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

Cover of Doing Mathematics with
Scientific WorkPlace & Scientific Word 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

Preface  xxi

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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