Symmetry, and Band Matrices: An Application. Numerical and Programming Considerations: Partial Pivoting, Overwriting Matrices, and Ill-Conditioned Systems. Review Exercises.
Giving great service since 2004: Buy from the Best! 4,000,000 items shipped to delighted customers. We have 1,000,000 unique items ready to ship! Find your Great Buy today!
$148.50 | 677.169 | 1 |
Calculus: One and Several Variables, Ninth Edition
Wiley is proud to publish a new revision of this successful classic text known for its elegant writing style, precision and perfect balance of theory and applications. This Ninth Edition is refined to offer students an even clearer understanding of calculus and insight into mathematics. It includes a wealth of rich problem sets which give relevance to calculus for students. Salas/Hille/Etgen is recognized for its mathematical integrity, accuracy, and clarity.
Customer Reviews:
Great book for learning
By C. Waldorf - March 23, 2005
This is a superb textbook and it's easy to see why the book is in its ninth edition. What I really enjoyed (yes, I know this may sound a little incongruous in relation to calculus) was the step-by-step build-up of knowledge with good, clear examples. Also, for the problems at the end of each section, all the odd problems have solutions, so one can get some practice (something that is unfortunately rare for many textbooks).
Before going through this book, I had minimal exposure to calculus and what I had seen wasn't very favorable. This book was a key reason why I now really enjoy the subject and feel very comfortable in this area.
Not for the mediocre
By A Customer - February 5, 2004
This book is a stunning rebuke to all attempts to dumb down the math curriculum in high schools and colleges. This book, in my opinion, expects the student to have mastered precalculus at the level set forth in, say, David Cohen's Precalculus with unit-circle trigonometry (ISBN 0-534-35275-8). It introduces mathematical rigor in the Calculus 101 semester (of a three semester calculus program) and thereby begins preparing the math major for the hard analysis courses that comes later on. There are no cute stories featuring 'How I Use Math In The Workplace' to inspire you - your self esteem will be hard won as you master the concepts as presented here (especially the problems). The book's greatest strength is that it is basic and traditional in its approach to calculus - no problem or example requires obscure special tricks from mathematical journals or Isaac Newton level ingenuity. This book is a must get!
Start with it but don't end with it!
By Mohammad - June 30, 2000
I used this book in my first engineering calculus course. The professor was incredibly theoretical and did not teach from the book which made matters somewhat difficult. However, he was showing us the meaning of math which I found refreshing. This book serves its purpose as one which teaches the mechanics of solving problems but very little in developing an intuitive feeling for mathematics. I must admit that the multitude of exercises were very helpful in getting comfortable with difficult mechanical problems. For single variable calculus it is a standard book with good examples, excellent diagrams, and some applications. Getting into multivariables, the ideas are not connected well and seem segragated from the rest of material. I guess as a brief overview, it makes its point but should not be used as a text for multivariable calculus. If you are interested in theory I recommend Apostol's Calculus which covers a great range of material with rigorous foundation... read more | 677.169 | 1 |
in a... more...
Mathematical Applications and Modelling is the second in the series of the yearbooks of the Association of Mathematics Educators in Singapore. The book is unique as it addresses a focused theme on mathematics education. The objective is to illustrate the diversity within the theme and present research that translates into classroom pedagogies.The book,...This book is an introduction to the subject of mean curvature flow of hypersurfaces with special emphasis on the analysis of singularities. This flow occurs in the description of the evolution of numerous physical models where the energy is given by the area of the interfaces. These notes provide a detailed discussion of the classical parametric approach... more... | 677.169 | 1 |
Calculus
Chapter 1 What Is Calculus? In This Chapter * You're only on page 1 and you've got a calc test already * Calculus — it's just souped-up regular math * Zooming in is the key * The world before and after calculus "My best day in Calc 101 at Southern Cal was the day I had to cut class to get a ...
0 1 2 1 2 3 4 We want the area underneath the stepped line, not the area underneath the smooth curve. It is ironic that we can easily determine the area underneath the smooth curve using the infinite calculus but have trouble determining the much more simple area under the stepped line.
Calculus The infinitesimal calculus, with its two branches, differential and integral calculus, has its roots in two special geometrical problems: (1) To find the tangent to a curve; (2) To find the area enclosed by a plane curve ("quadrature").
Curriculum Module: Calculus: Functions Defined by Integrals 1 AP Calculus Functions Defined by Integrals Scott Pass John H. Reagan High School Austin, TX Reasoning from the graph of the derivative function f in order to obtain information about the behavior of the function F defined by F ( x ... | 677.169 | 1 |
The Elementary
Algebra test measures your skills in three main categories
a. Operations with integers and rational numbers
This includes addition, subtraction, multiplication and division with integers
and negative rational numbers along with the use of absolute values and
ordering.
b. Operations with algebraic expressions.
This includes evaluations of simple formulas and expressions as well as adding
and subtracting monomials and polynomials. Also covered is evaluation of
positive rational roots and exponents, simplifying algebraic fractions and
factoring.
c. Equation solving, inequalities and word problems.
This includes solving verbal problems presented in algebraic context, geometric
reasoning, the translation of written phrases into algebraic expressions and
graphing.
Elementary Algebra Practice Test Materials
Print out the
Practice Exam linked below. Work through each problem and if you don't
understand a question, use the TEGRITY video solutions to watch a video
presentation of the worked out solution.
The following file is a
sample test made by the ARCC math department. Note: this test consists of 30
problems, but the actual exam is 20 problems. TEGRITY solutions are linked
below:
For
Video Presentation of
Solutions to each of
the questions on the Elementary Algebra Practice Exam, click
Accuplacer Elementary Algebra Practice Exam Solutions.
Each problem is explained in a short video-like presentation complete
with audio explanations. (Make sure your speakers are on!) Warning:
Disable pop-up blockersin order to view the video
solutions.
Two sample questions for the
Elementary Algebra test:
1.
A.
B.
C. D.
2. The width of a rectangle
is of the
length which is 10x. If the area of the rectangle is 100, what is the
value of x
A.
B.
C.
D.
The answer to sample question 1 is B and the answer to
sample question 2 is C.
You may wish to visit the following sites to refresh your arithmetic skills: | 677.169 | 1 |
Villanova University Computer Algebra Systems
A computer algebra system (CAS) is a mathematics software package which manipulates mathematical objects symbolically ("algebra"), as well as giving numerical and graphical computing capabilities, with typesetting options for making nice technical reports. There are two leading computer algebra systems: Maple from Maplesoft and Mathematica from Wolfram. These symbolic based computer algebra systems should not be confused with more specific purpose mathematical computation tools like MathCad or MatLab.
Mathematica
In the summer of 2008 we began a Mathematica site license experiment for 30 simultaneous users on campus to support the existing individual Mathematica users at Villanova and allow faculty members to explore the rich array of possibilities that Mathematica offers.
Mathematica offers amazing access to current data through remote servers in all kinds of fields not limited to more traditional mathematically based fields like mathematics, statistics, physics, astronomy, meteorology, chemistry, biology, engineering, etc., but also in the social and political sciences, economics, finance, geography, etc. AND in addition, this data may be fed into Mathematica for analysis or visualization without a great deal of expertise in using the product. For some demonstrations see:
Mathematica 7 has added genomic data, protein data, and current and historical weather data. The Learning Center is a good place to start nosing around Mathematica for the first time, after watching the short video
To access Mathematica, simply log in to the Villanova Citrix Server (which requires a quick client install if your computer image does not already have this software) and find Mathematica under Academic Applications, Math and Stat: you will see the Mathematica icon listed alphabetically right after Maple. If you do not, simply request that UNIT give you access by calling the Help Desk 9-7777. At present the Mathematica icon should be visible to all faculty and staff.
To get started actually using Mathematica if you are a new user, there is a 15 minute hands on introduction:
There are a few scattered Mathematica users on campus, some of whom may be willing to offer limited help in getting started using it. Email bob jantzen to try connecting with someone like this.
Maple
We adopted Maple for use in teaching mathematics at Villanova in the mid 1990s and it has evolved into a very user-friendly powerful tool for aiding learning in the college mathematics environment as well as for professional research and applications. In the summer of 2008 we upgraded out site license to an unlimited license which allows access to local copies of the Maple program to any faculty, staff or students in the Villanova community. For more information on Maple at Villanova, see:
Maple is also available through Citrix under Academic Applications, Math and Stat: choose the red Standard Maple icon. Simply log in to the Villanova Citrix Server (which requires a quick client install if your computer image does not already have this software) and find Maple under Academic Applications, Math and Stat: you will see the red Standard Maple icon listed alphabetically.
Mathematica videos
Mathematica is used in a variety of fields—from math, physics, and engineering to sociology, finance, and earth science. Two of the most popular Mathematica tutorials are the following
"Hands-on Start to Mathematica" is a free, two-part online screencast that introduces Mathematica basics to get you started with your first calculations, visualizations, and interactive examples. If you haven't already, be sure to check out Part 1 here:
Many students have asked for more in-depth training, so we now also offer "M10: A Student's First Course in Mathematica," a self-paced video training course providing step-by-step instructions on the basic features of Mathematica for students. Through the included videos and practice exercises, students learn how to navigate the user interface, build calculations, create graphics and dynamic models, work with data, and more—for under $30: | 677.169 | 1 |
Extra Examples shows
you additional worked-out examples that mimic the ones in
your book. These requirements include the benchmarks from
the Sunshine State Standards that are most relevant to this
course. The benchmarks printed in regular type are required
for this course. The portions printed in italic type
are not required for this course.
Students identify and use the arithmetic properties of subsets of integers and rational, irrational, and real numbers, including closure properties for the four basic arithmetic operations where applicable:
Students graph a linear equation and compute the x-and y-intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequality (e.g., they sketch the region defined by 2x + 6y < 4).
Students understand the concepts of parallel lines and perpendicular lines and how those slopes are related. Students are able to find the equation of a line perpendicular to a given line that passes through a given point.
Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets.
Students apply basic factoring techniques to second-and simple third-degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials.
Given a specific algebraic statement involving linear, quadratic, or absolute value expressions or equations or inequalities, students determine whether the statement is true sometimes, always, or never. | 677.169 | 1 |
In this section
Lifeskills Mathematics
New National Qualifications
Lifeskills Mathematics sits within the Mathematics curriculum area.
Finalised Course and Unit documents are now available for all the new qualifications, from National 2 to Advanced Higher. These documents contain both mandatory information (in the Specifications) and advice and guidance (in the Support Notes). You can download all of these documents for this subject from our download page, using the button below, or use our check-box facility to download a selection of documents.
Assessment support materials are also now available for all the new National 2, National 3, National 4 and National 5 qualifications. Information on Course assessment support material (such as Specimen Question Papers and coursework information) is available on the National 5 subject page and, for all National 2 to National 5 qualifications, information on how to access Unit assessment support materials can be found on each subject page.
Following the development of these support materials, some documents with mandatory information (Course Assessment Specifications, in particular) have been updated with further information and clarifications. In line with our standard practice, these documents contain version information and, if necessary, a note of changes. This ensures you can recognise the most up-to-date documents.
Development process
The final documents have been published following a lengthy engagement process. Find out how we got here. Considerable work has been carried out by the Curriculum Area Review Groups (CARGs), the Qualifications Design Teams (QDTs) and Subject Working Groups (SWGs) to develop the final documents.
At each stage of the qualification development process, we publish draft documents outlining our proposals and plans.
Visit our timeline to find out when the next documents for each qualification will be published.
Key points
National 3 to National 5
develops confidence and independence in being able to handle information and mathematical tasks in both personal life and in the workplace
motivates and challenges learners by enabling them to think through real-life situations involving mathematics
has mathematical skills underpinned by numeracy and is designed to develop learners' mathematical reasoning skills relevant to learning, life and work
provides opportunities in Units for combined assessment
has a hierarchical Unit structure that provides progression from National 2 to National 5
has a test as the added value assessment at National 4, and question papers at National 5
National 2 Lifeskills Mathematics
offers opportunities for flexible delivery through the use of Units which can be delivered sequentially, in parallel or in a combined way
offers increased opportunities for personalisation, choice and flexibility in Unit assessment, with opportunities for integrated assessment
provides increased opportunities for interdisciplinary and cross-curriculum working
includes both mathematical operational and reasoning skills in the Units
provides an opportunity to use mathematical skills in real-life contexts
The Unit titles have been changed to better reflect the Unit content, which has been reorganised. Units, including the National 4 Added Value Unit Outcomes, Assessment Standards and Evidence Requirement statements, have all been revised to increase flexibility. Additional information is provided in the Evidence Requirements for all Units.
At National 2, there has been a change of Unit title from Personal Mathematics to Shape, Space and Data to better reflect content. There is now a range of small optional Units, which aims to improve accessibility for learners at this level. | 677.169 | 1 |
Based on his two successful textbooks [Lineare Algebra. Eine Einführung für Studienanfänger. 16th revised and enlarged ed. Wiesbaden: Vieweg (2008; Zbl 1234.15001) see also ME 2001a.00676 and ME 2007f.00322] and [Analytische Geometrie. Eine Einführung für Studienanfänger. 7., durchgesehene Aufl. Wiesbaden: Vieweg (2001; Zbl 0980.51018)], the author presents another introduction for beginners. It is called a "Lernbuch" rather than a "Lehrbuch", which would be the usual term in German. (Reviewer's remark for those, who do not know German: "Buch" means "book", "lernen" means "to learn", but "lehren" means "to teach".) Indeed, the goal of the book is twofold: On the one hand it is to be a book which presents the most important parts of linear algebra and geometry. On the other hand it aims at assisting the first year student by providing very extensive explanations and a great number of examples. Both task are accomplished without any doubt. The text is very well written and accompanied by a big number of illustrations, thus emphasising the geometric nature of linear algebra. In order to sketch the contents of the book, here are the titles of its six chapters: 0.~Linear geometry in $n$-dimensions (over the real numbers); 1.~Basic notions; 2.~Vector spaces and linear mappings; 3.~Determinants; 4.~Eigenvalues; 5.~Bilinear algebra and geometry.
Reviewer:
Hans Havlicek (Wien) | 677.169 | 1 |
The Everything Guide to Calculus I
A step-by-step guide to the basics of calculus—in plain English!
By Greg Hill, National Council of Teachers of Mathema
Format:
SKU# Z9326
Details
Calculus is the basis of all advanced science and math. But it can be very intimidating, especially if you're learning it for the first time! If finding derivatives or understanding integrals has you stumped, this book can guide you through it. This indispensable resource offers hundreds of practice exercises and covers all the key concepts of calculus, including:
Greg Hill has more than twenty-five years of experience teaching AP Calculus and other advanced math classes. He is a two-time Illinois state finalist for the Presidential Award for Excellence in Mathematics and Science Teaching, and is a member of the Illinois and National Councils of Teachers of Mathematics. Hill has been a College Board consultant and AP Calculus exam grader for the past ten years. He currently teaches at Hinsdale Central High School, and also conducts day- and week-long professional development seminars for AP Calculus teachers. He is the author of CLEP Calculus, a test prep book for the College Board's College Level Entrance Exam.
Additional Information
SKU
Z9326
Author/Speaker/Editor
Greg Hill, National Council of Teachers of Mathema
File/Trim Size
9 x 9-1/4
Format
No
ISBN 13
9781440506291
Number Of Pages
320
Retail:
$16.95
Your price:
$11.53
You save: $5.42 | 677.169 | 1 |
Elementary MathematicsIn secondary school, the main topics in elementary mathematics are algebra and trigonometry. Calculus, even though it is often taught to advanced secondary schoolstudents, is usually considered college level mathematicsIn the United States, there has been considerable concern about the low level of elementary mathematics skills on the part of many students, as compared to students in other developed countries. The No Child Left Behind program was one attempt to address this deficiency, requiring that all American students be tested in elementary mathematics.
"A school is not a factory. Its raison d'être is to provide opportunity for experience." —J.L. (James Lloyd)
"Cloud-clown, blue painter, sun as horn, Hill-scholar, man that never is, The bad-bespoken lacker, Ancestor of Narcissus, prince Of the secondary men. There are no rocks And stones, only this imager." —Wallace Stevens (1879–1955)
"The longer we live the more we must endure the elementary existence of men and women; and every brave heart must treat society as a child, and never allow it to dictate." —Ralph Waldo Emerson (1803–1882)
"He taught me the mathematics of anatomy, but he couldn't teach me the poetry of medicine.... I feel that MacFarland had me on the wrong road, a road that led to knowledge, but not to healing." —Philip MacDonald, and Robert Wise. Fettes (Russell Wade) | 677.169 | 1 |
Simple Sets
Simple Sets
In this Algebra video, we introduce a few of the most fundamental aspects of mathematics and set theory. This lesson covers some of the terminology and concepts about sets that we will use in future algebra lessons. Below, we show some of the common ways we will represent sets.
The first set is the set of letters G, D, and Q and is given by the pictorial diagram and is more concrete than our other representations. The second shows the five integers 3, 56, 34, 6, and 38 inside of braces. Notice that these elements are not in order, even though we often order our lists for clarity. Like the second set, the third set is given in braces, but is more descriptive; this set is just {1,2,3,4}. The fourth and final set is read as "the set of elements x such that x is an integer and x is between 3 and 8." This set is just {4,5,6,7}, but it is given in formal notation that we will commonly use.
Although our first sets above are all finite, we will often use infinite sets like the set of integers. Also, we will refer to the set with no elements in it: this set is called the empty set and is denoted by a circle with a line through it. Sets with a single element like this, {2} are called singleton sets. Below, we show the infinite set integers, the empty set and a singleton set containing 7, respectively.
There is some additional notation that we will use. First, we will typically assign sets a letter name so that we can refer to them more succinctly. Second, we will use the character that looks like a strange "e" to specify that something "is an element of" a given set. We will use the same symbol with a strike through to denote the something "is not an element of" a given set.
Above, we have assigned the set {1,2,4} the name A. The second statement reads "1 is an element of A." The third statement reads "3 is not an element of A."
Furthermore, we define three set-wise operations that we will apply to pairs of sets: Intersection, Union, and Difference. Examples of these are given below. | 677.169 | 1 |
Content/concepts goals for this activity
Higher order thinking skills goals for this activity
Using algebra to derive the needed equation from other given equations. Analyzing a large data set using equations in the spreadsheet.
Other skills goals for this activity
Learning how to document a mathematical derivation of an equation and to document units and their consistency in the equations. Learning how to produce a correctly annotated graph with the spreadsheet software.
Description of the activity/assignment
Students download a comma-delimited data set that is a time series of stream discharge measurements and the concentration of a trace element in the stream. Given the concentration of this element in the precipitation and in the groundwater, the students analyze the data using spreadsheet software to separate the hydrograph into baseflow and quickflow components. Students produce a graph of their results. To do the analysis, students must derive an appropriate equation based on other equations presented in the text (Eqs. 1.2 and 1.3).
Determining whether students have met the goals
They are asked to derive and annotate the correct equation to use in the spreadsheet, and then must submit a graph showing their resulting hydrograph separation. | 677.169 | 1 |
This course is a review of elementary algebra. Topics include real numbers, exponents, polynomials, equation solving and factoring.
†
MATH 0099: Intermediate Algebra
4-0-4. Prerequisite: Satisfactory placement scores/MATH 0097
This course is a review of intermediate algebra. Topics include numbers, linear equations and inequalities, quadratic equations, polynomials and rational expressions and roots. Students must pass the class with a C or better and pass the statewide exit examination.
This course places quantitative skills and reasoning in the context of experiences that students will be likely to encounter. It emphasizes processing information in context from a variety of representations, understanding of both the information and the processing and understanding which conclusions can be reasonably determined. Topics covered include sets and set operations, logic, basic probability, data analysis, linear models, quadratic models and exponential and logarithmic models. This course is an alternative in area A of the core curriculum and is not intended to 1071: Mathematics I
3-0-3. Prerequisite: Satisfactory placement scores/MATH 0097
This course in practical mathematics is suitable for students in many career and certificate programs. Topics covered include a review of basic algebra, ratio and proportion, percent, graphing, consumer mathematics and the metric system.
†
MATH 1111: College Algebra
3-0-3. Prerequisite: Satisfactory placement scores/MATH 0099
This course is a functional approach to algebra that incorporates the use of appropriate technology. Emphasis will be placed on the study of functions and their graphs, inequalities, and linear, quadratic, piece-wise defined, rational, polynomial, exponential and logarithmic functions. Appropriate applications will be included. This course is an alternative in Area A of the core curriculum and does 1113: Precalculus
3-0-3. Prerequisite: MATH 1111 with a grade of C or better
This course is designed to prepare students for calculus, physics and related technical subjects. Topics include an intensive study of algebraic and trigonometric functions accompanied by analytic geometry as well as DeMoivreís theorem, polar coordinates and conic sections. Appropriate technology is utilized in the instructional process.
†
MATH 2008: Foundations of Numbers and Operations
3-0-3. Prerequisite: Math 1001, Math 1101, Math 1111, or Math 1113
This course is an Area F introductory mathematics course for early childhood education majors. This course will emphasize the understanding and use of the major concepts of number and operations. As a general theme, strategies of problem solving will be used and discussed in the context of various topics.
†
MATH 2200: Elementary Statistics
3-0-3. Prerequisites: MATH 1001/MATH 1111
This is a basic course in statistics at a level that does not require knowledge of calculus. Statistical techniques needed for research in many different fields are presented. Course content includes descriptive statistics, probability theory, hypothesis testing, ANOVA, Chi-square, regression and correlation.
Conic sections, translation and rotation of axes, polar coordinates, parametric equations, vectors in the plane and in three-space, the cross product, cylindrical and spherical coordinates, surfaces in three-space, vector fields, line and surface integrals, Stokeís theorem, Greenís theorem and differential equations are studied in this course.
†
MATH 2280: Discrete Mathematics
3-0-3. Prerequisite: MATH 1113 with a grade of C or better or permission of the instructor or permission of the academic dean. | 677.169 | 1 |
Computational Introduction To Number Theory And Algebra - 05 edition
Summary: Number theory and algebra play an increasingly significant role in computing and communications, as evidenced by the striking applications of these subjects to such fields as cryptography and coding theory. This introductory book emphasises algorithms and applications, such as cryptography and error correcting codes, and is accessible to a broad audience. The mathematical prerequisites are minimal: nothing beyond material in a typical undergraduate course in calculus...show more is presumed, other than some experience in doing proofs - everything else is developed from scratch. Thus the book can serve several purposes. It can be used as a reference and for self-study by readers who want to learn the mathematical foundations of modern cryptography. It is also ideal as a textbook for introductory courses in number theory and algebra, especially those geared towards computer science students | 677.169 | 1 |
What is the difference between matrix theory and linear algebra? - MathOverflow most recent 30 from is the difference between matrix theory and linear algebra?kolistivra2010-01-13T17:17:41Z2010-04-04T18:10:56Z
<p>Hi,</p>
<p>Currently, I'm taking matrix theory, and our textbook is Strang's Linear Algebra. Besides matrix theory, which all engineers must take, there exists linear algebra I and II for math majors. What is the difference,if any, between matrix theory and linear algebra?</p>
<p>Thanks!</p>
by Steve Huntsman for What is the difference between matrix theory and linear algebra?Steve Huntsman2010-01-13T17:23:10Z2010-01-13T17:23:10Z<p>The difference is that in matrix theory you have chosen a particular <a href=" rel="nofollow">basis</a>.</p>
by Qiaochu Yuan for What is the difference between matrix theory and linear algebra?Qiaochu Yuan2010-01-13T18:29:28Z2010-01-13T18:29:28Z<p>Let me elaborate a little on what Steve Huntsman is talking about. A matrix is just a list of numbers, and you're allowed to add and multiply matrices by combining those numbers in a certain way. When you talk about matrices, you're allowed to talk about things like the entry in the 3rd row and 4th column, and so forth. In this setting, matrices are useful for representing things like transition probabilities in a Markov chain, where each entry indicates the probability of transitioning from one state to another. You can do lots of interesting numerical things with matrices, and these interesting numerical things are very important because matrices show up a lot in engineering and the sciences.</p>
<p>In linear algebra, however, you instead talk about <strong>linear transformations,</strong> which are <strong>not</strong> (I cannot emphasize this enough) a list of numbers, although sometimes it is convenient to use a particular matrix to write down a linear transformation. The difference between a linear transformation and a matrix is not easy to grasp the first time you see it, and most people would be fine with conflating the two points of view. However, when you're given a linear transformation, you're not allowed to ask for things like the entry in its 3rd row and 4th column because questions like these depend on a <strong>choice of basis.</strong> Instead, you're only allowed to ask for things that don't depend on the basis, such as the rank, the trace, the determinant, or the set of eigenvalues. This point of view may seem unnecessarily restrictive, but it is fundamental to a deeper understanding of pure mathematics.</p>
by Anweshi for What is the difference between matrix theory and linear algebra?Anweshi2010-01-24T14:00:45Z2010-03-30T21:57:38Z<p>Matrix theory is the specialization of linear algebra to the case of finite dimensional vector spaces and doing explicit manipulations after fixing a basis. More precisely: The algebra of $n \times n$ matrices with coefficients in a field $F$ is isomorphic to the algebra of $F$-linear homomorphisms from an $n$-dimensional vector space $V$ over $F$, to itself. And the choice of such an isomorphism is precisely the choice of a basis for $V$. </p>
<p>Sometimes you need concrete computations for which you use the matrix viewpoint. But for conceptual understanding, application to wider contexts and for overall mathematical elegance, the abstract approach of vector spaces and linear transformations is better.</p>
<p>In this second approach you can take over linear algebra to more general settings such as modules over rings(PIDs for instance), functional analysis, homological algebra, representation theory, etc.. All these topics have linear algebra at their heart, or, rather, "is" indeed linear algebra..</p>
by Konrad Waldorf for What is the difference between matrix theory and linear algebra?Konrad Waldorf2010-03-30T22:06:44Z2010-03-30T22:06:44Z<p>Let me quote without further comment from Dieudonné's "Foundations of Modern Analysis, Vol. 1".</p>
<blockquote>
<p>There is hardly any theory which is more elementary [than linear algebra], in spite of the fact that generations of professors and textbook writers have obscured its simplicity by preposterous calculations with matrices. </p>
</blockquote>
by John Stillwell for What is the difference between matrix theory and linear algebra?John Stillwell2010-03-31T08:06:31Z2010-03-31T08:06:31Z<p>A counter-quotation to the one from Dieudonné:</p>
<blockquote>
<p>We share a philosophy about linear algebra: we think basis-free, we write basis-free, but when the chips are down we close the office door and compute with matrices like fury.</p>
</blockquote>
<p>(Irving Kaplansky, writing of himself and Paul Halmos)</p>
by zhaoliang for What is the difference between matrix theory and linear algebra?zhaoliang2010-04-01T14:45:47Z2010-04-01T14:45:47Z<p>My opinion: matrix theory mostly deals with matrix of a paticular kind , or a few relevant ones. But linear algebra cares about the general, underlying structrue. </p>
by Jon for What is the difference between matrix theory and linear algebra?Jon2010-04-01T21:08:12Z2010-04-04T18:10:56Z<p>Although some years ago I would have agreed with the above comments about the relationship between Linear Algebra and Matrix Theory, I DO NOT agree any more! </p>
<p>See, for example Bhatia's "Matrix Analysis" GTM book. For example, doubly-(sub)stochastic matrices arise naturally in the classification of unitarily-invariant norms. They also naturally appear in the study of quantum entanglement, which really has nothing to do with a basis. (In both instances, all sorts of NONarbitrary bases come into play, mainly after the spectral theorem gets applied.)</p>
<p>Doubly-stochastic matrices turn out to be useful to give concise proofs of basis-independent inequalities, such as the non-commutative Holder inequality:</p>
<p>tr |AB| $\le$ $||A||_p$ $||B||_q$</p>
<p>with 1/p+1/q=1, $|A|=(A^*A)^{1/2}$, and $||A||_p = (tr |A|^p)^{1/p}$</p>
by XX for What is the difference between matrix theory and linear algebra?XX2010-04-02T02:24:30Z2010-04-02T02:24:30Z<p>I'm with Jon. Matrices don't always appear as linear transformations. Yes, you can look at them as linear transformations, but there are times when it's better not to and study them for their own right. Jon already gave one example. Another example is the theory of positive (semi)definite matrices. They appear naturally as covariance matrices of random vectors. The notions like schur complements appear naturally in a course in matrix theory, but probably not in linear algebra.</p> | 677.169 | 1 |
Complex Numbers from A to ...Z2994
FREE
About the Book
It is impossible to imagine modern mathematics without complex numbers. Complex Numbers from A to . . . Z introduces the reader to this fascinating subject that, from the time of L. Euler, has become one of the most utilized ideas in mathematics.
The exposition concentrates on key concepts and then elementary results concerning these numbers. The reader learns how complex numbers can be used to solve algebraic equations and to understand the geometric interpretation of complex numbers and the operations involving them.
The theoretical parts of the book are augmented with rich exercises and problems at various levels of difficulty. A special feature of the book is the last chapter, a selection of outstanding Olympiad and other important mathematical contest problems solved by employing the methods already presented.
The book reflects the unique experience of the authors. It distills a vast mathematical literature, most of which is unknown to the western public, and captures the essence of an abundant problem culture. The target audience includes undergraduates, high school students and their teachers, mathematical contestants (such as those training for Olympiads or the W. L. Putnam Mathematical Competition) and their coaches, as well as anyone interested in essential mathematics. | 677.169 | 1 |
...Algebraic math is a major stepping stone to multiple sciences and must be mastered to facilitate future academic progress in the sciences. As algebra skills consolidate, we move toward calculus (differential and integration calculus) which employs extremely powerful math skills. Pre-calculus is the bed rock of algebraic math upon which calculus stands | 677.169 | 1 |
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more.
Course Hero has millions of course specific materials providing students with the best way to expand
their education.
Definite Integral Word ProblemsExample 1 An empty bucket is placed under a tap and filled with water. t minutes after the bucket has been placed under the tap. The rate of flow of water into the bucket is equal to 2.3 - 0.1t gallons per minute. How
Simple Linear RegressionLeast Square Curve Fitting1PurposeAssume that two quantitative variables that are measured on the same items are sampled from a population. We get n pairs of observations: (x1, y1),.,(xn ,yn) Our aim is to develop a mode
Cost, Revenue and Profit Functions Michael Cooney1Many business situations allow us to model how cost, revenue and vary with respect to different parameters, and how they combine to yield a functional expression for profit. In most cases, it is t
INTRODUCTION TO THE TI-83 AND TI-83 PLUS BasicsKeyboardEach key on the TI-83 and TI-83 Plus accesses up to three objects, operations, or menus. The primary object, operation, or menu is written on the key. Above each key are other objects, operatio
TI-83 (+) Keystrokes for Chapter 4 of Understanding Basic StatisticsItems in boxes are actual keys; other items are menu choices (selected with arrow keys, or the key). Some keys have text above them; this is given [in brackets]. A vertical line li
Just the Basics: Regression on the TI-83 Before You Do Your First Regression on the TI-83: Press CATALOG (2nd 0) and press the D key to jump down to the commands that start with the letter D. Use the down arrow to move the triangle down until it is t | 677.169 | 1 |
AcademicsNumber Theory
Course Outline: Number theory is primarily
concerned with the properties of and relationships between
whole numbers. Topics we will study:
1. Prime numbers
2. Modular arithmetic
3. Sums of squares
4. Pythagorean triples
5. Fermat's Last Theorem
6. Magic squares
7. Continued fractions
8. Approximation of reals by rationals
We will also spend a couple of weeks studying cryptography.
In particular, we will look at how the RSA system works.
This relies heavily on some of the number theory we will have
learnt and is behind almost all modern cryptographic systems.
You will need two books for the course: "A Pathway
Into Number Theory" by R.P. Burn and "An Introduction
To Number Theory" by H. Stark. Burn's book will
lead us to discover and prove for ourselves some of the main
results of number theory. Stark's book is more traditional. | 677.169 | 1 |
Trigonometry Smarts!
Buy ePub
List price:
$7.95
Our price:
$5.99
You save: $1.96 (25%)
Are you having trouble with trigonometry? Do you wish someone could explain this challenging subject in a clear, simple way? From triangles and radians to sine and cosine, this book takes a step-by-step approach to teaching trigonometry. This book is designed for students to use alone or with a tutor or parent, and provides clear lessons with easy-to-learn techniques and plenty of examples. Whether you are looking to learn this information for the first time, on your own or with a tutor, or you would like to review some trigonometry skills, this book will be a great choice. | 677.169 | 1 |
Description
By connecting applications, modeling, and visualization, Gary Rockswold motivates students to learn mathematics in the context of their experiences. In order to both learn and retain the material, students must see a connection between the concepts and their real-lives. In this new edition, connections are taken to a new level with "See the Concept" features, where students make important connections through detailed visualizations that deepen understanding.
Rockswold is also known for presenting the concept of a function as a unifying theme, with an emphasis on the rule of four (verbal, graphical, numerical, and symbolic representations). A flexible approach allows instructors to strike their own balance of skills, rule of four, applications, modeling, and technology. Additionally, incorporating technology with this edition has never been so exciting! Within the MyMathLab® course, new Interactive Figures help students visualize difficult topics. Also, Getting Ready integrated review allows students to remediate "just-in-time," by providing review of prerequisite material when needed to help them succeed in the course.
Table of Contents
1. Introduction to Functions and Graphs
1.1 Numbers, Data, and Problem Solving
1.2 Visualizing and Graphing Data
Checking Basic Concepts for Sections 1.1 and 1.2
1.3 Functions and Their Representations
1.4 Types of Functions and Their Rates of Change
Checking Basic Concepts for Sections 1.3 and 1.4
Chapter 1 Summary
Chapter 1 Review Exercises
Chapter 1 Extended and Discovery Exercises
2. Linear Functions and Equations
2.1 Equations of Lines
2.2 Linear Equations
Checking Basic Concepts for Sections 2.1 and 2.2
2.3 Linear Inequalities
2.4 More Modeling with Functions
Checking Basic Concepts for Sections 2.3 and 2.4
2.5 Absolute Value Equations and Inequalities
Checking Basic Concepts for Section 2.5
Chapter 2 Summary
Chapter 2 Review Exercises
Chapter 2 Extended and Discovery Exercises
Chapters 1-2 Cumulative Review Exercises
3. Quadratic Functions and Equations
3.1 Quadratic Functions and Models
3.2 Quadratic Equations and Problem Solving
Checking Basic Concepts for Sections 3.1 and 3.2
3.3 Complex Numbers
3.4 Quadratic Inequalities
Checking Basic Concepts for Sections 3.3 and 3.4
3.5 Transformations of Graphs
Checking Basic Concepts for Section 3.5
Chapter 3 Summary
Chapter 3 Review Exercises
Chapter 3 Extended and Discovery Exercises
4. More Nonlinear Functions and Equations
4.1 More Nonlinear Functions and Their Graphs
4.2 Polynomial Functions and Models
Checking Basic Concepts for Sections 4.1 and 4.2
4.3 Division of Polynomials
4.4 Real Zeros of Polynomial Functions
Checking Basic Concepts for Sections 4.3 and 4.4
4.5 The Fundamental Theorem of Algebra
4.6 Rational Functions and Models
Checking Basic Concepts for Sections 4.5 and 4.6
4.7 More Equations and Inequalities
4.8 Radical Equations and Power Functions
Checking Basic Concepts for Sections 4.7 and 4.8
Chapter 4 Summary
Chapter 4 Review Exercises
Chapter 4 Extended and Discovery Exercises
Chapters 1-4 Cumulative Review Exercises
5. Exponential and Logarithmic Functions
5.1 Combining Functions
5.2 Inverse Functions and Their Representations
Checking Basic Concepts for Sections 5.1 and 5.2
5.3 Exponential Functions and Models
5.4 Logarithmic Functions and Models
Checking Basic Concepts for Sections 5.3 and 5.4
5.5 Properties of Logarithms
5.6 Exponential and Logarithmic Equations
Checking Basic Concepts for Sections 5.5 and 5.6
5.7 Constructing Nonlinear Models
Checking Basic Concepts for Section 5.7
Chapter 5 Summary
Chapter 5 Review Exercises
Chapter 5 Extended and Discovery Exercises
6. Trigonometric Functions
6.1 Angles and Their Measure
6.2 Right Triangle Trigonometry
Checking Basic Concepts for Sections 6.1 and 6.2
6.3 The Sine and Cosine Functions and Their Graphs
6.4 Other Trigonometric Functions and Their Graphs
Checking Basic Concepts for Sections 6.3 and 6.4
6.5 Graphing Trigonometric Functions
6.6 Inverse Trigonometric Functions
Checking Basic Concepts for Sections 6.5 and 6.6
Chapter 6 Summary
Chapter 6 Review Exercises
Chapter 6 Extended and Discovery Exercises
Chapters 1-6 Cumulative Review Exercises
7. Trigonometric Identities and Equations
7.1 Fundamental Identities
7.2 Verifying Identities
Checking Basic Concepts for Sections 7.1 and 7.2
7.3 Trigonometric Equations
7.4 Sum and Difference Identities
Checking Basic Concepts for Sections 7.3 and 7.4
7.5 Multiple-Angle Identities
Checking Basic Concepts for Section 7.5
Chapter 7 Summary
Chapter 7 Review Exercises
Chapter 7 Extended and Discovery Exercises
8. Further Topics in Trigonometry
8.1 Law of Sines
8.2 Law of Cosines
Checking Basic Concepts for Sections 8.1 and 8.2
8.3 Vectors
8.4 Parametric Equations
Checking Basic Concepts for Sections 8.3 and 8.4
8.5 Polar Equations
8.6 Trigonometric Form and Roots of Complex Numbers
Checking Basic Concepts for Sections 8.5 and 8.6
Chapter 8 Summary
Chapter 8 Review Exercises
Chapter 8 Extended and Discovery Exercises
Chapters 1-8 Cumulative Review Exercises
9. Systems of Equations and Inequalities
9.1 Functions and Systems of Equations in Two Variables
9.2 Systems of Inequalities in Two Variables
Checking Basic Concepts for Sections 9.1 and 9.2
9.3 Systems of Linear Equations in Three Variables
9.4 Solutions to Linear Systems Using Matrices
Checking Basic Concepts for Sections 9.3 and 9.4
9.5 Properties and Applications of Matrices
9.6 Inverses of Matrices
Checking Basic Concepts for Sections 9.5 and 9.6
9.7 Determinants
Checking Basic Concepts for Section 9.7
Chapter 9 Summary
Chapter 9 Review Exercises
Chapter 9 Extended and Discovery Exercises
Chapters 1-9 Cumulative Review Exercises
10. Conic Sections
10.1 Parabolas
10.2 Ellipses
Checking Basic Concepts for Sections 10.1 and 10.2
10.3 Hyperbolas
Checking Basic Concepts for Section 10.3
Chapter 10 Summary
Chapter 10 Review Exercises
Chapter 10 Extended and Discovery Exercises
11. Further Topics in Algebra
11.1 Sequences
11.2 Series
Checking Basic Concepts for Sections 11.1 and 11.2
11.3 Counting
11.4 The Binomial Theorem
Checking Basic Concepts for Sections 11.3 and 11.4
11.5 Mathematical Induction
11.6 Probability
Checking Basic Concepts for Sections 11.5 and 11.6
Chapter 11 Summary
Chapter 11 Review Exercises
Chapter 11 Extended and Discovery Exercises
Chapters 1-11 Cumulative Review Exercises
R. Reference: Basic Concepts from Algebra and Geometry
R.1 Formulas from Geometry
R.2 Integer Exponents
R.3 Polynomial Expressions
R.4 Factoring Polynomials
R.5 Rational Expressions
R.6 Radical Notation and Rational Exponents
R.7 Radical Expressions
Appendix A: Using the Graphing Calculator
Appendix B: A Library of Functions
Appendix C: Partial Fractions
Appendix D: Percent Change and Exponential Functions
Appendix E: Rotation of Axes
Bibliography
Answers to Selected Exercises
Photo Credits | 677.169 | 1 |
This textbook is devoted to Combinatorics and Graph Theory, which are cornerstones of Discrete Mathematics. Every section begins with simple model problems. Following their detailed analysis, the reader is led through the derivation of definitions, concepts and methods for solving typical problems.
Best Internet Links
Since the 1980s, the theory of groups - in particular simple groups, finite and algebraic - has influenced a number of diverse areas of mathematics. Such areas include topics where groups have been traditionally applied, such as algebraic combinatorics, finite geometries, Galois theory and permutation groups, as well as several more recent developments.
Alan Tucker's newest issue of Applied Combinatorics builds on the previous editions with more in depth analysis of computer systems in order to help develop proficiency in basic discrete math problem solving. As one of the most widely used book in combinatorial problems, this edition explains how to reason and model combinatorically while stressing the systematic analysis of different possibilities, exploration of the logical structure of a problem, and ingenuity.
of set partitions from 1500 A.D. to today.
This book written by experts in their respective fields, and covers a wide spectrum of high-interest problems across these discipline domains. The book focuses on discrete mathematics and combinatorial algorithms interacting with real world problems in computer science, operations research, applied mathematics and engineering.
The articles collected here are the texts of the invited lectures given at the Eighth British Combinatorial Conference held at University College, Swansea. The contributions reflect the scope and breadth of application of combinatorics, and are up-to-date reviews by mathematicians engaged in current research. This volume will be of use to all those interested in combinatorial ideas, whether they be mathematicians, scientists or engineers concerned with the growing number of applications.
With examples of all 450 functions in action plus tutorial text on the mathematics, this book is the definitive guide to Experimenting with Combinatorica, a widely used software package for teaching and research in discrete mathematics. Three interesting classes of exercises are provided--theorem/proof, programming exercises, and experimental explorations--ensuring great flexibility in teaching and learning the material.
This book constitutes the refereed proceedings of the 16th Annual International Conference on Computing and Combinatorics, held in Dallas, TX, USA, in August 2011. The 54 revised full papers presented were carefully reviewed and selected from 136 submissions. Topics covered are algorithms and data structures; algorithmic game theory and online algorithms; automata, languages, logic, and computability; combinatorics related to algorithms and complexity; complexity theory; computational learning theory and knowledge discovery; cryptography, reliability and security, and database theory; computational biology and bioinformatics; computational algebra, geometry, and number theory; graph drawing and information visualization; graph theory, communication networks, and optimization; parallel and distributed computing.
This volume is a collection of survey papers in combinatorics that have grown out of lectures given in the workshop on Probabilistic Combinatorics at the Paul Erdös Summer Research Center in Mathematics in Budapest. The papers, reflecting the many facets of modern-day combinatorics, will be appreciated by specialists and general mathematicians alike: assuming relatively little background, each paper gives a quick introduction to an active area, enabling the reader to learn about the fundamental results and appreciate some of the latest developments. An important feature of the articles, very much in the spirit of Erdös, is the abundance of open problems.
Wherefore is this conference book on graphs and combinatorics different from other such books?
One way is that it is, frankly, a progress report on recent results in the field, and does not claim to be a definitive work. Another is that all contributions have been refereed. A third way is that it contains both expository review articles and research contributions. | 677.169 | 1 |
Book Description: Teaching Secondary Mathematics, Third Edition is practical, student-friendly, and solidly grounded in up-to-date research and theory. This popular text for secondary mathematics methods courses provides useful models of how concepts typically found in a secondary mathematics curriculum can be delivered so that all students develop a positive attitude about learning and using mathematics in their daily lives.A variety of approaches, activities, and lessons is used to stimulate the reader's thinking--technology, reflective thought questions, mathematical challenges, student-life based applications, and group discussions. Technology is emphasized as a teaching tool throughout the text, and many examples for use in secondary classrooms are included. Icons in the margins throughout the book are connected to strands that readers will find useful as they build their professional knowledge and skills: Problem Solving, Technology, History, the National Council of Teachers of Mathematics Principles for School Mathematics, and "Do" activities asking readers to do a problem or activity before reading further in the text. By solving problems, and discussing and reflecting on the problem settings, readers extend and enhance their teaching professionalism, they become more self-motivated, and they are encouraged to become lifelong learners.The text is organized in three parts:*General Fundamentals--Learning Theory, Curriculum; and Assessment; Planning; Skills in Teaching Mathematics;*Mathematics Education Fundamentals--Technology; Problem Solving; Discovery; Proof; and*Content and Strategies--General Mathematics; Algebra 1; Geometry; Advanced Algebra and Trigonometry; Pre-Calculus; Calculus.New in the Third Edition:*All chapters have been thoroughly revised and updated to incorporate current research and thinking.*The National Council of Teachers of Mathematics Standards 2000 are integrated throughout the text.*Chapter 5, Technology, has been rewritten to reflect new technological advances.*A Learning Activity ready for use in a secondary classroom has been added to the end of each chapter.*Two Problem-Solving Challenges with solutions have been added at the end of each chapter.*Historical references for all mathematicians mentioned in the book have been added within the text and in the margins for easy reference.*Updated Internet references and resources have been incorporated to enhance the use of the text. | 677.169 | 1 |
Elementary Algebra With Bca Tutorial, and Infotrac
9780534400415
ISBN:
0534400418
Publisher: Thomson Learning
Summary: Jerome Kaufmann and Karen Schwitters discuss algebra with clear and concise exposition, numerous examples, and plentiful problem sets. They reinforce the following common thread - learn a skill, use the skill to solve equations, and then apply this to solve application problems. | 677.169 | 1 |
From the Ishango Bone of central Africa and the Inca quipu of South America to the dawn of modern mathematics, The Crest of the Peacock makes it clear that human beings everywhere have been capable of advanced and innovative mathematical thinking. George Gheverghese Joseph takes us on a breathtaking multicultural tour of the roots and shoots of book treats all of the most commonly used theories of the integral. After motivating the idea of integral, we devote a full chapter to the Riemann integral and the next to the Lebesgue integral. Another chapter compares and contrasts the two theories. The concluding chapter offers brief introductions to the Henstock integral, the Daniell integral,...Many problems in science can be formulated in the language of optimization theory, in which case an optimal solution or the best response to a particular situation is required. In situations of interest, such classical optimal solutions are lacking, or at least, the existence of such solutions is far from easy to prove. So, non-convex optimization... more...
The subject of fractional calculus and its applications (that is, convolution-type pseudo-differential operators including integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past three decades or so, mainly due to its applications in diverse fields of science and engineering.... more... | 677.169 | 1 |
MATH 111 - College Algebra
A course in Algebra for college students with a strong emphasis on problem-solving and applications. Topics include: introduction to functions and their graphs; linear and quadratic functions; solution of a variety of types of equations and inequalities using algebraic, numeric and graphical techniques; systems of equations, operations with polynomials; rational, radical, exponential and logarithmic expressions; and exponential functions. Use of a graphing calculator may be an integral part of the course. Prerequisite: placement per high school transcript, completion of MATH101 or MATH101X with "C-" or higher, or by permission of the Mathematics Department. [Fall, Spring] | 677.169 | 1 |
M2= Math Mediator
Algebra 2 High School Lesson Plans
Click on this link: Lesson Plan Example
for an example of one of the lesson plans available. Notice that each lesson
plan offers:
Designed for 55 minutes of class time
Includes topics to meet math standards
Exercises are included with answers
Activities are used in many lesson plans
No text book required, lessons are "stand alone"
Available individually (email [email protected] for set)
Easy PayPal Express Payment for contributions
Currently, we are only offering Algebra 2 lessons in pdf format. We've recently added a few PowerPoint lessons and will continue to add them as they are created.
To view our list of lesson plans and place you order, just click on Algebra 2
Lesson Plans at the top of the page and you will be directed to all the information.
Application Enriched Lesson Plans
We developed stand alone Algebra 2 high school lesson plans that incorporate
real life examples and applications. In addition, many of these lesson plans
raise awareness to pertinent issues on health, career choices, budgeting
and social responsibilities. These plans have been structured for a 55
minute class duration with detailed instruction, projects and examples. Please
take a look at an example lesson by clicking on "Lesson Plan Example" above and view the example. | 677.169 | 1 |
Graphing Calculator MatchingManiaGraphing Calculator MatchingMania consists of 12 functions. Students work together using a graphing calculator to find the zeroes, minimums and/or maximums and the points of intesection of two functions. This is a great calculator review activity or learning activity to begin the school year with in advanced math classes.
PDF (Acrobat) Document File
Be sure that you have an application to open this file type before downloading and/or purchasing.
28.96 | 677.169 | 1 |
Aquasco ACT knows how to tap into each kid?s learning style. ItMy philosophy for studying is that it is insufficient to merely memorize formulas; it is necessary to understand the formula's derivation as well as its potential applications. This leads to a much greater understanding of the subject material.A solid algebra foundation is necessary for almost aIn essence, there is a systematic process that should be used to establish some of the fundamental principles young children need to become successful at future attempts at higher level mathematics. Ultimately, if there is a lack of sound mathematics skills, a student may experience some level o... | 677.169 | 1 |
Properties: Evaluate The learner will be able to
evaluate mathematical and algebraic expressions using the following properties: associative, commutative, identity, substitution, inverse and zero properties.
Data Collection: Organize The learner will be able to
collect, organize ,and display data with appropriate notation in tables, charts, and graphs (scatter plots, line graphs, bar graphs, and pie charts).
Figures: Two-/Three-Dimensional Objects The learner will be able to
use appropriate vocabulary to precisely explain, classify, and comprehend relationships among types of two- and three-dimensional objects by applying their defining properties.
Mathematical Reasoning: Explain The learner will be able to
apply many different methods to describe and communicate mathematical reasoning and concepts such as words, numbers, symbols, graphical forms, and/or models.
Area/Volume/Length: Differences The learner will be able to
identify the differences and relationships between length, area, and volume (capacity) measure in the metric and U.S. Customary measurement systems. | 677.169 | 1 |
The Mathematics curriculum follows the guiding principles of the Massachusetts Mathematics
Curriculum Framework:
1) mathematical ideas must be explored,
2) all students must have access to high quality mathematics programs,
3) mathematics learning is a lifelong process,
4) mathematics instruction must connect with other disciplines and move toward integration
of mathematical domains,
5) group work enhances the learning of mathematics,
6) technology is an essential tool, and
7) mathematics assessment must be multifaceted to monitor student performance, improve instruction,
enhance learning and encourage student self-reflection. The core subjects for all college
preparatory students include Algebra I, Geometry, and Algebra II. Beyond this, a full range
of opportunities exists for students to broaden and refine their mathematical skills through
specialized and advanced courses.
As active learners, students are expected to share in the responsibility of becoming mathematically
literate and technically competent. Students will explore, investigate, validate, discuss, represent,
and construct mathematics while teachers create the learning environment, guide, discuss, question,
listen, and clarify. Five learning standards integrated throughout the curricula include
1) mathematics as problem solving,
2) mathematics as communicating,
3) mathematics as reasoning,
4) mathematical connections, and
5) mathematical representations.
Five mathematic strands interwoven throughout the curricula are
1) number sense and operations,
2) patterns, relations, and algebra,
3) geometry,
4) measurement, and
5) data analysis, statistics and probability.
Those students wishing to take AP Calculus should successfully complete the Honors sequence.
Calculators may be used in all mathematics courses in order that students may
1) concentrate on the problem-solving process,
2) gain access to mathematics beyond the students' level of computational skills,
3) explore, develop, and reinforce concepts including estimation, computation, approximation, and properties,
4) experiment with mathematical ideas and discover patterns, and
5) perform those tedious computations that arise when working with real data in problem-solving situations.
Since scientific or graphing calculators are necessary for most courses, students should provide
their own calculators. Teachers will inform students of the recommended calculator at the
beginning of the school year. | 677.169 | 1 |
Curriculum and Requirements
Related Links
For well over two thousand years, mathematics has been a part of the human search for understanding. Mathematical discoveries have come both from the attempt to describe the natural world and from the desire to arrive at a form of inescapable truth through careful reasoning that begins with a small set of self-evident assumptions Learning to think in mathematical terms is an essential part of becoming a liberally educated person.
Mathematics is an engaging field, rich in beauty, with powerful applications to other subjects. Thus we strive to ensure that Kenyon students encounter and learn to solve problems using a number of contrasting but complementary mathematical perspectives: continuous and discrete, algebraic and geometric, deterministic and stochastic, theoretical and applied. In our courses we stress mathematical thinking and communication skills. And in courses where it makes sense to incorporate technological tools, our students learn to solve mathematical problems using computer algebra systems, statistical packages, and computer programming languages.
New Students
For those students who want only an introduction to mathematics, or perhaps a course to satisfy a distribution requirement, selection from MATH 105, 106, 111, 116, 128 and SCMP 118 is appropriate. Students who think they might want to continue the study of mathematics beyond one year, either by pursuing a major or minor in mathematics or as a foundation for courses in other disciplines, usually begin with the calculus sequence (MATH 111, 112, and 213). Students who have already had calculus or who want to take more than one math course may choose to begin with the Elements of Statistics (MATH 106) and Data Analysis (MATH 206) or Introduction to Programming (SCMP 118). A few especially well-prepared students take Linear Algebra (MATH 224) or Foundations (MATH 222) in their first year. (Please see the department chair for further information.)
MATH 111 is an introductory course in calculus. Students who have completed a substantial course in calculus might qualify for one of the successor courses, MATH 112 or 213. MATH 106 is an introduction to statistics, which focuses on quantitative reasoning skills and the analysis of data. SCMP 118 introduces students to computer programming.
To facilitate proper placement of students in calculus courses, the department offers placement tests that help students decide which level of calculus course is appropriate for them. This and other entrance information is used during the orientation period to give students advice about course selection in mathematics. We encourage all students who do not have Advanced Placement credit to take the placement exam that is appropriate for them.
The ready availability of powerful computers has made the computer one of the primary tools of the mathematician. Students will be expected to use appropriate computer software in many of the mathematics courses. However, no prior experience with the software packages or programming is expected, except in advanced courses that presuppose earlier courses in which use of the software or programming was taught.
Course Requirements for the Major
There are two concentrations within the mathematics major: classical mathematics and statistics. The coursework required for completion of the major in each concentration is given below.
Classical Mathematics A student must have credit for the following core courses:
Three semesters of calculus (MATH 111, 112, 213, or the equivalent)
One semester of statistics (MATH 106 or 436, or the equivalent)
SCMP 118 Introduction to Programming
MATH 222 Foundations
MATH 224 Linear Algebra I
MATH 335 Abstract Algebra I or MATH 341 Real Analysis I
In addition, majors must have credit for at least three otherelective courses selected with the consent of the department. MATH 110 may not be used to satisfy the requirements for the major.
Statistics A student must have credit for the following core courses:
Three semesters of calculus (MATH 111, 112, 213 or the equivalent)
SCMP 118 Introduction to Programming
MATH 222 Foundations
MATH 224 Linear Algebra I
MATH 336 Probability
MATH 341 Real Analysis I
MATH 416 Linear Regression Models or MATH 436 Mathematical Statistics
In addition to the core courses, majors must also have credit for two elective courses from the following list:
MATH 106 Elements of Statistics
MATH 206 Data Analysis
MATH 216 Nonparametric Statistics
MATH 236 Random Structures
MATH 416 Linear Regression Models
MATH 436 Mathematical Statistics
Applications of Math Requirement
Mathematics is a vital component in the methods used by other disciplines, and the applied math requirement is designed to expose majors to this vitality. There are two ways to satisfy the requirement:
a) Earn credit for two courses (at least 1 unit) from a single department or program that use mathematics in significant ways. Typically, majors will choose a two-course sequence from the following list; other two-course sequences require departmental approval:
PHYS 140/145
ECON 101/102
PSYC 200 together with a 400-level Research Methods in Psychology course
b) Earn credit for a single math course that focuses on the development and analysis of mathematical models used to answer questions arising in other fields. The following courses satisfy the requirement, but other courses may satisfy the requirement with approval of the department:
MATH 258 Mathematical Biology
MATH 347 Mathematical Models
Classical mathematics majors may also use MATH 206, MATH 216, MATH 226, or MATH 416 to satisfy the requirement. Additionally, students choosing this option may not use the applied math course as one of the elective courses required for the major.
Depth Requirement
Majors are expected to attain a depth of study within mathematics, as well as breadth. Therefore majors should earn credit in one of four two-course upper-level sequences:
MATH 335/435 Abstract Algebra I & II
MATH 341/441 Real Analysis I & II
MATH 336/436 Probability and Mathematical Statistics
MATH 336/416 Probability and Linear Regression Models
Other two-course sequences may satisfy the requirement with approval from the department.
Senior Exercise
The Senior Exercise begins promptly in the fall of the senior year with independent study on a topic of interest to the student and approved by the department. The independent study culminates in the writing of a paper, which is due in November. (Juniors are encouraged to begin thinking about possible topics before they leave for the summer.) Students are also required to take the Major Field Test in Mathematics produced by the Educational Testing Service. Evaluation of the Senior Exercise is based on the student's performance on the paper and the standardized exam. A detailed guide on the Senior Exercise is available on the math department Web site under the link "mathematics academic program."
Suggestions for Majoring in Mathematics
Students wishing to keep open the option of a major in mathematics typically begin with the study of calculus and normally complete the calculus sequence, MATH 222 (Foundations), and either SCMP 118 or MATH 106 by the end of the sophomore year. A major is usually declared no later than the second semester of the sophomore year. Those considering a mathematics major should consult with a member of the mathematics department to plan their course of study.
The requirements for the major are minimal. Anyone who is planning a career in the mathematical sciences, or who intends to read for honors, is encouraged to consult with one or more members of the department concerning further studies that would be appropriate. Similarly, any student who wishes to propose a variation of the major program is encouraged to discuss the plan with a member of the department prior to submitting a written proposal for a decision by the department.
Students who are interested in teaching mathematics at the high-school level should take MATH 230 (Geometry) and MATH 335 (Abstract Algebra I), since these courses are required for certification in most states, including Ohio.
Honors in Mathematics
Eligibility
To be eligible to enroll in the Mathematics Honors Seminar, by the end of junior year students must have completed one depth sequence (MATH 335-435, MATH 336-416, MATH 336-436, MATH 341-441) and have earned a GPA of at least 3.33, with a GPA in Kenyon mathematics courses of at least 3.6. The student must also have, in the estimation of the mathematics faculty, a reasonable expectation of fulfilling the requirements for Honors, listed below.
To earn Honors in mathematics, a student must: (1) Complete two depth sequences (see list above); (2) Complete at least six 0.5-unit courses in mathematics numbered 300 or above; (3) Pass the Senior Exercise in the fall semester; (4) Pass the Mathematics Honors Seminar MATH 498; (5) Present the results of independent work in MATH 498 to a committee consisting of an outside examiner and members of the Kenyon Mathematics Department; (6) successfully complete an examination written by an outside examiner covering material from MATH 498 and previous mathematics courses; (7) Maintain an overall Kenyon GPA of at least 3.33; (8) Maintain a Mathematics Department GPA of at least 3.6.
Awarding Honors
Based on performance in all of the above-mentioned areas, the department (in consultation with the outside examiner) can elect to award Honors, High Honors, or Highest Honors; or not to award honors at all.
Requirements for the Minors
There are two minors in mathematics. Each minor deals with core material of a part of the discipline, and each reflects the logically structured nature of mathematics through a pattern of prerequisites. A minor consists of satisfactory completion of the courses indicated.
Statistics Five courses in statistics from the following: MATH 106 or 116, 206, 216, 236, 336, 416, 436. (Students may count at most one statistics course from another department. For example, ECON 375 or PSYC 200 may be substituted for one of the courses listed above.)
Our goal is to provide a solid introduction to basic statistical methods, including data analysis, design and analysis of experiments, statistical inference, and statistical models, using professional software such as Minitab, SAS, Maple, and R.
Deviations from the list of approved minor courses must be ratified by the Mathematics Department. Students considering a minor in mathematics or statistics are urged to speak with a member of the department about the selection of courses.
Cross-listed course The following course is cross-listed in biology and will satisfy the natural science requirement: MATH 258 Mathematical Biology | 677.169 | 1 |
Testimonial
Bridgeway Math Book 1 Math Foundations
Bridgeway Math Foundations is the first book in a remedial math course written specifically to be used as a homeschool independent study course. No teacher's guides, no extra work, no extra instruction needed. Instead, it provides step by step easy to follow instructions to the student and plenty of practice to ensure that they are really mastering each concept. 728795162764
Bridgeway Math Foundations covers the topics of: basic operations, adding, subtracting, multiplying, and dividing, using whole numbers, place value, estimation, fractions, least common denominators, mixed numbers, decimals, and word problems. This book guides students through the first half of the Bridgeway course and prepares them for Bridgeway Pre-Algebra.
Somehow the way this remedial homeschool math course presents the material works! Kids just get it and find that they are able to succeed as they move on to more difficult concepts. The order makes sense, the easy to understand teaching and instructions, the number of practice questions for each concept… All carefully put together to ensure success.
A terrific foundations or remedial math course for children in 7th grade all the way up to high school seniors in need of extra instruction in math. And it works. | 677.169 | 1 |
Welcome to class. This is where you can get you daily classwork and assignments by clicking on the correct link below. I will also post handouts and review sheets under your specific class link from time to time. Please contact me by clicking on the e-mail link to the left or stop by rm. 65 (the portable by the football fields) if you have any questions about the class. I am available before and after school.
AP Calculus BC
Welcome to AP Calculus. This course covers differential, Integral, and Series Calculus. Students can receive credit for MTH 251, 252, 253 over the three terms.
Please see me as early as possible should problems arise. Remember that hotmath is not available for the calculus course.
This course applies mathematical concepts learned in previous math classes to various careers and situations. Management Science covers ways to look at routing problems used by delivery, communications, and transportation companies. Social Choice and Decision Making will look at election methods, fair division, and apportionment.
Game Theroy: The Mathematics of Competition looks at strategies for different types of games and situations that resemble them. Linear Programming deals with methods for optimizing production and profits. We will finish with Probability and where it applies.
This course is an extensive coverage of Euclidian geometry. Topics covered will be transformations and symmetry, angle relationships, similarity, right triangle trigonometry, special right triangles, laws of sine and cosine, proof, probability, constructions, polygons, 2-D dimensions and area, circles and chords, 3-D area and volume, constructions, and conic sections. This is a year long course and will be graded using a standards based system. Please see attachment.
Welcome to Pre-Calculus. This course replaces our college algebra and trigonometry classes and combines them into one year long class. In the first semester this course covers the algebra required for entry level math at the college level. Students may, if they wish receive, credit for MTH 111 in March (or Winter term at Chemeketa). The course covers advanced functions, logarithms, polynomial functions, rates, sequences, and series. The second semester covers trigonometry which is equivalent to MTH 112. Credit can be earned in June (spring term at Chemeketa). Topics include the radians and the unit circle, trigonometric functions and their graphs, identities, solving trig. equations, periodic fuctions, polar equations and complex numbers. A graphing calculator is required for much of the homework and classwork. A TI-84 plus is recommended but others will work. I won't be able to help with the programming and use of other brands of calculators. Please feel free to come in before or after school for additional help. The sooner you get help the more you will enjoy the course. | 677.169 | 1 |
Mathematics for Elementary Teachers, Third Edition offers an inquiry-based approach, which helps readers reach a deeper understanding of mathematics. Sybilla Beckmann, known for her contributions in math education, writes a text that encourages future teachers to find answers through exploration and group work. Fully integrated activities are found in her accompanying Activities Manual, which comes with every new copy of this text. As a result, readers engage, explore, discuss, and ultimately reach a true understanding of mathematics.
Customer Reviews:
excellent book for learning to think about elementary math
By mathwonk - August 29, 2010
The recent review of books and programs in mathematics teacher education in America by the National Council on Teacher Quality rated this book the best one in the country, and rated the mathematics education program at UGA where this book is used and was developed, as the ONLY "exemplary" program in the entire nation. Obviously not everyone finds it easy to read, but it does serve its purpose of helping, and requiring, the student to understand the concepts behind the mathematics. It is not easy to teach from this book, as I have learned by experience. The reason was not the fault of the book, but rather the extreme difficulty in getting "test oriented" students to stop looking at mathematics as just a list of formulas and procedures, and to begin trying to grasp the ideas which underly them. Understanding is harder than calculating, and having read it, I can easily understand why this book is professionally considered the best one in the country for making that transition in... read more
Satisfied Student
By Katie Driver "Katie" - October 12, 2011
This book, and the activity book that comes with it, was required for an education class I'm taking, and I found it very interesting. I plan on keeping it to use in the future when I become a teacher; the activity book included has a large variety of activities that can be used to help students learn. I highly recommend this book if you are interested in teaching any form of math in the future.
Great book for my course
By Monica Caropreso - January 9, 2013
This manual was invaluable in my course in college. Its activities really helped nail down the concepts my professor taught. | 677.169 | 1 |
New:
New 096241971061
FREE
New:
New BRAND NEW BOOK! Shipped within 24-48 hours. Normal delivery time is 5-12 days.
AwesomeBooksUK
OXON, GBR
$43.14
FREE
About the Book
Since everything assembled consists of either straight lines, curved lines, or a combination of both, the ability to calculate circles and right triangles is essential for anyone who works in a building trade. This simple and straightforward book explains the basic math used in construction, manufacturing, and design. Starting with fractions and decimals and moving to mitered turns and arcs, these principles are presented with detailed illustrations, practical applications, and in larger print for easy reading. The result is increased efficiency, productivity, and confidence in one's work from initial design to final product. | 677.169 | 1 |
The high school math curriculum helps students to prepare for college. Every student is required to accumulate 3 math credits in order to graduate. Every student must pass the Integrated Algebra Regents exam in order to graduate. If a student wishes to pursue an Advanced Academic diploma with designation, they must past the Geometry and Algebra II/Trig Regents exams also.
Choose the appropriate link to view teachers, courses, and links to other sites that will help students and parents succeed. | 677.169 | 1 |
This book is meant to be easily readable to engineers and scientists while still being (almost) interesting enough for mathematics students. Be advised that in-depth proofs of such matters as series convergence, uniqueness, and existence will not be given; this fact will appall some and elate others. This book is meant more toward solving or at the very least extracting information out of problems involving partial differential equations. The first few chapters are built to be especially simple to understand so that, say, the interested engineering undergraduate can benefit; however, later on important and more mathematical topics such as vector spaces will be introduced and used.
What follows is a quick intro for the uninitiated, with analogies to ordinary differential equations.
What is a Partial Differential Equation?
Ordinary differential equations (ODEs) arise naturally whenever a rate of change of some entity is known. This may be the rate of increase of a population, the rate of change of velocity, or maybe even the rate at which soldiers die on a battlefield. ODEs describe such changes of discrete entities. Respectively, this may be the capita of a population, the velocity of a particle, or the size of a military force.
More than one entity may be described with more than one ODE. For example, cloth is very often simulated in computer graphics as a grid of particles interconnected by springs, with Newton's law (an ODE) applied to each "cloth particle". In three dimensions, this would result in 3 second order ODEs written and solved for each particle.
Partial differential equations (PDEs) are analogous to ODEs in that they involve rates of change; however, they differ in that they treat continuous media. For example, the cloth could just as well be considered to be some kind of continuous sheet. This approach would most likely lead to only 3 (maybe 4) partial differential equations, which would represent the entire continuous sheet, instead of a set of ODEs for each particle.
This continuum approach is a very different way of looking at things. It may or may not be favorable: in the case of cloth, the resulting PDE system would be too difficult to solve, and so the computer graphics industry goes with a particle based approach (but a prime counterexample is a fluid, which would be represented by a PDE system most of the time).
While PDEs may not be straightforward to solve on a computer, they have a major advantage over ODEs when applicable: it is nearly impossible to gain any analytical insight from a huge system of particles, while a relatively small PDE system can reveal much insight, even if it won't yield an analytic solution.
But PDEs don't strictly describe continuum mechanics. As with anything mathematical, they are what you make of them.
The Character of Partial Differential Equations
The solution of an ODE can be represented as a function of one variable. For example, the position of the Earth may be represented by coordinates with respect to, say, the sun, and each of these coordinates would be functions of time. Note that the effects of other celestial bodies would certainly affect the solution, but it would still be expressible strictly as a function of time.
The solution of a PDE will, in general, depend on more than one variable. An example is a vibrating string: the deflection of the string will depend both on time and which part of the string you're looking at.
The solution of an ODE is called a trajectory. It may be represented graphically by one or more curves. The solution of a PDE, however, could be a surface, a volume, or something else, depending on how many variables are involved and how they're interpreted.
In general, PDEs are complicated to solve. Concepts such as separation of variables or integral transformations tend to work very differently. One significant difficulty is that the solution of a PDE depends very strongly on the initial/boundary conditions (ICs/BCs). An ODE typically yields a general solution, which involves one or more constants which may be determined from one or more ICs/BCs. PDEs, however, do not easily yield such general solutions. A solution method that works for one initial boundary value problem (IBVP) may be useless for a different IBVP.
PDEs tend to be more difficult to solve numerically as well. Most of the time, an ODE can be expressed in terms of its highest order derivative, and can be solved on a computer very easily with knowledge of the ICs (boundary value problems are a little more complicated), using well established and more or less generally applicable methods, such as Runge Kutta (RK). With this in mind, an ODE may be solved quickly by entering the equation and its ICs/BCs into the right application and pressing the solve button. An IBVP for a PDE, however, will typically require its own specialized solution, and it may take much effort to make the solution more than, say, second order accurate.
An Early Example
Many of the concepts of the previous section may be summarized in this example. We won't deal with the PDE just yet.
Consider heat flow along a laterally insulated rod. In other words, the heat is only flowing along the rod but not into the surrounding air. Let's call the temperature of the rod , and let , where is time and represents the position along the rod. As the temperature depends both on time and position along the rod, this is exactly what says. It is the change of the heat distribution over time. See the graphic below to get an idea.
Let's say that the rod has unitless length , and that its initial temperature (again unitless) is known to be . This states the initial condition, which depends on . The function is a simple hump between 0 and 1. Check for yourself with maxima ( or on android): plot2d(sin(x*%pi),[x,0,1])
Let's also say that the temperature is somehow fixed to at both ends of the rod, i.e. at and at . This would result in , which specifies boundary conditions. The BCs state that for all t, at and .
A PDE can be written to describe the situation. This and the IC/BCs form an initial boundary value problem (IBVP). The solution to this IBVP is (with a physical constant taken to be ):
Note that:
It also satisfies the PDE, but (again) that'll come later.
This solution may be interpreted as a surface, it's shown in the figure below with going from to , and going from to . That is, the distribution of heat is changing over time as the heat flows and dissipates.
from to and to .
Surfaces may or may not be the best way to convey information, and in this case a possibly better way to draw the picture would be to graph as a curve at several different choices of , this is portrayed below.
in the domain of interest for various interesting values of .
PDEs are extremely diverse, and their ICs and BCs can radically affect their solution method. As a result, the best (read: easiest) way to learn is by looking at many different problems and how they're solved.
Introductory Topics and Techniques
Parallel Plate Flow: Easy IC
Formulation
As with ODEs, separation of variables is easy to understand and works well whenever it works. For ODEs, we use the substitution rule to allow antidifferentiation, but for PDEs it's a very different process involving letting dependencies pass through the partial derivatives.
A fluid mechanics example will be used.
Consider two plates parallel to each other of huge extent, separated by a distance of 1. Fluid is smoothly flowing between these two plates in only one direction (call it x). This may be seen in the picture below.
Visualization of the parallel plate flow problem.
After some assumptions, the following PDE may be obtained to describe the fluid flow:
This linear PDE is the result of simplifying the Navier-Stokes equations, a large nonlinear PDE system which describes fluid flow. u is the velocity of the fluid in the x direction, ρ is the density of the fluid, ν is the kinematic viscosity (metaphorically speaking, how hard the molecules of the fluid rub against each other), and Px is the pressure gradient or pressure gradient vector field. Note that u = u(y, t), there is no dependence on x. In other words, the state of the fluid upstream is no different from the state downstream. Notice that we do not consider turbulences and that the state of the flow varies between the upper plate and the lower plate as the speed u is 0 close to the plates.
u(y, t) is a velocity profile. Fluid mechanics typically is concerned with velocity fields, contrary to rigid body mechanics in which the position of an object is what is important. In other words, with a rigid object all 'points' of the object move at the same speed. With a fluid we get a velocity field and each point is moving at its own speed.
The ratio Px/ρ describes the driving force; it's a pressure change (gradient) along the x direction. If Px is negative, then the pressure downstream (positive x) is smaller than the pressure upstream (negative x) and the fluid will flow left to right, i.e., u(y, t) will generally be positive.
Now on to create a specific problem: let's say that a constant negative pressure gradient was applied for a long time, until the velocity profile was steady (steady means "not changing with time"). Then the pressure gradient is suddenly removed, and without this driving force the fluid will slow down and stop. That is the assumption we are going to use for our example calculations. We assume the flow to be steady and then decaying as the pressure (expressed as Px/ρ) is removed. Hence we can remove the term -Px/ρ from our model as well, see (PDE) below.
Initial flow profile.
Let's say that before the pressure was removed, the velocity profile was u(y, t) = sin(π y). With velocity profile we mean, as the velocities are measured on a cross section of the flow, they form a hump. This would make sense: the friction dictates less motion near the plates (see next paragraph), so we could expect a maximum velocity near the centerline (y = 1/2). This assumed profile isn't really correct, but will serve as an example for now. It's graphed at right in the domain of interest.
Before getting into the math, one more thing is needed: boundary conditions. In this case, the BC is called the no slip condition, which states that the velocity of a fluid at a wall (boundary) is equal to the velocity of the wall. If we weren't making this assumption, the fluid would be moving like a rigid object. Since the velocities of the walls (or plates) in this problem are both zero, the velocity of the fluid must be zero at these two boundaries. The BCs are then u(0, t) = 0 (bottom plate) and u(1, t) = 0 (top plate).
The IBVP is:
Notice that the initial condition IC is the same hump from the first section of this book, just turned sideways. This IC implies that the flow is already flowing when we start our calculations. We are kind of calculating the decay of the flow speed.
Separation
Variables are separated the following way: we assume that , where Y and T are (unknown) functions respectively of y and t. This form is substituted into the PDE:
Using yields:
Look carefully at the last equation: the left side of the equation depends strictly on t, and the right side strictly on y, and they are equal. t may be varied independently of y and they'd still be equal, and y may be varied independently of t and they'd still be equal. This can only happen if both sides are constant. This may be shown as follows:
Taking the derivative of both sides, turns the right-hand-side into a constant, 0. The left-hand-side is left as is. Then taking the integrals of both sides yields:
Integration of the ordinary derivative recovers the left side but leaves the right side a constant. It follows by similarity that Y''/Y is a constant as well.
The constant in question is called the separation constant. We could simply give it a letter, such as A, but a good choice of the constant will make work easier later. In this case the best choice is -k2. This will be justified later (but it should be reemphasized that it may be notated any way you want, assuming it can span the domain).
The variables are now separated. The last two equations are two ODEs which may be solved independently (in fact, the Y equation is an eigenvalue problem), though they both contain an unknown constant. Note that ν was kept for the T equation. This choice makes the solution slightly easier, but is again completely arbitrary.
Rearrange and note that the notion buried in the expression below is that a function is equal to its own derivative, which kind of strikes the Euler number bell.
Then solve.
Here is a solution plucked from Ted Woollett's 'Maxima by Example', Chapter 3, '3.2.3 Exact Solution Using desolve'. Maxima's output is not shown but should be easily reconcilable with what has been written.
(%i1) de:(-%k^2*v*T(t))-('diff(T(t),t)) = 0;
(%i2) gsoln:desolve(de,T(t));
The same goes for our right-hand-side.
And solve.
In Maxima the solution goes like this. When Maxima is asking for 'zero' or 'nonzero' then type 'nonzero;':
(%i1)de:(-%k^2*Y(y))-('diff(Y(y),y,2));
(%i2)atvalue('diff(Y(y),y), y=0, Y(0)*%k )$
(%i3)gsoln:desolve(de,Y(y));
The overall solution will be , still with unknown constants, and as of now the product of Y and T. So we are plugging the partial solutions for and back in:
Note that C1 has been multiplied into C2 and C3, reducing the number of arbitrary constants. Because an unknown constant multiplied by an unknown constant yields still an unknown constant.
The IC or BCs should now be applied. If the IC was applied first, coefficients would be equated and all of the constants would be determined. However, the BCs may or may not have been fulfilled (in this case they would, but you're not generally so lucky). So to be safe, the BCs will be applied first:
So A being zero eliminates the term.
If we took B = 0, the solution would have just been u(y, t) = 0 (often called the trivial solution), which would satisfy the BCs and the PDE but couldn't possibly satisfy the IC. So, we take k = nπ instead, where n is any integer. After applying the BCs we have:
Decaying flow.
Then we need to apply the IC to it. Per the IC from above is:
Per the BC is, see above. Setting those two definitions of equal is:
Since t = 0 as per the IC assumption, is becoming . The equality can only hold if B = 1 and n = 1, allowing us to simply remove those two constants from the function in its BC incarnation. That's it! The complete solution is:
It's worth verifying that the IC, BCs, and PDE are all satisfied by this. Also notice that the solution is a product of a function of t and a function of y. The graph at the right is illustrating this. Observe that the profile is plotted for different values of νt, rather than specifying some ν and graphing different values for t. Remember that v is the kinematic viscosity. Hence the decay of flow speed also depends on it. Looking at the solution, t and ν appear only once and they're multiplying, so it's natural to do this. A dimensionless time could have been introduced from the beginning.
So what happens? The fluid starts with its initial profile and slows down exponentially. Note that with x replaced with y and t replaced with νt, this is exactly the same as the result of heat flow in a rod as shown in the introduction. This is not a coincidence: the PDE for the rod describes diffusion of heat, the PDE for the parallel plates describes diffusion of momentum.
Take a second look at the separation constant, -k2. The square is convenient, without it the solution for Y(y) would have involved square roots. Without the negative sign, the solution would have involved exponentials instead of sinusoids, so the constant would have come out imaginary.
The assumption that u(y, t) = Y(y)T(t) is justified by the physics of the problem: it would make sense that the profile would keep its general shape (due to Y(y)), only it'd get flattened over time as the fluid slows down (due to T(t)).
Quickly recap what we did. First we took a model PDE from Navier-Stokes and simplified it by making an assumption about the pressure being removed. Then we sort of first-stage solved it by separating Y and T. Then we applied the BC (boundary conditions) and the IC (initial condition) yielding our final solution.
Parallel Plate Flow: Realistic IC
The Steady State
The initial velocity profile chosen in the last problem agreed with intuition but honestly came out of thin air. A more realistic development follows.
The problem stated that (to come up with an IC) the fluid was under a pressure difference for some time, so that the flow became steady aka flowing steadily. "Steady" is another way of saying "not changing with time", and "not changing with time" is another way of saying that:
Putting this into the PDE from the previous section:
Independent of , the PDE became an ODE with variables separated and thus we can integrate.
The no slip condition results in the following BCs: at and . We can plug the BC values into the integrated ODE and resolve the Cs.
Inserting the Cs and and simplifying yields:
For the sake of example, take (recall that a negative pressure gradient causes left to right flow). Also note that this is a constant gradient or slope. This gives a parabola which starts at , increases to a maximum of at , and returns to at .
This parabola looks pretty much identical to the sinusoid previously used (you must zoom in to see a difference). However, even more so on the narrow domain of interest, the two are very different functions (look at their taylor expansions, for example). Using the parabola instead of the sine function results in a much more involved solution.
So this derives the steady state flow, which we will use as an improved, realistic IC. Recall that the problem is about a fluid that's initially in motion that is coming to a stop due to the absence of a driving force. The IBVP (Initial Boundary Value Problem) is now subtly different:
Separation
Since the only difference from the problem in the last section is the IC, the variables may be separated and the BCs applied with no difference, giving:
But now we're stuck (after applying the BCs)! Applying the IC makes the term go away as t = 0, which is the IC. However, then the IC function can't be made to match:
What went wrong? It was the assumption that . The fact that the IC couldn't be fulfilled means that the assumption was wrong. It should be apparent now why the IC was chosen to be in the previous section.
We can proceed however, thanks to the linearity of the problem. Another detour is necessary, it gets long.
Linearity (the superposition principle specifically) says that if is a solution to the BVP (not the whole IBVP, only the BVP, Boundary Value Problem, the BCs applied) and so is another , then a linear combination, , is also a solution.
Let's take a step back and suppose that the IC was
This is no longer a realistic flow problem but it contains the first two terms of what is called a Fourier sine expansion, see these examples of Fourier sine expansions. We are going to generalize this below. Let's now use this expression and equate it to the half way solution (BCs applied) with being eliminated as t = 0:
And it still can't match. However, observe that the individual terms in the IC can. We simply set the constants to values making both sides match:
Note the subscripts are used to identify each term: they reflect the integer from the separation constant. Solutions may be obtained for each individual term of the IC, identified with :
Linearity states that the sum of these two solutions is also a solution to the BVP (no need for new constants):
So we added the solutions and got a new solution... what is this good for? Try setting :
Each component solution satisfies the BVP, and the sum of these just happened to satisfy our surrogate IC. The IBVP with IC is now solved. It would work the same way for any linear combination of sine functions whose half frequencies are . "Linear combination" means a sum of terms, each multiplied by a constant. The sum is assumed to converge and be term by term differentiable.
Let's do what we just did in a more generalized fashion. First, we make our IC a linear combination of sines (with eliminated as t = 0), in fact, infinitely many of them. But each successive term has to 'converge', it can't stray wildly all over the place.
Second, find the n and B for each term assuming t = 0 (the IC), then plug them back into each term making no assumptions about t, leaving t as is.
Third, sum up all the terms with their individual n and Bs.
Fourth, plug t = 0 into the sum of terms and recover the IC from the first step.
So we went full circle on this example but found the n and Bs because we were able to equate/satisfy each term with the IC. Now we can solve the problem if the IC is a linear combination of sine functions. But the IC for this problem isn't such a sum, it's just a stupid parabola. Or is it?
Series Construction
In the 19th century, a man named Joseph Fourier took a break from helping Napoleon take over the world to ask an important question while studying this same BVP (concerning heat flow): can a function be expressed as a sum of sinusoids, similar to a taylor series? The short answer is yes, if a few reasonable conditions apply as we have already indicated. The long answer follows, and this section is a longer answer.
A function meeting certain criteria may indeed be expanded into a sum of sines, cosines, or both. In our case, all that is needed to accomplish this expansion is to find the coefficients . A little trick involving an integral makes this possible.
The sine function has a very important property called orthogonality. There are many flavors of this, which will be served in the next chapter. Relevant to this problem is the following:
A quick hint may help. Orthogonality literally means two lines at a right angle to each other. These lines could be vectors, each with its own tuple of coordinates. If those two vectors are at a right angle to each other, multiplying and summing their coordinate tuples always yields zero (in Euclidean space). The method of multiplying and summing is also used to determine whether two functions are orthogonal. Using this definition, our multiplied and integrated functions above are orthogonal most of the time, but not always.
Let's call the IC to generalize it. We equate the IC with its expansion, meaning the linear combination of sines, and then apply some craftiness. And remember that our goal is to reproduce a parabolic function from linearly combined sines:
In the last step, all of the terms in the sum became except for the term where , the only case where we get for the otherwise orthogonal sine functions. This isolates and explicitly defines which is the same as as m = n. The expansion for is then:
Or equivalently:
Many important details have been left out for later in a devoted chapter; one noteworthy detail is that this expansion is only approximating the parabola (very superficially) on the interval , not say from to .
This expansion may finally be combined with the sum of sines solution to the BVP developed previously. Note that the last equation looks very similar to . Following from this:
So the expansion will satisfy the IC given as (surprised?). The full solution for the problem with arbitrary IC is then:
In this problem specifically, the IC is , so:
Sines and cosines appear from the integration dependent only on . Since is an integer, these can be made more aesthetic.
Note that for even , . Putting everything together finally completes the solution to the IBVP:
There are many interesting things to observe. To begin with, is not a product of a function of and a function of . Such a solution was assumed in the beginning, proved to be wrong, but eventually happened to yield a solution anyway thanks to linearity and what is called a Fourier sine expansion.
A careful look at the procedure reveals something that may be disturbing: this lengthy solution is strictly valid for the given BCs. Thanks to the definition of , the solution is generic as far as the IC is concerned (the IC doesn't even need to match the BCs), however a slight change in either BC would mandate starting over almost from the beginning.
The parabolic IC, which looks very similar to the sine function used in the previous section, is wholly to blame (or thank once you understand the beauty of a Fourier series!) for the infinite sum. It is interesting to approximate the first several numeric values of the sequence :
Recall that the even terms are all . The first term by far dominates, this makes sense since the first term already looks very, very similar to the parabola. Recall that appears in an exponential, making the higher terms even smaller for time not too close to .
Change of Variables
As with ODEs, a PDE (or more accurately, the IBVP as a whole) may be made more amenable with the help of some kind of modification of variables. So far, we've dealt only with boundary conditions that specify the value of u, which represented fluid velocity, as zero at the boundaries. Though fluid mechanics can get more complicated than that (understatement of the millennium), let's look at heat transfer now for the sake of variety.
As hinted previously, the one dimensional diffusion equation can also describe heat flow in one dimension. Think of how heat could flow in one dimension: one possibility is a rod that's completely laterally insulated, so that the heat will flow only along the rod and not across it (be aware, though, it is possible to consider heat loss/gain along the rod without going two dimensional).
If this rod has finite length, heat could flow in and out of the uninsulated ends. A 1D rod can have at most two ends (it can also have one or zero: the rod could be modeled as "very long"), and the boundary conditions could specify what happens at these ends. For example, the temperature could be specified at a boundary, or maybe the flow of heat, or maybe some combination of the two.
The equation for heat flow is usually given as:
Which is the same as the equation for parallel plate flow, only with ν replaced with α and y replaced with x.
Fixed Temperatures at Boundaries
Let's consider a rod of length 1, with temperatures specified (fixed) at the boundaries. The IBVP is:
φ(x) is the temperature at t = 0. Look at what the BCs say: For all time, the temperature at x = 0 is u0 and at x = 1 is u1. Note that this could be just as well a parallel plate problem: u0 and u1 would represent wall velocities.
The PDE is easily separable, in basically the same way as in previous chapters:
Now, substitute the BCs:
We can't proceed. Among other things, the presence of t in the exponential factor (previously divided out) prevents anything from coming out of this.
This is another example of the fact that the assumption that u(x, t) = X(x)T(t) was wrong. The only thing that prevents us from getting a solution would be the non-zero BCs. This is where changing variables will help: a new variable v(x, t) will be defined in terms of u which will be separable.
Think of how v(x, t) could be defined to make its BCs zero ("homogeneous"). One way would be:
This form is inspired from the appearance of the BCs, and it can be readily seen:
If h(0) = u0 and h(1) = u1, v(x, t) would indeed have zero BCs. Pretty much any choice of h(x) satisfying these conditions would do it, but only one is the best choice. Making the substitution into the PDE:
So now the PDE has been messed up by the new term involving h. This will thwart separation...
...unless that last term happens to be zero. Rather then hoping it's zero, we can demand it (the best choice hinted above), and put the other requirements on h(x) next to that:
Note that the partial derivative became an ordinary derivative since h is a function of x only. The above constitutes a pretty simple boundary value problem, with unique solution:
It's just a straight line. Note that this is what would arise if the steady state (time independent) problem were solved for u(x). In other words, h could've been pulled out of one's ass readily just looking at the physics of the situation.
But anyway. The problem now reduces to finding v(x, t). The IBVP for this would be:
Note that the IC changed under the transformation. The solution to this IBVP was found in a past chapter through separation of variables and superposition to be:
u(x, t) may now be found simply by adding h(x), according to how the variable change was defined:
This solution looks like the sum of a steady state portion (that's h(x)) and a transient portion (that's v(x)):
Visualization of the change of variables.
Time Varying Temperatures at Boundaries
Note that this wouldn't work so nicely with non-constant BCs. For example, if the IBVP were:
Which doesn't really make anything simpler, despite freedom in the choice of IC.
But this isn't completely useless. Note that the PDE for h was chosen to simplify the PDE for v(x, t) (would lead to the terms involving h to cancel out), which may lead to the question: Was this necessary?
The answer is no. If that were the case, the PDE we picked for h would not be satisfied, and that would result in extra terms in the PDE for v(x, t). The no-longer-separable IBVP for v(x, t) could, however, be solved via an eigenfunction expansion, whose full story will be told later. It's worth noting though, that an eigenfunction expansion would require homogenous BCs, so the transformation was necessary.
So this problem has to be put aside without any conclusion for now. I told you that BCs can mess everything up.
Pressure Driven Transient Parallel Plate Flow
Now back to fluid mechanics. Previously, we dealt with flow that was initially moving but slowing down because of resistance and the absence of a driving force. Maybe, it'd be more interesting if we had a fluid initially at rest (ie, zero IC) but set into motion by some constant pressure difference. The IBVP for such a case would be:
This PDE with the pressure term was described previously. That pressure term is what drives the flow; it is assumed constant.
The intent of the change of variables would be to remove the pressure term from the PDE (which prevents separation) while keeping the BCs homogeneous.
One path to take would be to add something to u(x, t), either a function of t or a function of y, so that differentiation would leave behind a constant that could cancel the pressure term out. Adding a function of t would be very unfavorable since it'd result in time dependent BCs, so let's try a function of y:
Substituting this into the PDE:
This procedure will simplify the PDE and preserve the BCs only if the following conditions hold:
The first condition, an ODE, is required to simplify the PDE for v(y, t), it will result in cancellation of the last two terms. The other two conditions are chosen to preserve the homogeneous BCs of the problem (note that if the BCs of u(y, t) weren't homogeneous, the BCs on f(y) would need to be picked to amend that).
The solution to the BVP above is simply:
So f(y) was successfully determined. Note that this function is symmetric about y = 1/2. The IBVP for v(y, t) becomes:
This is the same IBVP we've been beating to death for some time now. The solution for v(y, t) is:
And the solution for u(y, t) follows from how the variable change was defined:
This solution fits what we expect: it starts flat and approaches the parabolic profile quickly. This is the same parabola derived as the steady state flow in the realistic IC chapter; the integral was evaluated for integer n, simplifying it.
A careful look at the solution reveals something interesting: this is just decaying parallel plate flow "in reverse". Instead of the flow starting parabolic and gradually approaching u = 0, it starts with u = 0 and gradually approaches a parabola.
Time Dependent Diffusivity
In this example we'll change time, an independent variable, instead of changing the dependent variable. Consider the following IBVP:
Note that this is a separable; a transformation isn't really necessary, however it'll be easier since we can reuse past solutions if it can be transformed into something familiar.
Let's not get involved with the physics of this and just call it a diffusion problem. It could be diffusion of momentum (as in fluid mechanics), diffusion of heat (heat transfer), diffusion of a chemical (chemistry), or simply a mathematician's toy. In other words, a confession: it was purposely made up to serve as an example.
The (time dependent) factor in front of the second derivative is called the diffusivity. Previously, it was a constant α (called "thermal diffusivity") or constant ν ("kinematic viscosity"). Now, it decays with time.
To simplify the PDE via a transformation, we look for ways in which the factor could cancel out. One way would be to define a new time variable, call it τ and leave it's relation to t arbitrary. The chain rule yields:
Substituting this into the PDE:
Note now that the variable t will completely disappear (divide out in this case) from the equation if:
C is completely arbitrary. However, the best choice of C is the one that makes τ = 0 when t = 0, since this wouldn't change the IC which is defined at t = 0; so, take C = 0. Note that the BCs wouldn't change either way, unless they were time dependent, in which they would change no matter what C is chosen. The IBVP is turned into:
Digging up the solution and restoring the original variable:
Note that, unlike any of the previous examples, the physics of the problem (if there were any) couldn't have helped us. It's also worth mentioning that the solution doesn't limit to u = 0 for long time.
Concluding Remarks
Changing variables works a little differently for PDEs in the sense that you have a lot of freedom thanks to partial differentiation. In this chapter, we picked what seemed to be a good general form for the transformation (inspired by whatever prevented easy solution), wrote down a bunch of requirements, and defined the transformation to uniquely satisfy the requirements. Doing the same for ODEs can often degrade to a monkey with typewriter situation.
Many simple little changes go without saying. For example, we've so far worked with rods of length "1" or plates separated by a distance of "1". What if the rod was 5 m long? Then space would have to be nondimensionalized using the following transformation:
Simple nondimensionalization is, well, simple; however for PDEs with more terms it can lead to scale analysis which can lead to perturbation theory which will all have to be explained in a later chapter.
It's worth noting that the physics of the IBVP very often suggest what kind of transformation needs to be done. Even some nonlinear problems can be solved this way.
This topic isn't nearly over, changes of variables will be dealt with again in future chapters.
The Laplacian and Laplace's Equation
By now, you've most likely grown sick of the one dimensional transient diffusion PDE we've been playing with:
Make no mistake: we're not nearly done with this stupid thing; but for the sake of variety let's introduce a fresh new equation and, even though it's not strictly a separation of variables concept, a really cool quantity called the Laplacian. You'll like this chapter; it has many pretty pictures in it.
Graph of .
The Laplacian
The Laplacian is a linear operator in Euclidean n-space. There are other spaces with properties different from Euclidean space. Note also that operator here has a very specific meaning. As a function is sort of an operator on real numbers, our operator is an operator on functions, not on the real numbers. See here for a longer explanation.
We'll start with the 3D Cartesian "version". Let . The Laplacian of the function is defined and notated as:
So the operator is taking the sum of the nonmixed second derivatives of with respect to the Cartesian space variables . The "del squared" notation is preferred since the capital delta can be confused with increments and differences, and is too long and doesn't involve pretty math symbols. The Laplacian is also known as the Laplace operator or Laplace's operator, not to be confused with the Laplace transform. Also, note that if we had only taken the first partial derivatives of the function , and put them into a vector, that would have been the gradient of the function . The Laplacian takes the second unmixed derivatives and adds them up.
In one dimension, recall that the second derivative measures concavity. Suppose ; if is positive, is concave up, and if is negative, is concave down, see the graph below with the straight up or down arrows at various points of the curve. The Laplacian may be thought of as a generalization of the concavity concept to multivariate functions.
This idea is demonstrated at the right, in one dimension: . To the left of , the Laplacian (simply the second derivative here) is negative, and the graph is concave down. At , the curve inflects and the Laplacian is . To the right of , the Laplacian is positive and the graph is concave up.
Concavity may or may not do it for you. Thankfully, there's another very important view of the Laplacian, with deep implications for any equation it shows itself in: the Laplacian compares the value of at some point in space to the average of the values of in the neighborhood of the same point. The three cases are:
If is greater at some point than the average of its neighbors, .
If is at some point equal to the average of its neighbors, .
If is smaller at some point than the average of its neighbors, .
So the laplacian may be thought of as, at some point :
The neighborhood of .
The neighborhood of some point is defined as the open set that lies within some Euclidean distance δ (delta) from the point. Referring to the picture at right (a 3D example), the neighborhood of the point is the shaded region which satisfies:
With this mentality, let's examine the behavior of this very important PDE. On the left is the time derivative and on the right is the Laplacian. This equation is saying that:
The rate of change of at some point is proportional to the difference between the average value of around that point and the value of at that point.
For example, if there's at some position a "hot spot" where is on average greater then its neighbors, the Laplacian will be negative and thus the time derivative will be negative, this will cause to decrease at that position, "cooling" it down. This is illustrated below. The arrows reflect upon the magnitude of the Laplacian and, by grace of the time derivative, the direction the curve will move.
Visualization of transient diffusion.
It's worth noting that in 3D, this equation fully describes the flow of heat in a homogeneous solid that's not generating it's own heat (like too much electricity through a narrow wire would).
Laplace's Equation
Laplace's equation describes a steady state condition, and this is what it looks like:
Solutions of this equation are called harmonic functions. Some things to note:
Time is absent. This equation describes a steady state condition.
The absence of time implies the absence of an IC, so we'll be dealing with BVPs rather then IBVPs.
In one dimension, this is the ODE of a straight line passing through the boundaries at their specified values.
All functions that satisfy this equation in some domain are analytic (informally, an analytic function is equal to its Taylor expansion) in that domain.
Laplace's equation is separable in the Cartesian (and almost any other) coordinate system. So, we shouldn't have too much problem solving it if the BCs involved aren't too convoluted.
Laplace's Equation on a Square: Cartesian Coordinates
Steady state conditions on a square.
Imagine a 1 x 1 square plate that's insulated top and bottom and has constant temperatures applied at its uninsulated edges, visualized to the right. Heat is flowing in and out of this thing steadily through the edges only, and since it's "thin" and "insulated", the temperature may be given as . This is the first time we venture into two spatial coordinates, note the absence of time.
Let's make up a BVP, referring to the picture:
So we have one nonhomogeneous BC. Assume that :
As with before, calling the separation constant in favor of just (or something) happens to make the problem easier to solve. Note that the negative sign was kept for the equation: again, these choices happen to make things simpler. Solving each equation and combining them back into :
At edge D:
Note that the constants can be merged, but we won't do it so that a point can be made in a moment. At edge A:
Taking as would satisfy this particular BC, however this would yield a plane solution of , which can't satisfy the temperature at edge C. This is why the constants weren't merged a few steps ago, to make it obvious that may not be . So, we instead take to satisfy the above, and then combine the three constants into one, call it :
Now look at edge B:
It should go without saying by now that can't be zero, since this would yield which couldn't satisfy the nonzero BC. Instead, we can take :
As of now, this solution will satisfy 3 of the 4 BCs. All that is left is edge C, the nonhomogeneous BC.
Neither nor can be contorted to fit this BC.
Since Laplace's equation is linear, a linear combination of solutions to the PDE is also a solution to the PDE. Another thing to note: since the BCs (so far) are homogeneous, we can add the solutions without worrying about nonzero boundaries adding up.
Though as shown above will not solve this problem, we can try summing (based on ) solutions to form a linear combination which might solve the BVP as a whole:
It looks like it needs Fourier series methodology. Finding via orthogonality should solve this problem:
25 term partial sum of the series solution.
was changed to in the last step. Also, for integer , . Note that a Fourier sine expansion has been done. The solution to the BVP can finally be assembled:
That solves it!
It's finally time to mention that the BCs are discontinuous at the points and . As a result, the series should converge slowly at those points. This is clear from the plot at right: it's a 25 term partial sum (note that half of the terms are ), and it looks perfect except at , especially near the discontinuities at and .
Laplace's Equation on a Circle: Polar Coordinates
Now, we'll specify the value of on a circular boundary. A circle can be represented in Cartesian coordinates without too much trouble; however, it would result in nonlinear BCs which would render the approach useless. Instead, polar coordinates should be used, since in such a system the equation of a circle is very simple. In order for this to be realized, a polar representation of the Laplacian is necessary. Without going in to the details just yet, the Laplacian is given in (2D) polar coordinates:
This result may be derived using differentials and the chain rule; it's not difficult but it's a little long. In these coordinates Laplace's equation reads:
Note that in going from Cartesian to polar coordinates, a price was paid: though still linear, Laplace's equation now has variable coefficients. This implies that after separation at least one of the ODEs will have variable coefficients as well.
Let's make up the following BVP, letting :
This could represent a physical problem analogous to the previous one: replace the square plate with a disc. Note the apparent absence of sufficient BC to obtain a unique solution. The funny looking statement that u is bounded inside the domain of interest turns out to be the key to getting a unique solution, and it often shows itself in polar coordinates. It "makes up" for the "lack" of BCs. To separate, we as usual incorrectly assume that :
Once again, the way the negative sign and the separation constant are arranged makes the solution easier later on. These decisions are made mostly by trial and error.
The equation is probably one you've never seen before, it's a special case of the Euler differential equation (not to be confused with the Euler-Lagrange differential equation). There are a couple of ways to solve it, the most general method would be to change the variables so that an equation with constant coefficients is obtained. An easier way would be to note the pattern in the order of the coefficients and the order of the derivatives, and from there guess a power solution. Either way, the general solution to this simple case of Euler's ODE is given as:
This is a very good example problem since it goes to show that PDE problems very often turn into obscure ODE problems; we got lucky this time since the solution for was rather simple though its ODE looked pretty bad at first sight. The solution to the equation is:
Combining:
Now, this is where the English sentence condition stating that u must be bounded in the domain of interest may be invoked. As , the term involving is unbounded. The only way to fix this is to take . Note that if this problem were solved between two concentric circles, this term would be nonzero and very important. With that term gone, constants can be merged:
Only one condition remains: on , yet there are 3 constants. Let's say for now that:
Then, it's a simple matter of equating coefficients to obtain:
Now, let's make the frequencies differ:
Equating coefficients won't work. However, if the IC were broken up into individual terms, the sum of the solution to the terms just happens to solve the BVP as a whole:
Verify that the solution above is really equal to the BC at :
And, since Laplace's equation is linear, this must solve the PDE as well. What all of this implies is that, if some generic function may be expressed as a sum of sinusoids with angular frequencies given by , all that is needed is a linear combination of the appropriate sum. Notated:
To identify the coefficients, substitute the BC:
The coefficients and may be determined by a (full) Fourier expansion on . Note that it's implied that must have period since we are solving this in a domain (a circle specifically) where .
You probably don't like infinite series solutions. Well, it happens that through a variety of manipulations it's possible to express the full solution of this particular problem as:
This is called Poisson's integral formula.
Derivation of the Laplacian in Polar Coordinates
Though not necessarily a PDEs concept, it is very important for anyone studying this kind of math to be comfortable with going from one coordinate system to the next. What follows is a long derivation of the Laplacian in 2D polar coordinates using the multivariable chain rule and the concept of differentials. Know, however, that there are really many ways to do this.
Three definitions are all we need to begin:
If it's known that , then the chain rule may be used to express derivatives in terms of and alone. Two applications will be necessary to obtain the second derivatives. Manipulating operators as if they meant something on their own:
Applying this to itself, treating the underlined bit as a unit dependent on and :
The above mess may be quickly simplified a little by manipulating the funny looking derivatives:
This may be made slightly easier to work with if a few changes are made to the way some of the derivatives are written. Also, the variable follows analogously:
Now we need to obtain expressions for some of the derivatives appearing above. The most direct path would use the concept of differentials. If:
Then:
Solving by substitution for and gives:
If , then the total differential is given as:
Note that the two previous equations are of this form (recall that and , just like above), which means that:
Equating coefficients quickly yields a bunch of derivatives:
There's an easier but more abstract way to obtain the derivatives above that may be overkill but is worth mentioning anyway. The Jacobian of the functions and is:
Note that the Jacobian is a compact representation of the coefficients of the total derivative; using as an example (bold indicating vectors):
So, it follows then that the derivatives that we're interested in may be obtained by inverting the Jacobian matrix:
Though somewhat obscure, this is very convenient and it's just one of the many utilities of the Jacobian matrix. An interesting bit of insight is gained: coordinate changes are senseless unless the Jacobian is invertible everywhere except at isolated points, stated another way the determinant of the Jacobian matrix must be nonzero, otherwise the coordinate change is not one-to-one (note that the determinant will be zero at in this example. An isolated point such as this is not problematic.).
Either path you take, there should now be enough information to evaluate the Cartesian second derivatives. Working on :
Proceeding similarly for :
Now, add these tirelessly hand crafted differential operators and watch the result collapse into just 3 nontrigonometric terms:
That was a lot of work. To save trouble, here is the Laplacian in other two other popular coordinate systems:
Fundamentals
Introduction and Classifications
The intent of the prior chapters was to provide a shallow introduction to PDEs and their solution without scaring anyone away. A lot of fundamentals and very important details were left out. After this point, we are going to proceed with a little more rigor; however, knowledge past one undergraduate ODE class alongside some set theory and countless hours on Wikipedia should be enough.
Some Definitions and Results
An equation of the form
is called a partial differential equation if is unknown and the function involves partial differentiation. More concisely, is an operator or a map which results in (among other things) the partial differentiation of . is called the dependent variable, the choice of this letter is common in this context. Examples of partial differential equations (referring to the definition above):
Note that what exactly is made of is unspecified, it could be a function, several functions bundled into a vector, or something else; but if satisfies the partial differential equation, it is called a solution. If it doesn't, everyone will laugh at you.
Another thing to observe is seeming redundancy of , its utility draws from the study of linear equations. If , the equation is called homogeneous, otherwise it's nonhomogeneous or inhomogeneous.
It's worth mentioning now that the terms "function", "operator", and "map" are loosely interchangeable, and that functions can involve differentiation, or any operation. This text will favor, not exclusively, the term function.
The order of a PDE is the order of the highest derivative appearing, but often distinction is made between variables. For example the equation
is second order in and fourth order in (fourth derivatives will result regardless of the form of ).
Linear Partial Differential Equations
Suppose that , and that satisfies the following properties:
for any scalar . The first property is called additivity, and the second one is called homogeneity. If is additive and homogeneous, it is called a linear function, additionally if it involves partial differentiation and
then the equation above is a linear partial differential equation. This is where the importance of shows up. Consider the equation
where is not a function of . Now, if we represent the equation through
then fails both additivity and homogeneity and so is nonlinear (Note: the equation defining the condition is 'homogeneous', but in a distinct usage of the term). If instead
then is now linear. Note then that the choice of and is generally not unique, but if an equation could be written in a linear form it is called a linear equation.
Linear equations are very popular. One of the reasons for this popularity is a little piece of magic called the superposition principle. Suppose that both and are solutions of a linear, homogeneous equation (here onwards, will denote a linear function), ie
for the same . We can feed a combination of and into the PDE and, recalling the definition of a linear function, see that
for some constants and . As stated previously, both and are solutions, which means that
What all this means is that if both and solve the linear and homogeneous equation , then the quantity is also a solution of the partial differential equation. The quantity is called a linear combination of and . The result would hold for more combinations, and generally,
The Superposition Principle
Suppose that in the equation
the function is linear. If some sequence satisfies the equation, that is if
then any linear combination of the sequence also satisfies the equation:
where is a sequence of constants and the sum is arbitrary.
Note that there is no mention of partial differentiation. Indeed, it's true for any linear equation, algebraic or integro-partial differential-whatever. Concerning nonhomogeneous equations, the rule can be extended easily. Consider the nonhomogeneous equation
Let's say that this equation is solved by and that a sequence solves the "associated homogeneous problem",
where is the same between the two. An extension of superposition is observed by, say, the specific combination :
More generally,
The Extended Superposition Principle
Suppose that in the nonhomogeneous equation
the function is linear. Suppose that this equation is solved by some , and that the associated homogeneous problem
is solved by a sequence . That is,
Then plus any linear combination of the sequence satisfies the original (nonhomogeneous) equation:
where is a sequence of constants and the sum is arbitrary.
The possibility of combining solutions in an arbitrary linear combination is precious, as it allows the solutions of complicated problems be expressed in terms of solutions of much simpler problems.
This part of is why even modestly nonlinear equations pose such difficulties: in almost no case is there anything like a superposition principle.
Classification of Linear Equations
A linear second order PDE in two variables has the general form
If the capital letter coefficients are constants, the equation is called linear with constant coefficients, otherwise linear with variable coefficients, and again, if = 0 the equation is homogeneous. The letters and are used as generic independent variables, they need not represent space. Equations are further classified by their coefficients; the quantity
is called the discriminant. Equations are classified as follows:
Note that if coefficients vary, an equation can belong to one classification in one domain and another classification in another domain. Note also that all first order equations are parabolic.
Smoothness of solutions is interestingly affected by equation type: elliptic equations produce solutions that are smooth (up to the smoothness of coefficients) even if boundary values aren't, parabolic equations will cause the smoothness of solutions to increase along the low order variable, and hyperbolic equations preserve lack of smoothness.
Generalizing classifications to more variables, especially when one is always treated temporally (ie associated with ICs, but we haven't discussed such conditions yet), is not too obvious and the definitions can vary from context to context and source to source. A common way to classify is with what's called an elliptic operator.
Definition: Elliptic Operator
A second order operator of the form
is called elliptic if , an array of coefficients for the highest order derivatives, is a positive definite symmetric matrix. is the imaginary unit. More generally, an order elliptic operator is
if the dimensional array of coefficients of the highest () derivatives is analogous to a positive definite symmetric matrix.
Not commonly, the definition is extended to include negative definite matrices.
The negative of the Laplacian, , is elliptic with . The definition for the second order case is separately provided because second order operators are by a large margin the most common.
Classifications for the equations are then given as
for some constant k. The most classic examples of these equations are obtained when the elliptic operator is the Laplacian: Laplace's equation, linear diffusion, and the wave equation are respectively elliptic, parabolic, and hyperbolic and are all defined in an arbitrary number of spatial dimensions.
Other classifications
Quasilinear
The linear form
was considered previously with the possibility of the capital letter coefficients being functions of the independent variables. If these coefficients are additionally functions of which do not produce or otherwise involve derivatives, the equation is called quasilinear. It must be emphasized that quasilinear equations are not linear, no superposition or other such blessing; however these equations receive special attention. They are better understood and are easier to examine analytically, qualitatively, and numerically than general nonlinear equations.
A common quasilinear equation that'll probably be studied for eternity is the advection equation
which describes the conservative transport (advection) of the quantity in a velocity field . The equation is quasilinear when the velocity field depends on , as it usually does. A specific example would be a traffic flow formulation which would result in
Despite resemblance, this equation is not parabolic since it is not linear. Unlike its parabolic counterparts, this equation can produce discontinuities even with continuous initial conditions.
General Nonlinear
Some equations defy classification because they're too abnormal. A good example of an equation is the one that defines a minimal surface expressible as :
Vector Spaces: Mathematic Playgrounds
The study of partial differential equations requires a clear definition of what kind of numbers are being dealt with and in what way. PDEs are normally studied in certain kinds of vector spaces, which have a number of properties and rules associated with them which make possible the analysis and unifies many notions.
The Real Field
A field is a set that is bundled with two operations on the set called addition and multiplication which obey certain rules, called axioms. The letter will be used to represent the field, and from definition a field requires the following ( and are in ):
Closure under addition and multiplication: the addition and multiplication of field members produces members of the same field.
Addition and multiplication are associative: and .
Addition and multiplication are commutative: and .
Addition and multiplication are distributive: and .
Existence of additive identity: there is an element in notated 0, sometimes called the sum of no numbers, such that .
Existence of multiplicative identity: there is an element in notated 1 different from 0, sometimes called the product of no numbers, such that .
Existence of additive inverse: there is an element in associated with notated such that .
Existence of multiplicative inverse: there is an element in associated with (if is nonzero), notated such that .
These are called the field axioms. The field that we deal with, by far the most common one, is the real field. The set associated with the real field is the set of real numbers, and addition and multiplication are the familiar operations that everyone knows about.
Another example of a set that can form a field is the set of rational numbers, numbers which are expressible as the ratio of two integers. An example of a common set that doesn't form a field is the set of integers: there generally is no multiplicative inverse since the reciprocal of an integer generally is not an integer.
Note that when we say that an object is in, what is meant is that the object is a member of the set associated in the field and that it complies with the field axioms.
The Vector
Most non-mathematics students are taught that vectors are ordered groups ("tuples") of quantities. This is not complete, vectors are a lot more general than that. Informally, a vector is defined as an object that can be scaled and added with other vectors. This will be made more specific soon.
The integers, at least when scaled by real numbers (since the result will not necessarily be an integer).
An interesting (read: confusing) fact to note is that, by the definition above, matrices and even tensors qualify as vectors since they can be scaled or added, even though these objects are considered generalizations of more "conventional" vectors, and calling a tensor a vector will lead to confusion.
The Vector Space
A vector space can be thought of as a generalization of a field.
Letting represent some field, a vector space over is a set of vectors bundled with two operations called vector addition and scalar multiplication, notated:
Vector addition: , where .
Scalar multiplication: , where and .
The members of are called vectors, and the members of the field associated with are called scalars. Note that these operations imply closure (see the first field axiom), so that it does not have to be explicitly stated. Note also that this is essentially where a vector is defined: objects that can be added and scaled. The vector space must comply with the following axioms ( and are in ; and are in ):
Addition is associative: .
Addition is commutative: .
Scalar multiplication is distributive over vector addition: .
Scalar multiplication is distributive over field addition: .
Scalar and field multiplication are compatible: .
Existence of additive identity: there is an element in notated 0 such that .
Existence of additive inverse: there is an element in associated with notated such that .
Existence of multiplicative identity: there is an element in notated 1 different from 0 such that .
An example of a vector space is one where polynomials are vectors over the real field. An example of a space that is not a vector space is one where vectors are rational numbers over the real field, since scalar multiplication can lead to vectors that are not rational (implied closure under scalar multiplication is violated).
By analogy with linear functions, vectors are linear by nature, hence a vector space is also called a linear space. The name "linear vector space" is also used, but this is somewhat redundant since there is no such thing as a nonlinear vector space. It's now worth mentioning an important quantity called a linear combination (not part of the definition of a vector space, but important):
where is a sequence of field members and is a sequence of vectors. The fact that a vector can be formed by a linear combination of other vectors is much of the essence of the vector field.
Note that a field over itself qualifies as a vector space. Fields of real numbers and other familiar objects are sometimes called spaces, since distance and other useful concepts apply.
The definition of a vector space is quite general. Note that, for example, there is no mention of any kind of product between vectors, nor is there a notion of the "length" of a vector. The vector space as defined above is very primitive, but it's a starting point: through various extensions, specific vector spaces can have a lot of nice properties and other features that make our playgrounds fun and comfortable. We'll discuss bases (plural of basis) and then take on some specific vector spaces.
The Basis
A nonempty subset of is called a linear subspace of if is itself a vector space. The requirement that be a vector space can be safely made specific by saying that is closed under vector addition and scalar multiplication, since the rest of the vector space properties are inherited.
The linear span of a set of vectors in may then be defined as:
Where . The span is the intersection over all choices of . This concept may be extended so that is not necessarily finite. The span of is the intersection of all of the linear subspaces of .
Now, think of what happens if a vector is removed from the set . Does the span change? Not necessarily, it may be possible that the remaining vectors in the span are sufficient to "fill in" for the missing vector through linear combination of the remaining vectors.
Let be a subset of . If the span of is the same as the span of , and if removing a vector from necessarily changes its span, then the set of vectors is called a basis of , and the vectors of are called linearly independent. It can be proven that a basis can be constructed for every vector space.
Note that the basis is not unique. This obscure definition of a basis is convenient because it is very broad, it is worth understanding fully. An important property of a vector space is that it necessarily has a basis, and that any vector in the space may be written in terms of a linear combination of the members of the basis.
A more understandable (though less elemental) explanation is provided: for a vector space over the field , the vectors form a basis of (where are in and the following are in ), satisfying the following properties:
The basis vectors are linearly independent: if
then
without exception.
The basis vectors span : for some given in , it is possible to choose so that
The basis vectors of a vector space are usually notated with as .
Euclidean n-Space
As most students are familiar with Euclidean n-space, this section serves more of an example than anything else.
Let be the field of real numbers, then the vector space over is defined to be the space of n-tuples of members of . In other words, more clearly:
If , then , where are the vectors of the vector space .
These vectors are called n-dimensional coordinates, and the vector space is called the real coordinate space; note that coordinates (unlike more general vectors) are often notated in boldface, or else with an arrow over the letter. They are also called spatial vectors, geometric vectors, just "vectors" if the context allows, and sometimes "points" as well, though some authors refuse to consider points as vectors, attributing a "fixed" sense to points so that points can't be added, scaled, or otherwise messed with. Part of the reason for this is that it allows one to say that some vector space is bound to a point, the point being called the origin.
The Euclidean n-space is the special real coordinate n-space with some additional structure defined which (finally) gives rise to the geometric notion of (specifically these) vectors.
To begin with, an inner product is first defined, notated with either angle braces or a dot:
This quantity, which turns two vectors into a scalar (a member of ) doesn't have a great deal of geometric meaning until some more structure is defined. In a coordinate space, the dot notation is favored, and this product is often called the "dot product", especially when or . The definition of this inner product qualifies as an inner product space.
Next comes the norm, in terms of the inner product:
The notation involving single pipes around the letter x is common, again, when or , due to analogy with absolute value, for real and especially complex numbers. For a coordinate space, the norm is often called the length of . This quickly leads to the notion of the distance between two vectors:
Which is simply the length of the vector "from" to .
Finally, the angle between and is defined through, for ,
The motivation for this definition of angle, valid for any , comes from the fact that one can prove that the literal measurable angle between two vectors in satisfies the above (the norm is motivated similarly). Discussing these 2D angles and distances of course mandates making precise the notion of a vector as an "arrow" (ie, correlating vectors to things you can draw on a sheet of paper), but that would get involved and most are already subconsciously familiar with this and it's not the point of this introduction.
This completes the definition of . A thorough introduction to Euclidean space isn't very fitting in a text on Partial Differential equations, it is included so that one can see how a familiar vector space can be constructed ground-up through extensions called "structure".
Banach Spaces
Banach spaces are more general than Euclidean space, and they begin our departure from vectors as geometric objects into vectors as toys in the crazy world of functional analysis.
To be terse, a Banach space is defined as any complete normed vector space. The details follow.
The Inner Product
The inner product is a vector operation which results in a scalar. The vectors are members of a vector space , and the scalar is a member of the field associated with . A vector space on which an inner product is defined is said to be "equipped" with an inner product, and the space is an inner product space. The inner product of and is usually notated .
A truly general definition of the inner product would be long. Normally, if the vectors are real or complex in nature (eg, complex coordinates or real valued functions), the inner product must satisfy the following axioms:
Distributive in the first variable: .
Associative in the first variable: .
Nondegeneracy and nonnegativity: , equality will hold only when .
Conjugate symmetry: .
Note that if the space is real, the last requirement (the overbar indicates complex conjugation) simplifies to , and then the first two axioms extend to the second variable.
A desirable property of an inner product is some kind of orthogonality. Two nonzero vectors are said to be orthogonal only if their inner product is zero. Remember that we're talking about vectors in general, not specifically Euclidean.
Inner products are by no means unique, good definitions are what add quality to specific spaces. The Euclidean inner product, for example, defines the Euclidean distance and angle, quantities which form the foundation of Euclidean geometry.
The Norm
The norm is usually, though not universally, defined in terms of the inner product, which is why the inner product was discussed first (to be technically correct, a Banach space doesn't necessarily need to have an inner product). The norm is an operation (notated with double pipes) which takes one vector and generates one scalar, necessarily satisfying the following axioms:
Scalability: .
The triangle inequality: .
Nonnegativity: , equality only when .
The fact that can be proven from the first two statements above.
Definition requires that only when (compare this to the inner product, which can be zero even if fed nonzero vectors); if this condition is relaxed so that is possible for nonzero vectors, the resulting operation is called a seminorm.
The distance between two vectors and is a useful quantity which is defined in terms of the norm:
The distance is often called the metric, and a vector space equipped with a distance is called a metric space.
Completeness
A Cauchy sequence shown in blue.
A sequence that is not Cauchy. The elements of the sequence fail to get close to each other as the sequence progresses.
As stated before, a Banach space is defined as a complete normed vector space. The norm was described above, so that all that is left to establish the definition of a Banach space is completeness.
Consider a sequence of vectors in a vector space . This sequence of vectors is called a Cauchy sequence if these vectors "tend" toward some "destination" vector, as shown in the pictures at right. Stated precisely, a sequence is a Cauchy sequence if it is always possible to make the distance arbitrarily small by picking larger values of and .
The limit of a Cauchy sequence is:
A vector space is called complete if every Cauchy sequence has a limit that is also in . A Banach space is, finally, a vector space equipped with a norm that is complete. Note that completeness implies existence of distance, which means that every Banach space is a metric space.
An example of a vector space that is complete is Euclidean n-space. An example of a vector space that isn't complete is the space of rational numbers over rational numbers: it is possible to form a sequence of rational numbers which limit to an irrational number.
Hilbert spaces
Note that the inner product was defined above but not subsequently used in the definition of a Banach space. Indeed, a Banach space must have a norm but doesn't necessarily need to have an inner product. However, if the norm in a Banach space is defined through the inner product by
then the resulting special Banach space is called a Hilbert space. Hilbert spaces are important in the study of partial differential equations (some relevance finally!) because many theorems and important results are valid only in Hilbert spaces.
Nondimensionalization
Introduction
You may have noticed something possibly peculiar about all of the problems so far dealt with: "simple" numbers like or keep appearing in BCs and elsewhere. For example, we've so far dealt with BCs such as:
Is this meant to simplify this book because the author is lazy? No. Well, actually, the author is lazy, but it's really the result of what's known as nondimensionalization.
On a basic level, nondimensionalization does two things:
Gets all units out of the problem.
Makes relevant variables range from to or so.
The second point has very serious implications which will have to wait for later. We'll talk about getting units out of the problem for now: important because most natural functions don't have any meaning when fed a unit. For example, is a goofy expression which doesn't mean anything at all (consider its Taylor expansion if you're not convinced).
Do not misunderstand: you can solve any problem you like keeping units in place. That's why angular velocity has units of Hz (), so then has meaning if is in seconds (or can be made to be).
A motivation for nondimensionalization can be seen by noting that ratios of variables to dimensions ("dimension" includes both a size and a unit) have a tendency to show up again and again. Examine what happens if steady state parallel plate flow (an ODE) with the walls separated by , not , is solved:
Let's keep the dimensions of and unspecified for now. Solving this BVP:
Note that we have showing up. Not a coincidence; this implies that the dimensionless problem (or at least halfway dimensionless. We haven't discussed the dimensions of ) could be setup by altering the variable:
is the normalized version of , it ranges from to where varies from to . It is said that is scaled by .
This new variable may be substituted into the problem:
Since the new variable contains no unit, it should be obvious that the rational coefficient must have units of velocity if is to have units of velocity. With this in mind, the coefficient may be divided:
We may define another new variable, a nondimensional velocity:
Substituting this into the equation:
It's finally time to ask an important question: Why?
There are many benefits. The original problem involved 4 parameters: viscosity, density, pressure gradient, and wall separation distance. In this fully nondimensionalized solution, there happens to be none such parameters. The shortened equation above completely describes the behavior of the solution, it contains all of the relevant information.
The solution of one nondimensional problem is far more useful then the solution of a specific dimensional problem. This is especially true if the problem only yields a numeric solution: solving the nondimensional problem greatly reduces the number of charts and graphs that need to be made since you've reduced the number of parameters that could affect the solution.
This leads to another important question, and the culmination of this chapter: here, we first solved a generic dimensional problem and then nondimensionalized it. This luxury wouldn't be available with a more complicated problem. Could it have been nondimensionalized beforehand? Yep. Recalling the BVP:
Note that varies from to in the domain we're interested in. For this reason, it is natural to scale with :
Note that we could have just as well scaled with numbers like or e10.0687 D and still ended up with nondimensionalized and everything mathematically sound. However, alone was the best choice since the resulting variable would vary from to . With this choice of scale, the variable is called normalized in addition to being nondimensional; being normalized is a desirable attribute for mathematic simplicity, accurate numeric evaluation, sense of scale, and other reasons.
What about ? The character of was known, the same can't be said for (why are we solving problems in the first place?). Let's come up with a name for the unknown scale of , say , and normalize using this unknown constant:
Using the chain rule, the new variables may be put into the ODE:
So we have our derivative now. It may be substituted into the ODE:
Remember that was some constant pulled out of thin air. Hence, it can be anything we want it to be. To nondimensionalize the equation and simplify it as much as possible, we may pick:
So that the ODE will become:
The BCs are homogeneous, so they simplify easily. Noting that when :
This may now be quickly solved:
So this isn't quite the same as the nondimensional solution developed from the dimensional solution: there's a factor of on the right side. Consequently, is missing the . It's not a problem, both developments solve the problem and nondimensionalize it. Note that in doing this, we got the following result before even solving the BVP:
This tells much about the size of the velocity.
Before closing this chapter, it's worth mentioning that, generally, if and , where , , , and are all constants,
The pieces that make up f(x), f'(x), and f''(x) are continuous over closed subintervals.
The first requirement is most significant; the last two requirements can, to an extent, be partly eased off in most cases without any trouble. An interesting thing happens at discontinuities. Suppose that f(x) is discontinuous at x = a; the expansion will converge to the following value:
So the expansion converges to the average of the values to the left and the right of the discontinuity. This, and the fact that it converges in the first place, is very convenient. The Fourier series looks unfriendly but it's honestly working for you.
The information needed to express f(x) as a Fourier series are the sequences An and Bn. This is done using orthogonality, which for the sinusoids may be derived easily using a few identities. The following are some useful orthogonality relations, with m and n restricted to integers:
δm,n is called the Kronecker delta, defined by:
The Kronecker delta may be thought of as a discrete version of the Dirac delta "function". Relevant to this topic is its sifting property:
Derivation of the Fourier Series
We're now ready to find An and Bn.
This is supposed to hold for an arbitrary integer m. If m = 0, note that the sum doesn't allow n = 0 and so the sum would be zero since in no case does m = n. This leads to:
This secures A0. Now suppose that m > 0. Since m and n are now in the same domain, the Kronecker delta will do its sifting:
In the second to the last step, sin(mπ) = 0 for integer m. In the last step, m was replaced with n. This defines An for n > 0. For the case n = 0,
Which happens to match the previous development (now you know why it's A0/2 and not just A0). So the sequence An is now completely defined for any value of n of interest:
To get Bn, nearly the same routine is used.
The Fourier series expansion of f(x) is now complete. To have it all in one place:
f(x): a square wave.
It's finally time for an example. Let's derive the Fourier series representation of a square wave, pictured at the right:
In the last bit we used the fact that all of the even terms happened to be absent, and the odd numbers are given by 2n - 1 for integer n. The sum will indeed converge to the square wave, except at the discontinuities where it'll converge to zero (the average of 1 and -1).
Graphs of partial sums are shown at right. Note that this particular expansion doesn't converge too quickly, and that as an approximation of the square wave it's poorest near the discontinuities.
There's another interesting thing to note: all of the cosine terms are absent. It's no coincidence, and this may be a good time to introduce the Fourier sine and cosine expansions for, respectively, odd and even functions.
Periodic Extension and Expansions for Even and Odd functions
Two important expansions may be derived from the Fourier expansion: the Fourier sine series and the Fourier cosine series, the first one was used in the previous section. Before diving in, we must talk about even and odd functions.
Suppose that feven(x) is an even function and fodd(x) is an odd function. That is:
Some interesting identities hold for such functions. Relevant ones include:
This is all very relevant to Fourier series. Suppose that an even function is expanded. Recall that sine is odd and cosine is even. Then:
(whole integrand is even)
(whole integrand is odd)
So the Fourier cosine series (note that all sine terms disappear) is just the Fourier series for an even function, given as:
A Fourier expansion may be similarly built for an odd function:
(whole integrand is odd)
(whole integrand is even)
And the Fourier sine series is:
At this point, the periodic extension may be considered. In the previous chapter, the problem mandated a sine expansion of a parabola. A parabola is by no means a periodic function, and yet a Fourier sine expansion was done on it. What actually happened was that the function was expanded as expected within its domain of interest: the interval 0 ≤ x ≤ 1. Inside this interval, the expansion truly is a parabola. Outside this interval, the expansion is periodic, and as a whole is odd (just like the sine functions it's built on).
The parabola could've been expanded just as well using cosines (resulting in an even expansion) or a full Fourier expansion on, say, -1 ≤ x ≤ 1.
Note that we weren't able to pick which expansion to use, however. While the parabola could be expanded any way we want on any interval we want, only the sine expansion on 0 ≤ x ≤ 1 would solve the problem. The ODE and BCs together picked the expansion and the interval. In fact, before the expansion was even constructed we had:
Which is a Fourier sine series only at t = 0. That the IC was defined at t = 0 allowed the expansion. For t > 0, the solution has nothing in common with a Fourier series.
What's trying to be emphasized is flexibility. Knowledge of Fourier series makes it much easier to solve problems. In the parallel plate problem, knowing what a Fourier sine series is motivates the construction of the sum of un. In the end it's the problem that dictates what needs to be done. For the separable IBVPs, expansions will be a recurring nightmare theme and it is most important to be familiar and comfortable with orthogonality and its application to making sense out of infinite sums. Many functions have orthogonality properties, including Bessel functions, Legendre polynomials, and others.
The keyword is orthogonality. If an orthogonality relation exists for a given situation, then a series solution is easily possible. As an example, the diffusion equation used in the previous chapter can, with sufficiently ugly BCs, require a trigonometric series solution that is not a Fourier series (non-integer, not evenly spaced frequencies of the sinusoids). Sturm-Liouville theory rescues us in such cases, providing the right orthogonality relation.
Numeric Methods
Finite Difference Method
The finite difference method is a basic numeric method which is based on the approximation of a derivative as a difference quotient. We all know that, by definition:
The basic idea is that if is "small", then
Similarly,
It's a step backwards from calculus. Instead of taking the limit and getting the exact rate of change, we approximate the derivative as a difference quotient. Generally, the "difference" showing up in the difference quotient (ie, the quantity in the numeriator) is called a finite difference which is a discrete analog of the derivative and approximates the derivative when divided by .
Replacing all of the derivatives in a differential equation ditches differentiation and results in algebraic equations, which may be coupled depending on how the discretization is applied.
For example, the equation
may be discretized into:
This discretization is nice because the "next" value (temporally) may be expressed in terms of "older" values at different positions.
Method of Lines
Introduction
The method of lines is an interesting numeric method for solving partial differential equations. The idea is to semi-discretize the PDE into a huge system of (continuous and interdependent) ODEs, which may in turn be solved using the methods you know and love, such as forward stepping Runge-Kutta. This is where the name "method of lines" comes from: the solution is composed of a juxtaposition of lines (curves, more properly).
The method of lines is applicable to IBVPs only. The variables (or variable) that have boundary constraints are discretized, and the (one) variable that is associated with the initial value(s) becomes the independent variable for the ODE solution. Put more formally, the method of lines is obtained when all of the variables are discretized except for one of them, resulting in a coupled system of ODEs.
Pure BVPs, such as Laplace's equation on some terrible boundary, can be solved in a manner similar to Jacobi iteration known as the method of false transients.
Example Problem
Consider the following nonlinear IBVP, where :
This intimidating nondimensional system describes the flow of liquid in an interior corner ("groove"), where is the height of the liquid and is the distance along the axis of the corner. The IBVP states that the corner is initially empty, and that at one end () the height is fixed to 1. Naturally, fluid will flow along , increasing until the other boundary (), where is always zero, is reached. It's worth noting that though this IBVP can't be solved analytically, similarity solutions to the nonlinear PDE exist for some other situations, such as fixed height at z = 0 and no constraint to the right of that.
The boundary values are associated with the spatial variable, . Suppose that is discretized in space but not time to points on a uniform grid. Then is approximated as follows:
So is a sequence (or vector) that has an index instead of a continuous dependence on ; note however that the whole sequence still has a continuous dependence on time. is an index that runs from 0 to , note that zero based indexing is being used, so that corresponds to and corresponds to . Looking at the boundaries, we know that:
What about the points of the sequence inside the boundaries? Initially (at ),
Suppose now that the right side of the PDE is discretized however you like, so that:
If, for example, central differences were used, one would eventually arrive at:
To construct the method of lines approximation for this problem, first is replaced with , and then the right side of the equation is replaced with its discretization (the exact equality is a slight abuse of notation):
Since depends continuously on time and nothing else, this differential equation becomes:
Putting everything together in one place, the solution to is the solution to the following IVP:
Solving this problem gives the approximation for . Note that if a second order forward stepping method is used, such as second order Runge Kutta, this solution method will have approximately the same accuracy as the Crank-Nicolson method, but without simultaneous solution of a mess of algebraic equations, which would make a nonlinear problem prohibitively difficult since a tidy matrix solution wouldn't work.
The method of lines is especially popular in electromagnetics, for example, using the Helmholtz equation to simulate the passage of light through a lens for better lens design; the reason for the popularity is that the system of ODEs can (in this case) be solved analytically, so that the accuracy is limited only by the spatial discretization.
A word of caution: an explicit forward stepping method of lines solution bears similarity to a forward stepping finite difference method; so there is no reason to believe that the method of lines doesn't suffer the same stability issues.
A Third Order TVD RK Scheme in C
What follows is an efficient TVD RK scheme implemented in C. Memory is automatically allocated per need (but it will just assume that there is no problem in memory allocation) and freed when n = 0. Note that the solution will not be TVD (total variation diminishing) unless the discretization of is TVD also.
This isn't strictly a method of lines solver, of course; it may find use whenever some large, interdependent vector ODE must be stepped forward.
Other Topics
Scale Analysis
In the chapter on nondimensionalization, variables (both independent and dependent) were nondimensionalized and at the same time scaled so that they ranged from something like to . "Something like to " is the mentality.
Scale analysis is a tool that uses nondimensionalization to:
Understand what's important in an equation and, more importantly, what's not.
Gain insight into the size of unknown variables, such as velocity or temperature, before (even without) actually solving.
Simplify solution process (nondimensional variables ranging for to are very amiable).
Reduce dependence of the solution on physical parameters.
Allow higher quality numeric solution since variables that are of the same range maintain accuracy better on a computer.
Scale analysis is very common sense driven and not too systematic. For this reason, and since it is somewhat unnatural and hard to describe without endless example, it may be difficult to learn.
Before going into the concept, we must discuss orders of magnitude.
Orders of Magnitude and Big O Notation
Suppose that there are two functions and . It is said (and notated) that:
Visualization of the limit superior and limit inferior of f(x) as x increases without bound.
It's worth understanding fully this possibly obscure definition. , short for limit superior, is similar to the "regular" limit, only it is the limit of the upper bound. This concept, alongside limit inferior, is illustrated at right. This intuitive analysis will have to suffice here, as the precise definition and details of these special limits are rather complicated.
As a further example, the limits of the cosine function as increases without bound are:
With this somewhat off topic technicality hopefully understood, the statement that:
Is saying that near , the order (or size, or magnitude) of is bounded by . It's saying that isn't crazily bigger then near , and this is precisely notated by saying that the limit superior is bounded (the "regular" limit wouldn't work since oscillations would ruin everything). The notation involving the big O is rather surprisingly called "big O notation", it's also known as Landau notation.
Take, for example, at different points:
In the first case, the term will easily dominate for large . Even if the coefficient on that term is very near zero, for large enough x that term will dominate. Hence, the function is of order for large .
In the second case, near the first two terms are limiting to zero while the constant term, , isn't changing at all. It is said to be of order , notated as order above. Why O(1) and not O(2)? Both are correct, but O(1) is preferred since it is simpler and more similar to .
This may put forth an interesting question: what would happen if the constant term was dropped? Both of the remaining terms would limit to zero. Since we are looking at x near zero and not at zero,
This is because as approaches zero, the quadratic term gets smaller much faster then the linear term. It would also be correct, though kind of useless, to call the quantity O(1). It would be incorrect to state that the quantity is of order zero since the limit would not exist, not under any circumstance.
As implied above, is by no means a unique function. All of the following statements are true, simply because the limit superior is bounded:
While technically correct, these are very misleading statements. Normally, the simplest, smallest magnitude function g(x) is selected.
Before ending the monotony, it should also be mentioned that it's not necessary for to be smaller then near , only the limit superior must exist. The following two statements are also true:
But again, these are misleading and it's most proper to state that:
A relatively simple concept has been beaten to death, to the point of being confusing. It'll be more clear in context, and it'll be used more in later chapters for different purposes.
Scale Analysis on a Two Term ODE
Previously, the following BVP was considered:
Wipe away any memory of solving this simple problem, the concepts of this chapter do not look at the actual solution. The variables are nondimensionalized by defining new variables:
So that is scaled by , and is scaled by an unknown scale . Now note that, thanks to the scaling:
These are both true near zero. will be O(1) (this is read "of order one") when its scale is properly chosen. Using the chain rule, the ODE was turned into the following:
Now, if both and are of order one, then it is reasonable to assume that, at least at some point in the domain of interest:
This is by no means guaranteed to be true, however it is reasonable.
To identify the velocity scale, we can set the derivative equal to one and solve. There is nothing "illegal" about purposely setting the derivative equal to one since all we need is some equation to specify an unknown constant, . There is much freedom in defining this scale, because what this constant is and how it's found has no effect on the validity of the solution of the BVP (as long as it's not something stupid like ).
Since:
It follows that:
This velocity scale may be thought of as a characteristic velocity. It's a number that shows us what to expect the velocity to be like. The velocity could actually be larger or smaller, but this gives a general idea. Furthermore, this scale tells us how chaging various physical parameters will affect the velocity; there are four of them summarized into one constant.
Compare this result to the coefficient (underlined) on the complete solution, with u dimensional and nondimensional:
They differ by a factor of , but they are of the same order of magnitude. So, indeed, characterizes the velocity.
Words like "reasonable" and "assume" were used a few times, words that would normally lead to the uglier word "approximate". Relax: the BVP itself hasn't been approximated or otherwise violated in any way. We just used scale analysis to pick a velocity scale that:
Turned the ODE into something very easy to look at:
Gained good insight into what kind of velocity the solution will produce without finding the actual solution.
Note that a zero pressure gradient can no longer show itself in the ODE. This is by no means a restriction, since a zero pressure gradient would result in a zero velocity scale which would unconditionally result in zero velocity.
Scale Analysis on a Three Term PDE
The last section was still more of nondimensionalization then it was scale analysis. To just begin getting deeper into the subject, we'll consider the pressure driven transient parallel plate IBVP, identical to the above only with a time component:
See the change of variables chapter to recall the origins of this problem. Scales are defined as follows:
Again, the scale on is picked to make it an order one quantity (based on the BCs), and the scales on and are just letters representing unknown quantities.
The chain rule has been used to define derivatives in terms of the new variables. Instead of taking this path, recall that, given variables and (for the sake of example) and their respective scales and :
So that makes things much easier. Performing the change of variables:
In the previous section, there was one unknown scale and one equation, so the unknown scale could be easily and uniquely isolated. Now, there are two unknown scales but only one equation (no, the BCs/IC will not help). What to do?
The physical meaning of scales may be taken into consideration. Ask: "What should the scales represent?"
There is no unique answer, but good answers for this problem are:
characterizes the steady state velocity.
characterizes the response time: the time to establish steady state.
Once again, these are picked (however, for this problem there really aren't any other choices). In order to determine the scales, the physics of each situation is considered. There may not be unique choices, but there are best choices, and these are the "correct" choices. An understanding of what each term in the PDE represents is vital to identifying these "correct" choices, and this is notated below:
For the velocity scale, a steady state condition is required. In that case, the time derivate (acceleration) must small. We could obtain the characteristic velocity associated with a steady state condition by requiring that the acceleration be something small (read: zero), stating that the second derivative is O(1), and solving:
This is the same as the velocity scale found in the previous section. This is expected since both situations are describing the same steady state condition. The neglect of acceleration equates to what's called a balance between driving force and viscosity since driving force and viscosity are all that remain.
Getting the time scale may be a little more elusive. The time associated with achieving steady state is dictated by the acceleration and the viscosity, so it follows that the time scale may be obtained by considering a balance between acceleration and viscosity. Note that this statement has nothing to do with pressure, so it should apply to a variety of disturbances. To balance the terms, pretend that the derivatives are O(1) quantities and disregard the pressure:
This is a statement that:
The smaller the viscosity, the longer you wait for steady state to be achieved.
The smaller the separation distance, the less you wait for steady state to be achieved.
Hence, the scale describes what will affect the transient time and how. The results may seem counterintuitive, but they are verified by experiment if the pressure is truly a constant capable of combating possibly huge viscosity forces for a high viscosity fluid.
Compare these scales to constants seen in the full, dimensional solution:
The velocity scales match in order of magnitude, nothing new there. But examine the time constant (extracted from the exponential factor) and compare to the time scale:
They are of the same order with respect to the physical parameters, though they'll differ by nearly a factor of 10 when n = 1. This result is more useful then it looks. Note that after determining the velocity scale, all three terms of the equation may have been considered to isolate a time scale. This would've been a poor choice that wouldn't have agreed with the time constant above since it wouldn't be describing the required settling between viscosity and acceleration.
Suppose that, for some problem, a time dependent PDE is too hard to solve, but the steady state version is easier and it is what you're interested in. A natural question would be: "How long do I wait until steady state is achieved?"
The time scale provided by a proper scale analysis will at least give an idea. In this case, assuming that the first term of the sum in the solution is dominant, the time scale will overestimate the response time by nearly a factor of 10, which is priceless information if you're otherwise clueless. This overestimate is actually a good (safe) overestimate, it's always better to wait longer and be certain of the steady state condition. Scales in general have a tendency to overestimate.
Before closing this section, consider the actual nondimensionalization of the PDE. During the scale analysis, the coefficients of the last two terms were equated and later the coefficients of the first two terms were equated. This implies that the nondimensionalized PDE will be:
And this may be verified by substituting the expressions found for the scales into the PDE. This dimensionless PDE, too, turned out to be completely independent of the physical parameters involved, which is very convenient.
Heat Flow Across a Thin Wall
Now, an important utility of scale analysis will be introduced: determining what's important in an equation and, better yet, what's not.
As mentioned in the introduction to the Laplacian, steady state heat flow in a homogeneous solid may be described by, in three dimensions:
Now, suppose we're interested in the heat transfer inside a large, relatively thin wall, with differing temperatures (not necessarily uniform) on different sides of the wall. The word 'thin' is crucial, write it down on your palm right now. You should suspect that if the wall is indeed thin, the analysis could be simplified somehow, and that's what we'll do.
Not caring about what happens at the edges of the wall, a BVP may be written:
is the thickness of the wall (implication: is the coordinate across the wall). Suppose that the wall is a boxy object with dimensions x x . Using the box dimensions as scales,
Only the scale of is unknown. Substituting into the PDE,
Note that the scale on divided out — so a logical choice must be made for it's scale; in this case it'd be an extreme boundary value (ie, the maximum value of ), let's say it's chosen and taken care of. Thanks to this scaling and the rearrangement that followed, we may get a good idea of the magnitude of each term in the equation:
Each derivative is approximately O(1). But what about the squared ratio of dimensions? This is called a dimensionless parameter. Look at your palm now (the one you don't write with), recall the word "thin". "Thin" in this case means exactly the same thing as:
And if the ratio above is much smaller then , then the square of this ratio is even smaller. Our dimensionless parameter is called a small parameter. When a parameter is small, there are many opportunities to simplify analysis; the simplest would be to state that it's too small to matter, so that:
What was just done couldn't have been justified without scaling variables so that their derivatives are (likely) O(1), since you have no idea what order they are otherwise. We know that each derivative is hopefully O(1), but some of these O(1) derivatives carry a very small factor. Only then can terms be righteously dropped. The dimensionless BVP becomes:
Note that it's still a partial differential equation (the and varialbes haven't been made irrelevant – look at the BCs). Also note that scaling on is undone since it cancels out anyway (the scale could've still been picked as, say, a maximum boundary value). This problem may be solved very simply by integrating the PDE twice with respect to , and then considering the BCs:
and are integration "constants". The first BC yields:
And the second:
The solution is:
It's just saying that the temperature varies linearly from one wall face to the other. It's worth noting that in practice, once scaling is complete, the hats on variables are "dropped" for neatness and to prevent carpal tunnel syndrome.
Words of Caution
Failure of one dimensional flow approximation.
"Extreme caution" is more fitting.
In the wall heat transfer problem, we took the partial derivatives in and to be O(1), and this was justified by the scaling: , and are O(1), so the derivatives must be so as well. Right?
Not necessarily. That they're O(1) is a linear approximation, however if the function is significantly nonlinear with respect to a variable of interest, then the derivatives may not be as O(1) as thought. In this problem, one way that this can happen is if the temperature at each wall face (the functions and ) have large and differing Laplacians. This will result in three dimensional heat conduction.
Examine carefully the image at right. Suppose that side length is ten times the wall thickness; and have zero Laplacians everywhere except along circles where temperatures suddenly change. At these locations, the Laplacian can be huge (unbounded if the sudden changes are discontinuities). This will suggest that the derivatives in question are not O(1) but much greater, so that these terms become important even though in this case:
Which is as required by the scale analysis: the wall is clearly thin. But apparently, the small thinness ratio multiplied by the large derivatives leads to significant quantities.
Both the exact solution and the solution to the problem approximated through scaling are shown at the location of a cutting plane. The exact solution shows at least two dimensional heat transfer, while the solution of the simplified solution shows only one dimensional heat transfer and is substantially different.
It's easy to see why the 1D approximation fails even without knowing what a Laplacian is: this is a heat transfer problem involving the diffusion of temperature, and the temperature will clearly need to diffuse along near the sudden changes within the wall (can't say the same about the BCs since they're fixed).
The caption of the figure starts with the word "failure". Is it really a failure? That depends on what you're looking for, it may or may not be. Note that if the wall were even thinner and the sudden jumps not discontinuities, the exact and 1D solutions could again eventually become indistinguishable. | 677.169 | 1 |
Then one day in a store he saw a book titled "Calculus for the Practical Man" by Silvanus P. Thompson. Curious, and thinking himself quite practical, he opened the book and on the second page, all by itself, thus giving it great reverence, was the quote: "What one fool can do, another can. - Ancient Simian Proverb".As is true with any engineering major, I was required to take many math and computer programming classes, so you can be assured that I have extensive knowledge and practical understanding of these disciplines. Having coached my three high school students to As in math, I know I can help you sign... | 677.169 | 1 |
Synopsis
Peterson's Master the SAT: Functions and Intermediate Algebra Review gives you the review and expert tips you need to help improve your score on the these types of questions on the Math part of the SAT. Here you can review functions, integer and rational expressions, solving complex equations, linear and quadratic functions, and more. In addition, the feature "Top 10 Strategies to Raise Your Score" offers expert tips to help you score high on rest of this important test. Master the SAT: Functions and Intermediate Algebra Review is part of Master the SAT 2011, which offers readers 6 full-length practice tests and in-depth review of the Critical Reading; Writing, and Math sections, as well as top test-taking tips to score high on the SAT | 677.169 | 1 |
Mathematics
Pupils who decide to study sixth form mathematics should be prepared for a very exciting, nerve-wracking, fretful, fun-filled, occasionally frustrating, but ultimately rewarding couple of years. At least, that is what we aim for in the King's Mathematics Department.
AS and A2 mathematics and further mathematics are not easy courses, in spite of what some newspapers, and talking heads seem to indicate. Thus, for a student to be successful they will require, most obviously and importantly, an interest and enjoyment in the subject. This, allied with superior algebraic skills, should allow the student to emerge triumphant at the end of their course.
Sixth form mathematics is taught by experienced teachers in a generally relaxed setting, where students will be expected to contribute to their own learning, both in the classroom and via independent study. | 677.169 | 1 |
Resources
Here are some
lecture notes for Introduction to Algebra, a prerequisite for this course. If
you feel unsure about proof by induction, the definition of a group, or
the relation of congruence, you may find some help in these notes.
Here is some advice about study (lectures,
coursework, classes, and exams).
Here are links to some Theorems of the Day on
number theory. You may have seen some of these on the screens in the
Mathematics building. You might also like to read about the Congruent Numbers
problem here. | 677.169 | 1 |
Aims: To consolidate and extend topics met at A-level.
To improve students' fluency and understanding of the basic techniques required for engineering analysis.
Learning Outcomes: After taking this unit the student should be able to:
Handle circular and hyperbolic functions, and sketch curves. Differentiate and integrate elementary functions, products of functions etc.
Use complex numbers.
Employ standard vector and matrix techniques for geometrical purposes. Determine the Fourier series of a periodic function.
Understand power series representations of functions and their convergence properties. | 677.169 | 1 |
I am only a first year graduate student, but I am very interested in mathematics education. My own approach to teaching is very much problem based: give students interesting problems which lead to the development of the concepts you want them to have. Even if they can't come up with all of the needed concepts on their own, if you give it to them after they have wrestled with a problem they will be much more likely to be able to apply the concept in novel situations in the future. Why couldn't this approach be carried through in a math grad situation? Design a sequence of problems, varying in difficulty, which in total cover need material from most of the "first year curriculum".
Before writing this off as a crazy idea, I would like to point out Cornell's vet school. They use exactly the model given above: Every week or two there is a new case. In each of your classes (anatomy, pharmacology, radiology, ...etc) you cover general information which is pertinent to the case of the week, but it is up to you and your team to do research, come up with a diagnosis and a method of treatment. So all of the classes you take are integrated together in the context of solving some real problems. Cornell is turning out some amazing vets. Why couldn't the same model work for mathematics? | 677.169 | 1 |
This book contains a large amount of information not found in standard textbooks. Written for the advanced undergraduate/beginning graduate student, it combines the modern mathematical standards of numerical analysis with an understanding of the needs of the computer scientist working on practical applications. Among its many particular features are: (bullet) fully worked-out examples (bullet) many carefully selected and formulated problems (bullet) fast Fourier transform methods (bullet) a thorough discussion of some important minimization methods (bullet) solution of stiff or implicit ordinary differential equations and of differential algebraic systems (bullet) modern shooting techniques for solving two-point boundary value problems (bullet) basics of multigrid | 677.169 | 1 |
General: We will follow the course outline of Prof. Richard Froese's class in previous term, which you can download from here: lecturenotes. However, the detailed topics may differ as will the homework and grading (we have no text book). Related information for section 201 of this course (by Prof. Joel Friedman) can be found here. Some simple programming is needed for this couse in Matlab or Octave. A page on Matlab/Octave in the math department is available here. For documentation you might try: MATLAB documentation page and GNU Octave page. GNU Octave is free, and is (usually) not too difficult to install on any Linux, MacOS, or Windows system (Linux systems usually come with Octave installed). Richard Froese has created a UBC Wiki page to help you install Octave if you'd like to give this a try; see also the GNU Octave download instructions.
Objective: We learn basic knowledge in linear algebra first (theoretically), together with programming using Matlab/Octave (which is suitable for matrix-matrix computation); during learning, related problem about practical application will be inserted, such as interpolation, finite difference approximation, power method, network connectivity, recursion relations, Fourier transform, etc. | 677.169 | 1 |
Hello there, I'm a student in high school and I'm tormented by my assignments. One of my issues is addressing online textbook mcdougal littell; Will someone on the web help me in understanding what it's all about? I need to complete this immediately! Thanks to allfor assisting.
I think I understand what you are searching) for. Try out Algebra Buster. This is a wonderful tool that helps you get your assignments completed quicker as well as correct. It can help out with courses in online textbook mcdougal littell, inequalities plus more.
Registered: 24.10.2003
From: Where the trout streams flow and the air is nice
Posted: Sunday 21st of Nov 18:29
It is great to understand that you desire to enhance your algebra skills as well as being demonstrating attempts to accomplish that. I reckon you could try Algebra Buster. This is not precisely just some tutoring tool however it provides results to algebra homework questions in a truly | an immensely step-by-step way. The strongest thing regarding this product is that it's extraordinarily easy to learn. There are several demos presented under assorted themes which are especially helpful to learn more about a specific content. Examine it. Wish you good luck with mathematics. | 677.169 | 1 |
MATH 127: Precalculus IISolve problems using the distance formula and the pythagorean theorem.
Compute trigonometric functions of special angles and use them to solve for the unknown part(s) of right triangles.
Use the Laws of Sines and Cosines to solve for the unknown parts of triangles.
Compute vector sums and differences.
Solve trigonometric identities and equations.
Graph trigonometric functions.
Compute the values of inverse trigonometric functions.
Graph equations and functions in polar coordinates.
Use DeMoivre's theorem to find the powers and roots of complex numbers.
III: Course Linkage
Linkage of course to educational program mission and at least one educational program outcome.
General Education Mission: This course addresses the fourth bullet under goal one of the college's mission to, "Provide instruction that contributes to a student's abilities to think critically and solve problems; to reason mathematically and apply computational skills."
Math 127 satisfies the General Education Requirement for any degree or certificate program and
addresses the following learning objectives of the General Education Requirement by ensuring that successful students:
Are able to apply appropriate college-level mathematical skills to real life applications | 677.169 | 1 |
Calculators as an Aid to Problem Solving
A common complaint among teachers is that students have great difficulty with computation and, what is worse, they are very weak in problem solving. Unfortunately this complaint is a self-perpetuating ill. Students who cannot succeed with arithmetic computation are constantly told to drill these skills and are rarely allowed to practice any problem-solving skills. Those who do go on to working on some elementary problems often do not get near an answer because of computational deficiencies. Their only exposure to problem solving is one of frustration, and they rarely realize success because of computational obstacles. Here the calculator can be of significant assistance. Selective use of the calculator to bypass potential computational barriers will allow students to concentrate on problem-solving skills without fear of meeting frustration previously caused by their computational deficiencies. Such activities should be carefully designed and monitored to be effective. After realizing success in problem solving, students should then be intrinsically motivated to conquer their computational deficiencies.
Although continuously nurtured on typical textbook problems, students usually find them boring and unrealistic. Traditionally, textbook authors design the problems in a way that will make the arithmetic computations as simple as possible so as not to detract from the problem. Real-life situations frequently are quite different. The numbers used are generally not simple. With the aid of a calculator, a teacher can provide realistic situations for problem solving and not worry about computational distractions. A uniform-motion problem, for example, can involve fractional quantities and yield an answer that is not an integer and still cause no displeasure for the student who has a calculator available. Furthermore, students using a calculator can be encouraged to create problems based on their own experiences (e.g., calculating their average speed walking to school). New vistas are opened up when a calculator is used to assist in problem solving bypassing arithmetic.
Problems in advanced secondary school mathematics courses can often involve extensive calculations. Not many years ago the slide rule or logarithms were used to solve such problems. Even Napier's rods and the abacus played a role at one point in the history of people's attempts to be free of the burden of onerous manual calculations. The abacus is still used in some parts of the less technologically advanced world. Today, the logical method of computation at this level is the calculator. A scientific calculator (i.e., one that, among other features, includes trigonometric functions) and a graphing calculator are very useful aids to instruction, but by no means replace instruction. | 677.169 | 1 |
This course is a continuation of Math 2030.03 with an emphasis on foundations and the theory of vector spaces and linear transformations. Additional topics include symmetric and orthogonal transformations, bilinear forms, inner product spaces, and various applications in mathematics, physics and computer science. | 677.169 | 1 |
COMPLETE MATH PRODUCTION PACKAGE --- DISCONTINUED. (Verified 02/2009) RETAINED IN DATABASE FOR REFERENCE. --- The Complete Math Production Package is a mathematical Braille translation program designed for use by individuals who are blind or have low vision. This package includes the Duxbury Braille Translator for Windows (see separate entry) and provides the additional capability of handling all types of equations from simple arithmetic to complex science and engineering notation....[More Information]
MATHEMATIX --- DISCONTINUED. (Verified 3/2001) RETAINED IN DATABASE FOR REFERENCE. --- MathematiX is a software program that works with BEX (see entry) to enable printing or verbalizing of Nemeth Code mathematical braille data. If a speech synthesizer is used, literary text is spoken as words and mathematical material is spelled out sign for sign. MathematiX can also be used to prepare regular-print documents that include text and fractions, square roots, chemical symbols, and other technical material. Mat | 677.169 | 1 |
Algebra 1/2 represents a culminatin of prealgebra mathematics, covering all topics normally taught in prealgebra, as well as additional topics from geometry and discrete mathematics. This program is recommended for seventh graders who plan to take first-year algebra in the eighth grade or for eighth-graders who plan to take first-year algebra in the ninth grade. | 677.169 | 1 |
Loci: Resources
Images of F
by Steve Phelps (Madeira High School)
Applet Description
This interactive Geogebra applet allows exploration of a linear transformation in terms of images of a closed figure that happens to be in the shape of the letter F initially. The Geogebra interface allows dragging of points and vectors to make for versatile explorations of basic linear algebra ideas. Suggested activities and exercises using the tool are included on page 2 of this posting and as a separate pdf file for easy printing.
Steve Phelps
Madeira High School
& GeoGebra Institute of OhioClick here or on the screen shot above to open the applet in a separate window.
Investigations
In the Images of F applet on page 1, the columns of the matrix are the elementary vectors e1 and e2. The blue figure is a pre-image initially in the shape of an F. The green figure is the image of the blue F under the transformation given by the matrix.
To answer the questions below, you can drag the tips of the elementary vector to set up the appropriate matrices. You may also need to drag the vertices of the blue F as well.
Warm Up: Set up the following matrices one at a time. Pay particular attention to the lattice points of F and to the lattice points of the image of F.
1.
\left[ \begin{array}{cc} 2 & 3 \\ 0 & 1 \end{array} \right]
2.
\left[ \begin{array}{cc} 1 & 0 \\ 3 & -1 \end{array} \right]
3.
\left[ \begin{array}{cc} 1 & 2 \\ 3 & 1 \end{array} \right]
4.
\left[ \begin{array}{cc} 2 & -1 \\ 2 & 1 \end{array} \right]
5.
\left[ \begin{array}{cc} -2 & 1 \\ 2 & -1 \end{array} \right]
6.
\left[ \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right]
Investigation 1: Drag the tips of the elementary vectors to set up the following matrices. Discuss the transformations and the resulting image of F under these matrix transformations.
transformations with matrices of the form
\left[ \begin{array}{cc} k & 0 \\ 0 & 1 \end{array} \right]
transformations with matrices of the form
\left[ \begin{array}{cc} 1 & 0 \\ 0 & k \end{array} \right]
transformations with matrices of the form
\left[ \begin{array}{cc} k & 0 \\ 0 & k \end{array} \right]
transformations with matrices of the form
\left[ \begin{array}{cc} 0 & k \\ k & 0 \end{array} \right]
transformations with matrices of the form
\left[ \begin{array}{cc} 1 & 0 \\ k & 1 \end{array} \right]
transformations with matrices of the form
\left[ \begin{array}{cc} 1 & k \\ 0 & 1 \end{array} \right]
Investigation 2: Drag the tips of the elementary vectors to set up matrices that will perform the following transformations. Pay attention to the orientation of the vectors.
Reflection over the x – axis
Reflection over the y – axis
90-degree clockwise rotation around the origin
Half-turn around the origin
90-degree counterclockwise rotation around the origin
Reflection over the line y = x
Reflection over the line y = -x
Copyright 2013. All rights reserved. The Mathematical Association of America. | 677.169 | 1 |
fromMath in Society is a free, open textbook. This book is a survey of mathematical topics, most non-algebraic, appropriate for a college-level topics course for liberal arts majors. The text is designed so that most chapters are independent, allowing the instructor to choose a selection of topics to be covered. Emphasis is placed on the applicability of the mathematics. Material for each topic is covered in the main text, with additional depth available through exploration exercises appropriate for in-class, group, or individual investigation. This book is appropriate for Math 107 (Washington State Community Colleges common course number).
The purpose of this course is to expose you to the wider world of mathematical thinking. There are two reasons for this. First, for you to understand the power of quantitative thinking and the power of numbers in solving and dealing with real world scenarios. Secondly, for you to understand that there is more to mathematics then expressions and equations. The core course is a complete, ready to run, fully online course, featuring 9 topics: Problem solving, voting theory, graph theory, growth models, consumer finance, collecting data, describing data, probability, and historical counting. Additional optional topics are provided. The course materials can easily be used with a face-to-face course.
This tool allows the individual or the classroom to explore several representations of fractions. After selecting numerator and denominator, any number from 1 to 100, learners see the fraction itself, a visual model, as well as decimal and percent equivalents. They can choose the model to be a circle, a rectangle, or a set model. | 677.169 | 1 |
Matlab: Linear Algebra
Linear algebra is about the solution of simultaneous linear
equations, linear eigensystems etc., and is based on numerical matrices.
This course starts by covering Matlab's basic matrix facilities, fairly
quickly, for people who are not experts with them. It then covers the
basics of linear algebra using real and complex matrices.
It is intended for people who can use Matlab, but need to know what it
can do with matrices and linear algebra.
Prerequisite
Matrix arithmetic and how matrices are used in linear algebra
(e.g. the solution of linear equations) is no longer taught in the
ordinary mathematics A-level, but only Further Pure mathematics. If you
do not know this, you MUST learn it first. For further
information, see Matrix
Prerequisites. | 677.169 | 1 |
...
Magic Box is a collection of applications. In it you will find some popular jokes and puzzles such as magic square, magic eye, latent image fading dollar and math transformation. Will the program learn the unknown number? Will the program learn divisible by both a and b. For example,...
Standard deviation is a widely used measurement of variability or diversity used in statistics and probability theory. It shows how precise your data is. The standard deviation is the square root of its variance. A low standard deviation...
If you already know, area can be calculated by multiplying length by width. To make things easier, we created this free tool that converts instantly between acres, square feet, square miles and other units. Enter the value, select the unit and...
Lite version converts several units of length. Plus version converts length, weight and capacity measures. By typing a number into box provided will instantly display the results without the user having to search through a confusing menu of...
......GRE Calculator is a simple application that will help you familiarize yourself with calculator which you'll be using on actual GRE revised General Test. Since the most important and crucial aspect on GRE is time management, there are also included...This software offers a solution for users who create graph paper using a printer. There is a full range of options for page size, margins, orientation and position on page. The user can then choose from a wide range of measurement scales...
Mandelx is a very fast fractal generator that uses highly optimized assembly routines for calculations (multithreaded). Mandelx runs best on AMD processors. While the precision is enough MandelX uses the appropriate 32 bit SIMD floating point code...
GeoGebra is a powerful mathematics tool for everybody, who is concerned with geometry. This program allows you to build drawings of almost any complexity. You can add points, lines, vectors, polygons, circles, angles, text and even images from...
Microsoft Math is a set of tools designed to help students to get their math homework done more quickly. It can quickly evaluate solve and graphic equations. Microsoft Match can also be used to evaluate ordinary numeric expressions. The screen is...
SineWaves is used for Mathematics and Physics lessons in Secondary Schools to demonstrate the harmonic make up of audio sounds. Students are able to view and hear constructed formulae. Four waveforms may be displayed at once and each waveform features: - mental arithmetic...
- Innovative design to help you pass APICS exams. You solve the exercise and get immediate feedback. This feature is perfect for instructor classroom demonstration. - Easy for instructors to use in the classroom. Use included datasets, adapt to...
Efofex's FX MathPack contains four of the most powerful and useful mathematical tools available for teachers and students. An FX MathPack subscription lets you use all of the products for one very low yearly rate. The Products FX Draw - The Only...
The Magic Math Wand is a math manipulative that you can use to show any addition, subtraction, multiplication or division problem. You can even do larger Addition - Division problems with The Magic Math Wand. It is a program for kids to practice...
Mouse Math is a verbal mathematics drill program that gradually uncovers whimsical photographs as answers to spoken problems are clicked with a mouse. Setup options allow you to select problem ranges appropriate for young children, older kids and...
Aplusix is an innovative software, developed to help students learn arithmetic and algebra. Aplusix reinforces students' skills, diminishes calculation mistakes, and shows students how to solve exercises. Good arithmetic and algebra skills | 677.169 | 1 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.