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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. |
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
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
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
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
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
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
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
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
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
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
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
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