Mathematica® by Example

Preface


1 Getting Started


1.1 Introduction to Mathematica


1.2 Getting Started with Mathematica


1.3 Loading Packages

Two Words of Caution


1.4 Getting Help from Mathematica

Help Commands

Mathematica Help


2 Mathematical Operations on Numbers, Expressions, and Functions


2.1 Numerical Calculations and Built-in Functions

Numerical Calculations

Built-In Constants

Built-In Functions

The Absolute Value, Exponential and Logarithmic Functions

Trigonometric Functions

Inverse Trigonometric Functions

A Word of Caution


2.2 Expressions and Functions

Basic Algebraic Operations on Expressions

Naming and Evaluating Expressions

A Word of Caution

Defining and Evaluating Functions

Additional Ways to Evaluate Functions and Expressions

Composition of Functions

A Word of Caution


2.3 Graphing Functions, Expressions, and Equations

Graphing Functions of a Single Variable

Graphing Several Functions

Piecewise-Defined Functions

Graphs of Parametric Functions in Two Dimensions

Three-Dimensional Graphics

Graphing Level Curves of Functions of Two Variables

Graphing Parametric Curves and Surfaces in Space

A Word of Caution


2.4 Exact and Approximate Solutions of Equations

Exact Solutions of Equations

Numerical Approximation of Solutions of Equations

Application: Intersection Points of Graphs of Functions


3 Calculus


3.1 Computing Limits

Computing Limits

One-Sided Limits

A Word of Caution


3.2 Differential Calculus

Calculating Derivatives of Functions and Expressions

Tangent Lines

Locating Critical Points and Inflection Points

Using Derivatives to Graph Functions

Graphing Functions and Derivatives

Approximations with FindRoot

Application: Rolle's Theorem and The Mean-Value Theorem

Application: Graphing Functions and Tangent Lines

Application: Maxima and Minima


3.3 Implicit Differentiation

Computing Derivatives of Implicit Functions

Other Methods to Compute Derivatives of Implicit Functions

Other Methods to Graph Equations


3.4 Integral Calculus

Estimating Areas

Computing Definite and Indefinite Integrals

Approximating Definite Integrals

Application: Area Between Curves

Application: Arc Length

Application: Volume of Solids of Revolution

Application: The Mean-Value Theorem for Integrals

A Word of Caution


3.5 Series

Introduction to Series

Determining the Interval of Convergence of a Power Series

Computing Power Series

Application: Approximating the Remainder

Application: Series Solutions to Differential Equations

Other Series


3.6 Multivariable Calculus

Limits of Functions of Two Variables

Partial Differentiation

Other Methods of Computing Derivatives

Application: Classifying Critical Points

Application: Tangent Planes

Application: The Method of Lagrange Multipliers

Double Integrals

Application: Volume

Triple Integrals

Higher-Order Integrals


4 Introduction to Lists and Tables


4.1 Defining Lists

A Word of Caution


4.2 Operations on Lists

Extracting Elements of Lists

Graphing Lists of Points and Lists of Functions

Evaluation of Lists by Functions

Evaluation of Parts of Lists by Functions

Other List Operations

Alternative Way to Evaluate Lists by Functions


4.3 Mathematics of Finance

Application: Compound Interest

Application: Future Value

Application: Annuity Due

Application: Present Value

Application: Deferred Annuities

Application: Amortization

Application: Financial Planning


4.4 Other Applications

Application: Secant Lines, Tangent Lines, and Animations

Application: Approximating Lists with Functions

Application: Introduction to Fourier Series

Application: The One-Dimensional Heat Equation


5 Nested Lists: Matrices and Vectors


5.1 Nested Lists: Introduction to Matrices, Vectors, and Matrix Operations

Defining Nested Lists: Matrices and Vectors

Extracting Elements of Matrices

Basic Computations with Matrices and Vectors


5.2 Linear Systems of Equations

Calculating Solutions of Linear Systems of Equations

Gauss-Jordan Elimination


5.3 Selected Topics from Linear Algebra

Fundamental Subspaces Associated with Matrices

The Gram-Schmidt Process

Linear Transformations

Application: Rotations

Eigenvalues and Eigenvectors

Jordan Canonical Form

The QR Method


5.4 Maxima and Minima Using Linear Programming

The Standard Form of a Linear Programming Problem

The Dual Problem

Application: A Transportation Problem


5.5 Vector Calculus

Definitions and Notation

Application: Green's Theorem

Application: The Divergence Theorem

Application: Stoke's Theorem


6 Applications Related to Ordinary and Partial Differential Equations


6.1 First-Order Ordinary Differential Equations

Separable Differential Equations

Homogeneous Differential Equations

Exact Equations

Linear Equations

Numerical Solutions of First-Order Ordinary Differential Equations

Application: Population Growth and the Logistic Equation

Application: Newton's Law of Cooling

Application: Free-Falling Bodies


6.2 Higher-Order Ordinary Differential Equations

The Homogeneous Second-Order Equation with Constant Coefficients

Nonhomogeneous Equations with Constant Coefficients Variation of Parameters

Cauchy-Euler Equations

Application: Harmonic Motion

Numerical Solutions of Higher-Order Ordinary Differential Equations

Application: The Simple Pendulum


6.3 Power Series Solutions of Ordinary Differential Equations

Power Series Solutions about Ordinary Points

Power Series Solutions about Regular Singular Points


6.4 Using the Laplace Transform to Solve Ordinary Differential Equations

Definition of the Laplace Transform

Solving Ordinary Differential Equations with the Laplace Transform

Application: The Convolution Theorem

Application: The Dirac Delta Function


6.5 Systems of Ordinary Differential Equations

Homogeneous Linear Systems with Constant Coefficients

Variation of Parameters

Nonlinear Systems, Linearization, and Classification of Equilibrium Points

Numerical Solutions of Systems of Ordinary Differential Equations

Application: Predator-Prey

Application: The Double Pendulum


6.6 Some Partial Differential Equations

The One-Dimensional Wave Equation

Application: Zeros of the Bessel Functions

Application: The Two-Dimensional Wave Equation


7 Some Graphics Packages


7.1 ComplexMap


7.2 ContourPlot3D


7.3 Graphics

Graphing in Polar Coordinates

Creating Charts


7.4 ImplicitPlot


7.5 MultipleListPlot and Graphics3D


7.6 PlotField and PlotField3D


7.7 Polyhedra and Shapes

Selected References

Index