Mathematical Methods in the Physical Sciences (with fixed cover thing)

Title Page......Page 4
Copyright Page......Page 5
Dedication Page......Page 6
Preface......Page 8
To the Student......Page 12
CONTENTS......Page 14
1. The Geometric Series......Page 20
2. Definitions and Notation......Page 23
4. Convergent and Divergent Series......Page 25
5. Testing Series for Convergence; the Preliminary Test......Page 28
A. The Comparison Test......Page 29
B. The Integral Test......Page 30
C. The Ratio Test......Page 32
D. A Special Comparison Test......Page 34
7. Alternating Series......Page 36
8. Conditionally Convergent Series......Page 37
9. Useful Facts About Series......Page 38
10. Power Series; Interval of Convergence......Page 39
12. Expanding Functions in Power Series......Page 42
13. Techniques for Obtaining Power Series Expansions......Page 44
A. Multiplying a Series by a Polynomial or by Another Series......Page 45
B. Division of Two Series or of a Series by a Polynomial......Page 46
C. Binomial Series......Page 47
F. Taylor Series Using the Basic Maclaurin Series......Page 49
G. Using a Computer......Page 50
14. Accuracy of Series Approximations......Page 52
15. Some Uses of Series......Page 55
16. Miscellaneous Problems......Page 63
1. Introduction......Page 65
3. The Complex Plane......Page 66
4. Terminology and Notation......Page 68
A. Simplifying to x+iy form......Page 70
B. Complex Conjugate of a Complex Expression......Page 71
C. Finding the Absolute Value of z......Page 72
E. Graphs......Page 73
F. Physical Applications......Page 74
6. Complex Infinite Series......Page 75
7. Complex Power Series; Disk of Convergence......Page 77
8. Elementary Functions of Complex Numbers......Page 79
9. Euler’s Formula......Page 80
10. Powers and Roots of Complex Numbers......Page 83
11. The Exponential and Trigonometric Functions......Page 86
12. Hyperbolic Functions......Page 89
13. Logarithms......Page 91
14. Complex Roots and Powers......Page 92
15. Inverse Trigonometric and Hyperbolic Functions......Page 93
16. Some Applications......Page 95
17. Miscellaneous Problems......Page 99
1. Introduction......Page 101
2. Matrices; Row Reduction......Page 102
3. Determinants; Cramer’s Rule......Page 108
4. Vectors......Page 115
5. Lines and Planes......Page 125
6. Matrix Operations......Page 133
7. Linear Combinations, Linear Functions, Linear Operators......Page 143
8. Linear Dependence and Independence......Page 151
9. Special Matrices and Formulas......Page 156
10. Linear Vector Spaces......Page 161
11. Eigenvalues and Eigenvectors; Diagonalizing Matrices......Page 167
12. Applications of Diagonalization......Page 181
13. A Brief Introduction to Groups......Page 191
14. General Vector Spaces......Page 198
15. Miscellaneous Problems......Page 203
1. Introduction and Notation......Page 207
2. Power Series in Two Variables......Page 210
3. Total Differentials......Page 212
4. Approximations using Differentials......Page 215
5. Chain Rule or Differentiating a Function of a Function......Page 218
6. Implicit Differentiation......Page 221
7. More Chain Rule......Page 222
8. Application of Partial Differentiation to Maximum and Minimum Problems......Page 230
9. Maximum and Minimum Problems with Constraints; Lagrange Multipliers......Page 233
10. Endpoint or Boundary Point Problems......Page 242
11. Change of Variables......Page 247
12. Differentiation of Integrals; Leibniz’ Rule......Page 252
13. Miscellaneous problems......Page 257
1. Introduction......Page 260
2. Double and Triple Integrals......Page 261
3. Applications of Integration; Single and Multiple Integrals
......Page 268
4. Change of Variables in Integrals; Jacobians......Page 277
5. Surface Integrals......Page 289
6. Miscellaneous Problems......Page 292
2. Applications of Vector Multiplication
......Page 295
3. Triple Products......Page 297
4. Differentiation of Vectors......Page 304
5. Fields......Page 308
6. Directional Derivative; Gradient......Page 309
7. Some Other Expressions Involving ∇......Page 315
8. Line Integrals......Page 318
9. Green’s Theorem in the Plane......Page 328
10. The Divergence and the Divergence Theorem......Page 333
11. The Curl and Stokes’ Theorem......Page 343
12. Miscellaneous Problems......Page 355
2. Simple Harmonic Motion and Wave Motion; Periodic Functions......Page 359
3. Applications of Fourier Series......Page 364
4. Average Value of a Function......Page 366
5. Fourier Coefficients......Page 369
6. Dirichlet Conditions......Page 374
7. Complex Form of Fourier Series......Page 377
8. Other Intervals......Page 379
9. Even and Odd Functions......Page 383
10. An Application to Sound......Page 391
11. Parseval’s Theorem......Page 394
12. Fourier Transforms......Page 397
13. Miscellaneous Problems......Page 405
1. Introduction
......Page 409
2. Separable Equations......Page 414
3. Linear First-Order Equations......Page 420
4. Other Methods for First-Order Equations......Page 423
5. Second-Order Linear Equations with Constant Coefficients and Zero Right-Hand Side......Page 427
6. Second-Order Linear Equations with Constant Coefficients and Right-Hand Side Not Zero......Page 436
7. Other Second-Order Equations......Page 449
8. The Laplace Transform......Page 456
9. Solution of Differential Equations by Laplace Transforms......Page 459
10. Convolution......Page 463
11. The Dirac Delta Function......Page 468
12. A Brief Introduction to Green Functions......Page 480
13. Miscellaneous Problems......Page 485
1. Introduction......Page 491
2. The Euler Equation......Page 493
3. Using the Euler Equation......Page 497
4. The Brachistochrone Problem; Cycloids......Page 501
5. Several Dependent Variables; Lagrange’s Equations......Page 504
6. Isoperimetric Problems......Page 510
7. Variational Notation......Page 512
8. Miscellaneous Problems......Page 513
1. Introduction......Page 515
2. Cartesian Tensors......Page 517
3. Tensor Notation and Operations......Page 521
4. Inertia Tensor......Page 524
5. Kronecker Delta and Levi-Civita Symbol......Page 527
6. Pseudovectors and Pseudotensors......Page 533
7. More About Applications......Page 537
8. Curvilinear Coordinates......Page 540
9. Vector Operators in Orthogonal Curvilinear Coordinates......Page 544
10. Non-Cartesian Tensors......Page 548
11. Miscellaneous Problems......Page 554
1. Introduction......Page 556
3. Definition of the Gamma Function; Recursion Relation......Page 557
4. The Gamma Function of Negative Numbers......Page 559
5. Some Important Formulas Involving Gamma Functions......Page 560
6. Beta Functions......Page 561
7. Beta Functions in Terms of Gamma Functions......Page 562
8. The Simple Pendulum......Page 564
9. The Error Function......Page 566
10. Asymptotic Series......Page 568
11. Stirling’s Formula......Page 571
12. Elliptic Integrals and Functions......Page 573
13. Miscellaneous Problems......Page 579
1. Introduction......Page 581
2. Legendre’s Equation......Page 583
3. Leibniz’ Rule for Differentiating Products......Page 586
4. Rodrigues’ Formula......Page 587
5. Generating Function for Legendre Polynomials......Page 588
6. Complete Sets of Orthogonal Functions......Page 594
7. Orthogonality of the Legendre Polynomials......Page 596
8. Normalization of the Legendre Polynomials......Page 597
9. Legendre Series......Page 599
10. The Associated Legendre Functions......Page 602
11. Generalized Power Series or the Method of Frobenius......Page 604
12. Bessel’s Equation......Page 606
13. The Second Solution of Bessel’s Equation......Page 609
14. Graphs and Zeros of Bessel Functions......Page 610
15. Recursion Relations......Page 611
16. Differential Equations with Bessel Function Solutions......Page 612
17. Other Kinds of Bessel Functions......Page 614
18. The Lengthening Pendulum......Page 617
19. Orthogonality of Bessel Functions......Page 620
20. Approximate Formulas for Bessel Functions......Page 623
21. Series Solutions; Fuchs’s Theorem......Page 624
22. Hermite Functions; Laguerre Functions; Ladder Operators......Page 626
23. Miscellaneous Problems......Page 634
1. Introduction......Page 638
2. Laplace’s Equation; Steady-State Temperature in a Rectangular Plate......Page 640
3. The Diffusion or Heat Flow Equation; the Schr¨odinger Equation......Page 647
4. The Wave Equation; the Vibrating String......Page 652
5. Steady-state Temperature in a Cylinder......Page 657
6. Vibration of a Circular Membrane......Page 663
7. Steady-state Temperature in a Sphere......Page 666
8. Poisson’s Equation......Page 671
9. Integral Transform Solutions of Partial Differential Equations......Page 678
10. Miscellaneous Problems......Page 682
1. Introduction......Page 685
2. Analytic Functions......Page 686
3. Contour Integrals......Page 693
4. Laurent Series......Page 697
5. The Residue Theorem......Page 701
6. Methods of Finding Residues......Page 702
7. Evaluation of Definite Integrals by Use of the Residue Theorem......Page 706
8. The Point at Infinity; Residues at Infinity......Page 721
9. Mapping......Page 724
10. Some Applications of Conformal Mapping......Page 729
11. Miscellaneous Problems......Page 737
1. Introduction......Page 741
2. Sample Space......Page 743
3. Probability Theorems......Page 748
4. Methods of Counting......Page 755
5. Random Variables......Page 763
6. Continuous Distributions......Page 769
7. Binomial Distribution......Page 775
8. The Normal or Gaussian Distribution......Page 780
9. The Poisson Distribution......Page 786
10. Statistics and Experimental Measurements......Page 789
11. Miscellaneous Problems......Page 795
REFERENCES......Page 798
ANSWERS TO SELECTED PROBLEMS......Page 800
INDEX......Page 830
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ch09......Page 904
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ch11......Page 911
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ch15......Page 927