Essentials of Mathematica: with applications to mathematics and physics
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| Main Author: | |
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| Format: | Kit Book |
| Language: | English |
| Published: |
New York, NY
Springer
2007
|
| Edition: | 1. ed. |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XXX, 539 S. Ill., graph. Darst. 1 CD-ROM (12 cm) |
| ISBN: | 9780387495132 0387495134 |
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| 245 | 1 | 0 | |a Essentials of Mathematica |b with applications to mathematics and physics |c Nino Boccara |
| 250 | |a 1. ed. | ||
| 264 | 1 | |a New York, NY |b Springer |c 2007 | |
| 300 | |a XXX, 539 S. |b Ill., graph. Darst. |e 1 CD-ROM (12 cm) | ||
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| adam_text |
Titel: Essentials of mathematica
Autor: Boccara, Nino
Jahr: 2007
Contents
Preface v
List of Figures xix
Part I Essential Commands
1 A Panorama of Mathematica 5
1.1 Notebooks and Cells 5
1.2 Basic Syntax 6
1.3 Basic Operations 6
1.4 Mathematica as a Functional Language 9
1.5 Getting Help 10
1.6 Logical Operators 12
1.7 Elementary Functions 14
1.8 User-Defined Functions 15
1.9 Rules and Delayed Rules 18
1.10 Built-in Nonelementary Functions 21
1.11 Plotting 21
1.11.1 2D plots 21
1.11.2 3D plots 22
1.12 Solving Equations 23
1.12.1 Exact Solutions 23
viii Contents
90
1.12.2 Numerical Solutions
1.13 Derivatives and Integrals 24
1.13.1 Exact Results 24
1.13.2 Numerical Integration 26
1.14 Series Expansions and Limits 27
1.15 Discrete Sums 2^
1.16 Ordinary Differential Equations 30
1.16.1 Symbolic Solutions 30
1.16.2 Numerical Solutions 31
1.17 Lists 32
1.18 Vectors and Matrices 36
1.19 Clear, ClearAll, and Remove . 40
1.20 Packages 42
1.21 Programming 43
1.21.1 Block and Module 43
1.21.2 Collatz Problem 47
1.21.3 Generalizing the Collatz Problem 49
2 Numbers 55
2.1 Characterizing Numbers 55
2.2 Real Numbers 56
2.3 Integers 58
2.4 Prime Numbers 61
2.5 Combinatorial Functions 62
2.5.1 Factorial 62
2.5.2 Binomial Coefficients 63
2.6 Rational Numbers 66
2.7 Complex Numbers 67
2.8 Different Bases 68
2.9 Calendars 7q
Contents ix
2.10 Positional Number Systems 71
2.11 Zeckendorf's Representation 73
3 Algebra 77
3.1 Algebraic Expressions 77
3.2 Trigonometric Expressions 82
3.3 Solving Equations 86
3.3.1 Solving Polynomial Equations Exactly 86
3.3.2 Numerical Solutions 89
3.4 Vectors and Matrices 95
4 Analysis 103
4.1 Differentiation 103
4.1.1 Partial Derivative 103
4.2 Total Derivative 105
4.3 Integration 106
4.3.1 Indefinite Integrals 106
4.3.2 Definite Integrals 107
4.3.3 Numerical Integration 109
4.3.4 Multiple Integrals 112
4.4 Differential Equations 113
4.4.1 Solving nonelementary ODE 114
4.4.2 Numerical Solutions 114
4.4.3 Series Solutions 117
4.4.4 Differential Vector Equations 119
4.5 Sum and Products 122
4.5.1 Exact Results 122
4.5.2 Numerical Results - - 123
4.6 Power Series 125
4.7 Limits 126
4.8 Complex Functions 130
xii Contents
097
7.3.4 Cauchy Distribution
OOQ
7.4 Descriptive Statistics
7.4.1 Poisson Distribution 229
7.4.2 Normal Distribution
; 7.4.3 Cosine Distribution 2^2
7.4.4 Uniform Distribution 2^2
8 Basic Programming 2^
8.1 The Matheinatica Language 235
8.2 Functional Programming 237
8.2.1 Applying Functions to Values 237
8.2.2 Defining Functions 239
8.2.3 Iterations 239
8.2.4 A Functional Program 242
8.3 Replacement Rules 247
8.3.1 The Two Kinds of Rewrite Global Rules 247
8.3.2 Local Rules 248
8.3.3 The Operators /. and // 249
8.3.4 Patterns . . 249
8.3.5 Example: the Fibonacci Numbers 252
8.4 Control Structures 257
8.4.1 Conditional Operations 257
8.4.2 Loops 259
8.5 Modules 262
8.5.1 Example 1 262
8.5.2 Example 2 263
8.5.3 Example 3 263
Contents xiii
Part II Applications
9 Axially Symmetric Electrostatic Potential 273
10 Motion of a Bead on a Rotating Circle 279
11 The Brachistochrone 285
12 Negative and Complex Bases 289
12.1 Negative Bases 289
12.2 Complex Bases 293
12.2.1 Arithmetic in Complex Bases 293
12.2.2 Fractal Images 295
13 Convolution and Laplace Transform 301
14 Double Pendulum 303
15 Duffing Oscillator 311
15.1 The Anharmonic Potential 311
15.2 Solving Duffing Equations 312
15.2.1 Single-Well Potential 312
15.2.2 Double-Well Potential 313
15.3 Oscillations in a Potential Well 314
15.3.1 Single-Well Potential 314
15.3.2 Double-Well Potential 315
15.4 Forced Duffing Oscillator with Damping 316
15.4.1 No Forcing Term 317
15.4.2 With Forcing Term 318
16 Egyptian Fractions 321
17 Electrostatics ^27
17.1 Potential and Field 327
. 327
.328
.330
.331
.333
.335
.341
.347
.348
.354
.357
.360
.369
.369
373
,374
375
,376
377
377
380
382
385
385
389
389
391
392
Contents
17.1.1 Useful Packages
17.1.2 Point Charge •
17.1.3 Dipole
17.1.4 Quadrupoles
17.1.5 Plots
17.1.6 Uniformly Charged Sphere
Foucault Pendulum . •
Fractals
19.1 Triadic Cantor Set
19.2 Sierpihski Triangle
19.3 Sierpihski Square
19.4 von Koch Curve
Iterated Function Systems
20.1 Chaos Game
20.2 Variations on the Chaos Game .
20.2.1 Example 1. . . ,
20.2.2 Example 2. . .
20.2.3 Example 3.
20.3 Barnsley Fern
20.3.1 The Original Barnsley Fern
20.3.2 Modifying the Probabilities
20.4 The Collage Theorem
Julia and Mandelbrot Sets
21.1 Julia Sets
21.2 Julia Sets of Different Functions .
21.2.1 z m 23 + c
21.2.2 Zy-+z4+c
21.3 Mandelbrot Sets
Contents xv
21.4 Mandelbrot Sets for Different Functions 397
21.4.1 2 )- z3 +c 397
21.4.2 z^z4+c 398
22 Kepler's Laws 399
23 Lindenmayer Systems 407
23.1 String Rewriting 407
23.2 von Koch Curve and Triangle 408
23.3 Hilbert Curve 412
23.4 Peano Curve 413
24 Logistic Map 417
24.1 Bifurcation Diagram 418
24.2 Exact Dynamics for r — 4 429
24.2.1 Conjugacy and Periodic Orbits 429
24.2.2 Exact Solution of the Recurrence Equation 433
24.2.3 Invariant Probability Density 434
25 Lorenz Equations 439
26 The Morse Potential 445
27 Prime Numbers 449
27.1 Primality 449
27.2 Mersenne Numbers 456
27.3 Perfect Numbers 458
28 Public-Key Encryption 461
28.1 The RSA Cryptosystem 461
28.1.1 ToCharacterCode and FromCharacterCode 462
28.1.2 Obtaining the Integer t 462
28.1.3 Choosing the Integer n — pq 464
28.1.4 Choosing the Public Exponent e 465
xvi Contents
28.1.5 Coding 465
28.1.6 Choosing the Secret Exponent d 466
28.1.7 Decrypting 4GG
28.2 Summing Up 4G^
29 Quadratrix of Hippias 469
29.1 Figure . 4G^
29.2 Trisecting an Angle -J?1
29.3 Squaring the Circle . , 472
30 Quantum Harmonic Oscillator *475
30.1 Schrodinger Equation 475
30.2 Creation and Annihilation Operators 479 ^
31 Quantum Square Potential 481
31.1 The Problem and Its Analytical Solution 481
31.2 Numerical Solution 482
31.2.1 Energy Levels for A = 16 483
31.2.2 Figure Representing the Potential and the Energy Levels . 485
31.2.3 Plotting the Eigcnfunctions 486
32 Skydiving 489
32.1 Terminal Velocity 489
32.2 Delaying Parachute Opening 490
^ 32.3 Taking into Account Time for Parachute to Open 493
33 Tautochrone 497
33.1 Involute and Evolute 497
33.2 The Cycloid . . 499
33.3 Fractional Calculus 59^
33.4 Other Tautochrone Curves 502
34 van der Pol Oscillator rnr
509
509
510
511
513
519
519
rians Moving in Opposite
520
strians 523
strians 524
th Types 526
526
529
533 |
| adam_txt |
Titel: Essentials of mathematica
Autor: Boccara, Nino
Jahr: 2007
Contents
Preface v
List of Figures xix
Part I Essential Commands
1 A Panorama of Mathematica 5
1.1 Notebooks and Cells 5
1.2 Basic Syntax 6
1.3 Basic Operations 6
1.4 Mathematica as a Functional Language 9
1.5 Getting Help 10
1.6 Logical Operators 12
1.7 Elementary Functions 14
1.8 User-Defined Functions 15
1.9 Rules and Delayed Rules 18
1.10 Built-in Nonelementary Functions 21
1.11 Plotting 21
1.11.1 2D plots 21
1.11.2 3D plots 22
1.12 Solving Equations 23
1.12.1 Exact Solutions 23
viii Contents
90
1.12.2 Numerical Solutions
1.13 Derivatives and Integrals 24
1.13.1 Exact Results 24
1.13.2 Numerical Integration 26
1.14 Series Expansions and Limits 27
1.15 Discrete Sums 2^
1.16 Ordinary Differential Equations 30
1.16.1 Symbolic Solutions 30
1.16.2 Numerical Solutions 31
1.17 Lists 32
1.18 Vectors and Matrices 36
1.19 Clear, ClearAll, and Remove . 40
1.20 Packages 42
1.21 Programming 43
1.21.1 Block and Module 43
1.21.2 Collatz Problem 47
1.21.3 Generalizing the Collatz Problem 49
2 Numbers 55
2.1 Characterizing Numbers 55
2.2 Real Numbers 56
2.3 Integers 58
2.4 Prime Numbers 61
2.5 Combinatorial Functions 62
2.5.1 Factorial 62
2.5.2 Binomial Coefficients 63
2.6 Rational Numbers 66
2.7 Complex Numbers 67
2.8 Different Bases 68
2.9 Calendars 7q
Contents ix
2.10 Positional Number Systems 71
2.11 Zeckendorf's Representation 73
3 Algebra 77
3.1 Algebraic Expressions 77
3.2 Trigonometric Expressions 82
3.3 Solving Equations 86
3.3.1 Solving Polynomial Equations Exactly 86
3.3.2 Numerical Solutions 89
3.4 Vectors and Matrices 95
4 Analysis 103
4.1 Differentiation 103
4.1.1 Partial Derivative 103
4.2 Total Derivative 105
4.3 Integration 106
4.3.1 Indefinite Integrals 106
4.3.2 Definite Integrals 107
4.3.3 Numerical Integration 109
4.3.4 Multiple Integrals 112
4.4 Differential Equations 113
4.4.1 Solving nonelementary ODE 114
4.4.2 Numerical Solutions 114
4.4.3 Series Solutions 117
4.4.4 Differential Vector Equations 119
4.5 Sum and Products 122
4.5.1 Exact Results 122
4.5.2 Numerical Results - - 123
4.6 Power Series 125
4.7 Limits 126
4.8 Complex Functions 130
xii Contents
097
7.3.4 Cauchy Distribution
OOQ
7.4 Descriptive Statistics
7.4.1 Poisson Distribution 229
7.4.2 Normal Distribution
; 7.4.3 Cosine Distribution 2^2
7.4.4 Uniform Distribution 2^2
8 Basic Programming 2^
8.1 The Matheinatica Language 235
8.2 Functional Programming 237
8.2.1 Applying Functions to Values 237
8.2.2 Defining Functions 239
8.2.3 Iterations 239
8.2.4 A Functional Program 242
8.3 Replacement Rules 247
8.3.1 The Two Kinds of Rewrite Global Rules 247
8.3.2 Local Rules 248
8.3.3 The Operators /. and // 249
8.3.4 Patterns . . 249
8.3.5 Example: the Fibonacci Numbers 252
8.4 Control Structures 257
8.4.1 Conditional Operations 257
8.4.2 Loops 259
8.5 Modules 262
8.5.1 Example 1 262
8.5.2 Example 2 263
8.5.3 Example 3 263
Contents xiii
Part II Applications
9 Axially Symmetric Electrostatic Potential 273
10 Motion of a Bead on a Rotating Circle 279
11 The Brachistochrone 285
12 Negative and Complex Bases 289
12.1 Negative Bases 289
12.2 Complex Bases 293
12.2.1 Arithmetic in Complex Bases 293
12.2.2 Fractal Images 295
13 Convolution and Laplace Transform 301
14 Double Pendulum 303
15 Duffing Oscillator 311
15.1 The Anharmonic Potential 311
15.2 Solving Duffing Equations 312
15.2.1 Single-Well Potential 312
15.2.2 Double-Well Potential 313
15.3 Oscillations in a Potential Well 314
15.3.1 Single-Well Potential 314
15.3.2 Double-Well Potential 315
15.4 Forced Duffing Oscillator with Damping 316
15.4.1 No Forcing Term 317
15.4.2 With Forcing Term 318
16 Egyptian Fractions 321
17 Electrostatics ^27
17.1 Potential and Field 327
. 327
.328
.330
.331
.333
.335
.341
.347
.348
.354
.357
.360
.369
.369
373
,374
375
,376
377
377
380
382
385
385
389
389
391
392
Contents
17.1.1 Useful Packages
17.1.2 Point Charge •
17.1.3 Dipole
17.1.4 Quadrupoles
17.1.5 Plots
17.1.6 Uniformly Charged Sphere
Foucault Pendulum . •
Fractals
19.1 Triadic Cantor Set
19.2 Sierpihski Triangle
19.3 Sierpihski Square
19.4 von Koch Curve
Iterated Function Systems
20.1 Chaos Game
20.2 Variations on the Chaos Game .
20.2.1 Example 1. . . ,
20.2.2 Example 2. . .
20.2.3 Example 3.
20.3 Barnsley Fern
20.3.1 The Original Barnsley Fern
20.3.2 Modifying the Probabilities
20.4 The Collage Theorem
Julia and Mandelbrot Sets
21.1 Julia Sets
21.2 Julia Sets of Different Functions .
21.2.1 z m 23 + c
21.2.2 Zy-+z4+c
21.3 Mandelbrot Sets
Contents xv
21.4 Mandelbrot Sets for Different Functions 397
21.4.1 2 )- z3 +c 397
21.4.2 z^z4+c 398
22 Kepler's Laws 399
23 Lindenmayer Systems 407
23.1 String Rewriting 407
23.2 von Koch Curve and Triangle 408
23.3 Hilbert Curve 412
23.4 Peano Curve 413
24 Logistic Map 417
24.1 Bifurcation Diagram 418
24.2 Exact Dynamics for r — 4 429
24.2.1 Conjugacy and Periodic Orbits 429
24.2.2 Exact Solution of the Recurrence Equation 433
24.2.3 Invariant Probability Density 434
25 Lorenz Equations 439
26 The Morse Potential 445
27 Prime Numbers 449
27.1 Primality 449
27.2 Mersenne Numbers 456
27.3 Perfect Numbers 458
28 Public-Key Encryption 461
28.1 The RSA Cryptosystem 461
28.1.1 ToCharacterCode and FromCharacterCode 462
28.1.2 Obtaining the Integer t 462
28.1.3 Choosing the Integer n — pq 464
28.1.4 Choosing the Public Exponent e 465
xvi Contents
28.1.5 Coding 465
28.1.6 Choosing the Secret Exponent d 466
28.1.7 Decrypting 4GG
28.2 Summing Up 4G^
29 Quadratrix of Hippias 469
29.1 Figure . 4G^
29.2 Trisecting an Angle -J?1
29.3 Squaring the Circle . , 472
30 Quantum Harmonic Oscillator *475
30.1 Schrodinger Equation 475
30.2 Creation and Annihilation Operators 479 ^
31 Quantum Square Potential 481
31.1 The Problem and Its Analytical Solution 481
31.2 Numerical Solution 482
31.2.1 Energy Levels for A = 16 483
31.2.2 Figure Representing the Potential and the Energy Levels . 485
31.2.3 Plotting the Eigcnfunctions 486
32 Skydiving 489
32.1 Terminal Velocity 489
32.2 Delaying Parachute Opening 490
^ 32.3 Taking into Account Time for Parachute to Open 493
33 Tautochrone 497
33.1 Involute and Evolute 497
33.2 The Cycloid . . 499
33.3 Fractional Calculus 59^
33.4 Other Tautochrone Curves 502
34 van der Pol Oscillator rnr
509
509
510
511
513
519
519
rians Moving in Opposite
520
strians 523
strians 524
th Types 526
526
529
533 |
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| index_date | 2024-07-02T19:11:42Z |
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| language | English |
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| publisher | Springer |
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| spelling | Boccara, Nino Verfasser aut Essentials of Mathematica with applications to mathematics and physics Nino Boccara 1. ed. New York, NY Springer 2007 XXX, 539 S. Ill., graph. Darst. 1 CD-ROM (12 cm) Mathematica Programm (DE-588)4268208-3 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Mathematica Programm (DE-588)4268208-3 s DE-604 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=2867265&prov=M&dok_var=1&dok_ext=htm Inhaltstext HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016221290&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Boccara, Nino Essentials of Mathematica with applications to mathematics and physics Mathematica Programm (DE-588)4268208-3 gnd |
| subject_GND | (DE-588)4268208-3 (DE-588)4123623-3 |
| title | Essentials of Mathematica with applications to mathematics and physics |
| title_auth | Essentials of Mathematica with applications to mathematics and physics |
| title_exact_search | Essentials of Mathematica with applications to mathematics and physics |
| title_exact_search_txtP | Essentials of Mathematica with applications to mathematics and physics |
| title_full | Essentials of Mathematica with applications to mathematics and physics Nino Boccara |
| title_fullStr | Essentials of Mathematica with applications to mathematics and physics Nino Boccara |
| title_full_unstemmed | Essentials of Mathematica with applications to mathematics and physics Nino Boccara |
| title_short | Essentials of Mathematica |
| title_sort | essentials of mathematica with applications to mathematics and physics |
| title_sub | with applications to mathematics and physics |
| topic | Mathematica Programm (DE-588)4268208-3 gnd |
| topic_facet | Mathematica Programm Lehrbuch |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=2867265&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016221290&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT boccaranino essentialsofmathematicawithapplicationstomathematicsandphysics |