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CHAPTER 1 Getting Acquainted
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1.1 1.3 1.5 1.9 Notation and Conventions 1.2 The Kernel and the Front End Mathematica Quirks 1.4 Mathematica Gives Exact Answers Mathematica Basics 1.6 Cells 1.7 Getting Help 1.8 Packages A Preview of What Is to Come
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CHAPTER 2 Basic Concepts
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2.1 Constants 2.2 Built-In Functions 2.3 Basic Arithmetic Operations 2.4 Strings 2.5 Assignment and Replacement 2.6 Logical Relations 2.7 Sums and Products 2.8 Loops 2.9 Introduction to Graphing 2.10 User-Defined Functions 2.11 Operations on Functions
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CHAPTER 3 Lists
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3.1 Introduction 3.2 Generating Lists Theory 3.5 Tables and Matrices 3.3 List Manipulation 3.4 Set
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CHAPTER 4 Two-Dimensional Graphics
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4.1 Plotting Functions of a Single Variable 4.2 Additional Graphics Commands 4.3 Special Two-Dimensional Plots 4.4 Animation
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CHAPTER 5 Three-Dimensional Graphics
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5.1 Plotting Functions of Two Variables 5.2 Other Graphics Commands 5.3 Special Three-Dimensional Plots 5.4 Standard Shapes 3D Graphics Primitives
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CHAPTER 6 Equations
6.1 Solving Algebraic Equations 6.2 Solving Transcendental Equations
CHAPTER 7 Algebra and Trigonometry
7.1 Polynomials 7.2 Rational and Algebraic Functions 7.3 Trigonometric Functions 7.4 The Art of Simplification
Contents
CHAPTER 8 Differential Calculus
8.1 Limits 8.2 Derivatives 8.4 Power Series 8.3 Maximum and Minimum Values
CHAPTER 9 Integral Calculus
9.1 Antiderivatives 9.2 Definite Integrals Integrals 9.4 Riemann Sums 9.3 Functions Defined by
CHAPTER 10 Multivariate Calculus
10.1 Partial Derivatives 10.2 Maximum and Minimum Values Total Differential 10.4 Multiple Integrals 10.3 The
CHAPTER 11 Ordinary Differential Equations
11.1 Analytical Solutions Transforms 11.2 Numerical Solutions 11.3 Laplace
CHAPTER 12 Linear Algebra
12.1 Vectors and Matrices 12.2 Matrix Operations 12.3 Matrix Manipulation 12.4 Linear Systems of Equations 12.5 Orthogonality 12.6 Eigenvalues and Eigenvectors 12.7 Diagonalization and Jordan Canonical Form
Appendix Index
A.1 Pure Functions A.2 Patterns A.5 Commands Used in This Book
A.3 Contexts
A.4 Modules
332 353
Mathematica
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C HA PTE R 1
Getting Acquainted
1.1 Notation and Conventions
Mathematica is a language that is best learned by experimentation. Therefore, the reader is urged to try as many examples and problems as possible and experiment by changing options and parameters. In fact, this chapter may be considered a tutorial for those readers who want to get their hands on Mathematica right away. New commands are introduced with a bullet, and options associated with them are bulleted with a symbol for easy reference. In keeping with Mathematica s conventions, all commands and instructions will be written in Courier bold face type and Mathematica output in Courier light face type. This line is written in Courier bold face type.
This line is written in Courier light face type.
Menu commands in this text are described using double arrows ( ). For example, Format Style Input, written in Arial font, means go to the Format menu, then to the Style submenu, and then click on Input.
Mathematica occasionally uses a special symbol, `, which we call a backquote. Do not confuse this with an apostrophe. Finally, most Mathematica commands use an arrow, , to specify options within the command. You may use > ( followed by > ) as an alternate, if you wish. Mathematica will automatically convert this sequence to . In a similar manner, the sequence != is automatically converted to , <= is replaced by , and >= is changed to .
Getting Acquainted
The examples used in this book were executed using Mathematica versions 6 and 7. You may notice some differences on your computer if you are using earlier versions of the software. Most noticeably, graphics, particularly three-dimensional graphics, have been enhanced in the later version and many computational algorithms have been improved, resulting in greater efficiency and speed.
The Kernel and the Front End
The kernel is the computational engine of Mathematica. You input instructions and the kernel responds with answers in the form of numbers, graphs, matrices, and other appropriate displays. The kernel works silently in the background and, for the most part, is invisible. The interface between the user and the kernel is called the front end and the medium of the front end is the Mathematica notebook. The notebook not only enables you to communicate with the kernel, but is a convenient tool for documenting your work. To execute an instruction, type the instruction and then press [ENTER]. Most PCs have two [ENTER] keys, but only the [ENTER] key to the far right of the keyboard will execute instructions. The other [ENTER] key must be pressed with the [SHIFT] key held down; otherwise you will merely get a new line. This is especially important if you are using a laptop. If you are using a Macintosh computer, do not confuse the [ENTER] key with the [RETURN] key. The picture in Example 1 shows the standard Mathematica display. The symbols on the right-hand side form the Basic Math Input palette and allow access by mouse-click to the most common mathematical symbols. (If you don t see the palette on your screen, click on Palettes BasicMathInput or Palettes Other Basic Math Input and it should appear.) Other palettes such as Basic Math Assistant and Classroom Assistant (version 7 and above) are available for specialized purposes and can be accessed via the Palettes menu. Each symbol is accessed by clicking on the palette. If you use the palette, your notebooks will look like pages from a math textbook. Most examples in this book take full advantage of the Basic Math Input palette. However, each Mathematica symbol has an alternative descriptive format that can be typed manually. For example, can be represented as Pi and 5 can be written Sqrt[5]. These representations are useful for experienced Mathematica users who prefer not to use the mouse. The notebook in Example 1, labeled Untitled 1, is where you input your commands and where Mathematica places the result of its calculations. The picture shows the input and output of Example 1. (The display on a Macintosh computer will look slightly different.)
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