MathematicaŽ in Action: Problem Solving Through Visualization and Computation

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Springer Science & Business Media, Jun 29, 2010 - Mathematics - 580 pages
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This book is an example-based introduction to techniques, from elementary to advanced, of using Mathematica, a revolutionary tool for mathematical computation and exploration. By integrating the basic functions of mathematics with a powerful and easy-to-use programming language, Mathematica allows us to carry out projects that would be extremely laborious in traditional programming environments. And the new developments that began with version 6 — allowing the user to dyna- cally manipulate output using sliders or other controls — add amazing power to the interface. Animations have always been part of Mathematica, but the new design allows the manipulation of any number of variables, an important enhancement. Mathematica in Action illustrates this power by using demonstrations and ani- tions, three-dimensional graphics, high-precision number theory computations, and sophisticated geometric and symbolic programming to attack a diverse collection of problems.. It is my hope that this book will serve a mathematical purpose as well, and I have interspersed several unusual or complicated examples among others that will be more familiar. Thus the reader may have to deal simultaneously with new mat- matics and new Mathematica techniques. Rarely is more than undergraduate mathematics required, however. An underlying theme of the book is that a computational way of looking at a mathematical problem or result yields many benefits. For example: Well-chosen computations can shed light on familiar relations and reveal new patterns. One is forced to think very precisely; gaps in understanding must be eliminated if a program is to work.
 

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Contents

1 Plotting
23
2 Prime Numbers
53
3 Rolling Circles
77
4 ThreeDimensional Graphs
113
5 Dynamic Manipulations
141
6 The Cantor Set Real and Complex
168
7 The Quadratic Map
179
8 The Recursive Turtle
209
14 Differential Equations
363
15 Computational Geometry
399
16 Check Digits and the Pentagon
423
17 Coloring Planar Maps
430
18 New Directions For
473
19 The BanachTarski Paradox
491
20 The Riemann Zeta Function
505
21 Miscellany
523

9 Parametric Plotting of Surfaces
235
10 Penrose Tiles
267
Julia Sets and the Mandelbrot Set
276
12 Solving Equations
301
13 Optimization
329

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