Maple and Mathematica, A Problem Solving Approach for Mathematics

Preface\r......Page 7
ToC\r......Page 15
Part I:\rFoundations of Maple and Mathematica......Page 19
1.1.1History......Page 21
1.1.3Design......Page 22
1.2.2Help System......Page 23
1.2.3Worksheets and Interface......Page 24
1.2.5Numerical Evaluation......Page 25
1.3 Maple Language......Page 26
1.3.1Basic Principles......Page 27
1.3.2Constants......Page 29
1.3.3 Functions......Page 30
1.3.4Procedures andModules......Page 34
1.3.5Control Structures......Page 36
1.3.6Objects and Operations......Page 38
2.1.1History......Page 41
2.1.2Basic Features......Page 42
2.1.4Changes for New Versions......Page 43
2.2.2Help System......Page 44
2.2.3Notebook and Front End......Page 45
2.2.4Packages......Page 46
2.3 Mathematica Language......Page 47
2.3.1Basic Principles......Page 48
2.3.2Constants......Page 53
2.3.3 Functions......Page 54
2.3.4Modules......Page 58
2.3.5Control Structures......Page 59
2.3.6Objects and Operations......Page 61
2.3.7Dynamic Objects......Page 66
Part II:\rMathematics: Maple and Mathematica......Page 67
3.2 Various Options......Page 69
3.3 Multiple Graphs......Page 72
3.4 Text in Graphs......Page 75
3.5 Special Graphs......Page 76
3.6 Animations......Page 82
4.1 Finite Sets......Page 87
4.2 Infinite Sets......Page 90
4.3 Operations on Sets......Page 102
4.4 Equivalence Relations and Induction......Page 103
4.5 Mathematical Expressions......Page 104
4.6 Simplifying Mathematical Expressions......Page 107
4.7 Trigonometric and Hyperbolic Expressions......Page 109
4.8 Defining Functions orMappings......Page 113
4.9 Operations on Functions......Page 121
4.10 Univariate andMultivariate Polynomials......Page 126
4.11 Groups......Page 131
4.12 Rings, Integral Domains, and Fields......Page 143
5 Linear Algebra......Page 151
5.1 Vectors......Page 152
5.2 Matrices......Page 159
5.3 Functions ofMatrices......Page 169
5.4 Vector Spaces......Page 172
5.5 Normed and Inner Product Vector Spaces......Page 174
5.6 Systems of Linear Equations......Page 176
5.7 Linear Transformations......Page 178
5.8 Eigenvalues and Eigenvectors......Page 180
5.9 Matrix Decompositions and Equivalence Relations......Page 182
5.10 Bilinear and Quadratic Forms......Page 186
5.11 Linear Algebra withModular Arithmetic......Page 189
5.12 Linear Algebra over Rings and Fields......Page 192
5.13 Tensors......Page 196
6.1 Points in the Plane and Space......Page 207
6.2 Parametric Curves......Page 209
6.4 Curves in Polar Coordinates......Page 212
6.5 Secant and Tangent Lines......Page 214
6.6 Tubes and Knots......Page 215
6.7 Surfaces in Space......Page 216
6.8 Level Curves and Surfaces......Page 218
6.9 Surfaces of Revolution......Page 219
6.10 Vector Fields......Page 221
6.11 Cylindrical Coordinates......Page 222
6.13 Standard Geometric Shapes......Page 223
7.1 Real Functions......Page 225
7.2 Limits of Sequences and Functions......Page 227
7.3 Continuity of Functions......Page 228
7.4 Differential Calculus......Page 230
7.5 Integral Calculus......Page 236
7.6 Series......Page 246
7.7 Multivariate and Vector Calculus......Page 252
8.1 Complex Algebra......Page 263
8.2 Complex Functions and Derivatives......Page 266
8.3 Complex Integration......Page 271
8.4 Sequences and Series......Page 273
8.5 Singularities and Residue Theory......Page 276
8.6 Transformations andMappings......Page 277
9.1 Functions Defined by Integrals......Page 279
9.2 Orthogonal Polynomials......Page 281
9.4 Functions Defined by Differential Equations......Page 282
9.6 Generalized Functions or Distributions......Page 285
10.1 Laplace Transforms......Page 287
10.2 Integral Fourier Transforms......Page 292
10.3 Discrete Fourier Transforms......Page 296
10.4 Hankel Transforms......Page 299
11.1 Algebraic and Transcendental Equations......Page 303
11.2 Ordinary Differential Equations......Page 309
11.3 Partial Differential Equations......Page 318
11.4 Integral Equations......Page 334
12 Numerical Analysis and Scientific Computing......Page 359
12.1 Nonlinear Equations......Page 362
12.2 Approximation of Functions and Data......Page 374
12.3 Numerical Differentiation and Integration......Page 396
12.4 Linear Systems of Equations......Page 413
12.5 Differential Equations......Page 423
References......Page 459
General Index......Page 464
Maple Index......Page 477
Mathematica Index......Page 490
CD-ROM Contents......Page 502