Maple and Mathematica

A Problem Solving Approach for Mathematics

By: Inna K. Shingareva, Carlos LizÃ¡rraga-Celaya

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CD-ROM | 1 September 2009 | Edition Number 2

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504 Pages

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Revised

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22.86 x 15.88 x 2.54

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In the history of mathematics there are many situations in which cal- lations were performed incorrectly for important practical applications. Let us look at some examples, the history of computing the number ? began in Egypt and Babylon about 2000 years BC, since then many mathematicians have calculated ? (e. g. , Archimedes, Ptolemy, Vi` ete, etc. ). The ?rst formula for computing decimal digits of ? was disc- ered by J. Machin (in 1706), who was the ?rst to correctly compute 100 digits of ?. Then many people used his method, e. g. , W. Shanks calculated ? with 707 digits (within 15 years), although due to mistakes only the ?rst 527 were correct. For the next examples, we can mention the history of computing the ?ne-structure constant ? (that was ?rst discovered by A. Sommerfeld), and the mathematical tables, exact - lutions, and formulas, published in many mathematical textbooks, were not veri?ed rigorously [25]. These errors could have a large e?ect on results obtained by engineers. But sometimes, the solution of such problems required such techn- ogy that was not available at that time. In modern mathematics there exist computers that can perform various mathematical operations for which humans are incapable. Therefore the computers can be used to verify the results obtained by humans, to discovery new results, to - provetheresultsthatahumancanobtainwithoutanytechnology. With respectto our example of computing?, we can mention that recently (in 2002) Y. Kanada, Y. Ushiro, H. Kuroda, and M.

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Foundations of Maple and Mathematica	p. 1
Maple	p. 3
Introduction	p. 3
History	p. 3
Basic Features	p. 4
Design	p. 4
Basic Concepts	p. 5
First Steps	p. 5
Help System	p. 5
Worksheets and Interface	p. 6
Packages	p. 7
Numerical Evaluation	p. 7
Maple Language	p. 8
Basic Principles	p. 9
Constants	p. 11
Functions	p. 12
Procedures and Modules	p. 16
Control Structures	p. 18
Objects and Operations	p. 20
Mathematica	p. 23
Introduction	p. 23
History	p. 23
Basic Features	p. 24
Design	p. 25
Changes for New Versions	p. 25
Basic Concepts	p. 26
First Steps	p. 26
Help System	p. 26
Notebook and Front End	p. 27
Packages	p. 28
Numerical Evaluation	p. 29
Mathematica Language	p. 29
Basic Principles	p. 30
Constants	p. 35
Functions	p. 36
Modules	p. 40
Control Structures	p. 41
Objects and Operations	p. 43
Dynamic Objects	p. 48
Mathematics: Maple and Mathematica	p. 49
Graphics	p. 51
Simple Graphs	p. 51
Various Options	p. 51
Multiple Graphs	p. 54
Text in Graphs	p. 57
Special Graphs	p. 58
Animations	p. 64
Algebra	p. 69
Finite Sets	p. 69
Infinite Sets	p. 72
Operations on Sets	p. 84
Equivalence Relations and Induction	p. 85
Mathematical Expressions	p. 86
Simplifying Mathematical Expressions	p. 89
Trigonometric and Hyperbolic Expressions	p. 91
Denning Functions or Mappings	p. 95
Operations on Functions	p. 103
Univariate and Multivariate Polynomials	p. 108
Groups	p. 113
Rings, Integral Domains, and Fields	p. 125
Linear Algebra	p. 133
Vectors	p. 134
Matrices	p. 141
Functions of Matrices	p. 151
Vector Spaces	p. 154
Normed and Inner Product Vector Spaces	p. 156
Systems of Linear Equations	p. 158
Linear Transformations	p. 160
Eigenvalues and Eigenvectors	p. 162
Matrix Decompositions and Equivalence Relations	p. 164
Bilinear and Quadratic Forms	p. 168
Linear Algebra with Modular Arithmetic	p. 171
Linear Algebra over Rings and Fields	p. 174
Tensors	p. 178
Geometry	p. 189
Points in the Plane and Space	p. 189
Parametric Curves	p. 191
Implicitly Defined Curves	p. 194
Curves in Polar Coordinates	p. 194
Secant and Tangent Lines	p. 196
Tubes and Knots	p. 197
Surfaces in Space	p. 198
Level Curves and Surfaces	p. 200
Surfaces of Revolution	p. 201
Vector Fields	p. 203
Cylindrical Coordinates	p. 204
Spherical Coordinates	p. 205
Standard Geometric Shapes	p. 205
Calculus and Analysis	p. 207
Real Functions	p. 207
Limits of Sequences and Functions	p. 209
Continuity of Functions	p. 210
Differential Calculus	p. 212
Integral Calculus	p. 218
Series	p. 228
Multivariate and Vector Calculus	p. 234
Complex Functions	p. 245
Complex Algebra	p. 245
Complex Functions and Derivatives	p. 248
Complex Integration	p. 253
Sequences and Series	p. 255
Singularities and Residue Theory	p. 258
Transformations and Mappings	p. 259
Special Functions and Orthogonal Polynomials	p. 261
Functions Defined by Integrals
Orthogonal Polynomials	p. 263
Functions Defined by Trascendental Equations	p. 264
Functions Defined by Differential Equations	p. 264
Functions Defined by Infinite Series	p. 267
Generalized Functions or Distributions	p. 267
Integral and Discrete Transforms	p. 269
Laplace Transforms	p. 269
Integral Fourier Transforms	p. 274
Discrete Fourier Transforms	p. 278
Hankel Transforms	p. 281
Mathematical Equations	p. 285
Algebraic and Trascendental Equations	p. 285
Ordinary Differential Equations	p. 291
Partial Differential Equations	p. 300
Integral Equations	p. 316
Numerical Analysis and Scientific Computing	p. 341
Nonlinear Equations	p. 344
Approximation of Functions and Data	p. 356
Numerical Differentiation and Integration	p. 378
Linear Systems of Equations	p. 395
Differential Equations	p. 405
References	p. 441
General Index	p. 446
Maple Index	p. 459
Mathematica Index	p. 472
CD-ROM Contents	p. 484
Table of Contents provided by Ingram. All Rights Reserved.

Maple and Mathematica

A Problem Solving Approach for Mathematics

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CD-ROM

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