## Doing Mathematics with Scientific WorkPlace® & Scientific Notebook® Version 5.5

 Darel W. Hardy and Carol L. Walker ©2005 536 pages ISBN: 0-9741652-6-3 Read more about Creating Mathematics with Scientific WorkPlace® & Scientific Notebook® Version 5.5 or about other books from MacKichan Software.

### Table of Contents

#### 1  Basic Techniques for Doing Mathematics  1

Inserting Text and Mathematics  1

Basic Guidelines  1

Displaying Mathematics  4

Centering Plots, Graphics, and Text  4

Basic Guidelines for Computing  5

Evaluating Expressions  5

Interpreting Expressions  8

The Compute Menu and Toolbar    8

Selecting Mathematical Expressions  9

Computing in Place  12

Stopping a Computation  14

Computational Engine  14

Error Handling  15

Frequently Asked Questions  16

#### 2  Numbers, Functions, and Units  19

Integers and Fractions  19

Addition and Subtraction  19

Multiplication and Division  20

Mixed Numbers and Long Division  21

Elementary Number Theory  21

Prime Factorization  21

Greatest Common Divisor and Least Common Multiple  22

Factorials  23

Binomial Coefficients  23

Real Numbers  24

Basic Operations  24

Powers and Radicals  25

Rationalizing a Denominator  27

Numerical Approximations  28

Scientific Notation  29

Computation and Display of Numerical Results  29

Functions and Relations  32

Absolute Value  33

Maximum and Minimum  33

Greatest and Smallest Integer Functions  34

Checking Equality and Inequality  35

Union, Intersection, and Difference  37

Complex Numbers  38

Basic Operations  38

Real Powers and Roots of Complex Numbers  39

Real and Imaginary Parts of a Complex Number  40

Absolute Value  41

Complex Conjugate  42

Numerical Approximations of Complex Numbers  42

Units and Measurements  43

Units  43

Physical Quantities, Symbols and Keyboard Shortcuts  44

Compound Units  47

Arithmetic Operations With Units  48

Converting Units  48

Exercises  49

#### 3  Algebra  51

Polynomials and Rational Expressions  51

Sums, Differences, Products, and Quotients of Polynomials  51

Summation Notation  53

Sums and Differences of Rational Expressions  53

Partial Fractions  54

Products and Powers of Polynomials  55

Division by Polynomials  56

Collecting and Ordering Terms  56

Factoring Polynomials  57

Greatest Common Divisor of Two Polynomials  58

Roots of Polynomials  59

Defining Variables and Functions  63

Assigning Values to Variables  64

Defining Functions of One Variable  64

Defining Functions of Several Variables  66

Showing and Removing Definitions  66

Solving Polynomial Equations  67

Equations with One Variable  67

Equations with Several Variables  70

Systems of Equations  70

Numerical Solutions  71

Inequalities  73

Substitution  74

Substituting for a Variable  75

Evaluating at Endpoints  75

Exponents and Logarithms  76

Exponents and Exponential Functions  76

Logarithms and Logarithmic Functions  77

Solving Exponential and Logarithmic Equations  79

Exercises  80

#### 4  Trigonometry  85

Trigonometric Functions  85

Radians and Degrees  86

Solving Trigonometric Equations  87

Trigonometric Identities  89

Combining and Simplifying Trigonometric Expressions  91

Inverse Trigonometric Functions and Trigonometric Equations  93

Combining and Rewriting Inverse Trigonometric Functions  93

Trigonometric Equations and Inverse Trigonometric Functions  94

Hyperbolic Functions  95

Inverse Hyperbolic Functions  97

Complex Numbers and Complex Functions  98

Arguments of a Complex Number  98

Forms of a Complex Number  99

Complex Powers and Roots of Complex Numbers  100

DeMoivre's Theorem  101

Complex Trigonometric and Hyperbolic Functions  101

Exercises  103

#### 5  Function Definitions  109

Function and Expression Names  109

Valid Names for Functions and Expressions  109

Custom Names  110

Automatic Substitution  111

Defining Variables and Functions  112

Assigning Values to Variables, or Naming Expressions  112

Functions of One Variable  114

Subscripts as Function Arguments  116

Piecewise-Defined Functions  117

Defining Generic Functions  118

Defining Generic Constants  119

Functions of Several Variables  119

Handling Definitions  119

Showing and Removing Definitions  119

Saving and Restoring Definitions  120

Assumptions About Variables  121

Formula  125

External Functions  128

Accessing Functions in MuPAD Libraries  128

User-Defined MuPAD Functions  130

Tables of Equivalents  130

Constants  130

Compute Menu Items  131

Equivalents for Functions and Expressions  137

Trigtype Functions  142

Determining the Argument of a Trigtype Function  143

Exercises  144

#### 6  Plotting Curves and Surfaces  147

Getting Started With Plots  147

The Frame, the View, and the Plot Properties Dialog  148

Layout  150

Resizing the Frame  151

Frame Placement  151

Screen Display and Print Attributes  153

Plot Intervals and View Intervals for 2D Plots  153

Rectangular Coordinates  155

Polar Coordinates  155

Implicit Plots  156

Parametric Plots  156

Plotting Tools for 2D Plots  157

Zooming In and Out  157

Translating the View  158

Plot Coordinates Dialog Bar  159

Items Plotted  160

Expressions and Relations  160

Intervals and Sample Size  161

Plot Color and Style  162

Adjust Plot for Discontinuities  162

Axes and Axis Scaling  163

Plot Captions, Keys, and Names  164

Plot Labels  165

2D Plots of Functions and Expressions  166

Expressions  166

Defined Functions  168

Continuous and Discontinuous Plots  169

Plotting Piecewise-Defined Functions  170

Special Functions  171

Polygons and Point Plots  173

Log and Log-Log Plots  178

Parametric Plots  179

Envelopes  181

Implicit Plots  182

Polar Coordinates  184

Parametric Polar Plots  184

Animated 2D Plots and the VCAM Window  185

Animated Plots in Rectangular Coordinates  187

Animated Plots in Polar Coordinates  189

Animated Implicit Plots  190

The View for 3D Plots  191

Plotting Tools and Dialogs for 3D Plots  192

The Plot Orientation Tool  192

The 3D Plot Properties Dialog  192

3D Plots of Functions and Expressions  198

Defined Functions  199

Parametric Plots  200

Implicit Plots  204

Curves in Space  205

Polygonal Paths  208

Cylindrical Coordinates  210

Spherical Coordinates  214

The VCAM Window and 3D Plots  217

Animated 3D Plots  218

Animated Plots in Rectangular Coordinates  218

Animated Plots in Cylindrical Coordinates  220

Animated Plots in Spherical Coordinates  222

Animated Implicit Plot  223

Animated Tube Plot  223

Plot Snapshots  224

Snapshot Generation and Removal  224

Snapshot Resolution  225

Snapshots as Pictures  226

Setting Plot Default Options  227

Universal Default Options For Plots  227

Default Plot Options for a Document  229

Exercises  231

#### 7  Calculus  239

Evaluating Calculus Expressions  239

Limits  240

Notation for Limits  241

Special Limits  243

Tables of Values and Plots  243

Differentiation  246

Notation for Derivative  246

Plotting Derivatives  249

Generic Functions  251

Implicit Differentiation  252

Numerical Solutions to Equations  255

Optimization  259

Curve Sketching  261

Indefinite Integration  266

Interpreting an Expression  267

Sequences of Operations  268

Methods of Integration  268

Integration by Parts  268

Change of Variables  269

Partial Fractions  270

Definite Integrals  271

Entering and Evaluating Definite Integrals  272

Methods of Integration with Definite Integrals  274

Improper Integrals  275

Assumptions About Variables  277

Definite Integrals from the Definition  277

Pictures of Riemann Sums  278

Approximation Methods  281

Numerical Integration  288

Visualizing Solids of Revolution  290

Sequences and Series  295

Sequences  296

Series  297

Multivariable Calculus  302

Optimization  302

Taylor Polynomials in Two Variables  306

Total Differential  307

Iterated Integrals  308

Exercises  311

#### 8  Matrix Algebra  319

Introduction  319

Changing the Appearance of Matrices  319

Creating Matrices  320

Revising Matrices  326

Concatenating and Stacking Matrices  328

Reshaping Lists and Matrices  329

Standard Operations  330

Matrix Addition and Scalar Multiplication   330

Inner Products and Matrix Multiplication  331

Rows and Columns  331

Identity and Inverse Matrices  331

Polynomials with Matrix Values  333

Operations on Matrix Entries  334

Row Operations and Echelon Forms  335

Gaussian Elimination and Row Echelon Form  335

Elementary Row Operations  336

Equations  337

Systems of Linear Equations  337

Matrix Equations  338

Matrix Operators  340

Trace  340

Transpose and Hermitian Transpose  341

Determinant  342

Adjugate  343

Permanent  344

Maximum and Minimum Matrix Entries  345

Matrix Norms  345

Spectral Radius  347

Condition Number  348

Exponential Functions  348

Polynomials and Vectors Associated With a Matrix  349

Characteristic Polynomial and Minimum Polynomial  349

Eigenvalues and Eigenvectors  351

Positive Definite Matrices  352

Vector Spaces Associated With a Matrix  353

The Row Space  353

The Column Space  355

The Left and Right Nullspaces  355

Orthogonal Matrices  356

The QR Factorization and Orthonormal Bases  356

Rank and Dimension  358

Normal Forms of Matrices  358

Smith Normal Form  359

Hermite Normal Form  360

Companion Matrix and Rational Canonical Form  360

Jordan Form  363

Matrix Decompositions  365

Singular Value Decomposition (SVD)  365

PLU Decomposition  366

QR Decomposition  367

Cholesky Decomposition  367

Exercises  368

#### 9  Vector Calculus  371

Vectors  371

Notation for Vectors  371

Vector Sums and Scalar Multiplication  372

Dot Product  372

Cross Product  373

Vector Norms  376

Planes and Lines in R3  378

Gradient, Divergence, and Curl  381

Gradient  382

Divergence  383

Curl  384

Laplacian  385

Directional Derivatives  386

Plots of Vector Fields and Gradients  387

Plots and Animated Plots of 2D Vector Fields  387

Plots and Animated Plots of 3D Vector Fields  389

Plots and Animated Plots of 2D Gradient Fields  391

Plots and Animated Plots of 3D Gradient Fields  393

Scalar and Vector Potentials  395

Scalar Potentials  395

Vector Potential  396

Matrix-Valued Operators  397

Hessian  397

Jacobian  399

Wronskian  400

Plots of Complex Functions  402

Conformal Plots  402

Animated Conformal Plots  403

Exercises  404

#### 10  Differential Equations  409

Ordinary Differential Equations  409

Exact Solutions  409

Series Solutions  414

Heaviside and Dirac Functions  414

Laplace Transforms  416

Fourier Transforms  420

Initial-Value Problems and Systems of Ordinary Differential Equations  422

Exact Solutions  422

Series Solutions  425

Numerical Methods For Ordinary Differential Equations  425

Numerical Solutions for Initial-Value Problems  425

Graphical Solutions to Initial-Value Problems  426

Numerical Solutions to Systems of Differential Equations  427

Graphical Solutions to Systems of ODEs  4428

Bessel Functions  429

Exercises  432

#### 11  Statistics  435

Introduction to Statistics  435

Lists and Matrices  435

Importing Data from an ASCII File  436

Measures of Central Tendency  438

Arithmetic Mean  438

Median  439

Quantile  440

Mode  440

Geometric Mean  441

Harmonic Mean  442

Measures of Dispersion  443

Mean Deviation  443

Variance and Standard Deviation  444

Covariance  445

Moment  446

Correlation  447

Distributions and Densities  448

Cumulative Distribution Functions  448

Inverse Distribution Functions  449

Distribution Tables  449

Families of Continuous Distributions  449

Gamma Function   449

Normal Distribution   450

Student's t Distribution   451

Chi-Square Distribution   452

F Distribution  453

Exponential Distribution  454

Weibull Distribution  455

Gamma Distribution  456

Beta Distribution  457

Cauchy Distribution  457

Uniform Distribution  458

Families of Discrete Distributions  459

Binomial Distribution  459

Poisson Distribution  460

Hypergeometric Distribution  461

Random Numbers  462

Curve Fitting  463

Linear Regression  463

Polynomial Fit  465

Overdetermined Systems of Equations  469

Exercises  470

#### 12  Applied Modern Algebra  473

Solving Equations  473

Integer Solutions  473

Continued Fractions  473

Recursive Solutions  474

Integers Modulo m  475

Multiplication Tables Modulo m  476

Inverses Modulo m  478

Solving Congruences Modulo m    479

Pairs of Linear Congruences  479

Systems of Linear Congruences  480

Extended Precision Arithmetic  480

Powers Modulo m  482

Generating Large Primes  482

Other Systems Modulo m  483

Matrices Modulo m   483

Polynomials Modulo m  485

Poynomials Modulo Polynomials  486

Greatest Common Divisor of Polynomials  487

Multiplicity of Roots of Polynomials  487

The Galois Field GFpn  489

Linear Programming  492

The Simplex Algorithm  492

Feasible Systems  493

Standard Form  494

The Dual of a Linear Program  494

Exercises  495

#### Index  501

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