Mathematical Methods in The Physical Scinces

PREFACE

This book is particularly intended for the student with a year (or a year and a half) of calculus who wants to develop, in a short time, a basic competence in each of the many areas of mathematics needed in junior to senior-graduate courses in physics, chemistry, and engineering. 

Thus it is intended to be accessible to sophomores (or freshmen with AP calculus from high school). It may also be used effectively by a more advanced student to review half-forgotten topics or learn new ones, either by independent study or in a class. 

Although the book was written especially for students of the physical sciences, students in any field (say mathematics or mathematics for teaching) may find it useful to survey many topics or to obtain some knowledge of areas they do not have time to study in depth. Since theorems are stated carefully, such students should not need to unlearn anything in their later work.

BOOKS CONTENTS

1 INFINITE SERIES, POWER SERIES 1

1. The Geometric Series 1
2. Definitions and Notation 4
3. Applications of Series 6
4. Convergent and Divergent Series 6
5. Testing Series for Convergence; the Preliminary Test 9
6. Convergence Tests for Series of Positive Terms: Absolute Convergence 10

• A. The Comparison Test 10
• B. The Integral Test 11
• C. The Ratio Test 13
• D. A Special Comparison Test 15

7. Alternating Series 17
8. Conditionally Convergent Series 18
9. Useful Facts About Series 19
10. Power Series; Interval of Convergence 20
11. Theorems About Power Series 23
12. Expanding Functions in Power Series 23
13. Techniques for Obtaining Power Series Expansions 25

• A. Multiplying a Series by a Polynomial or by Another Series 26
• B. Division of Two Series or of a Series by a Polynomial 27
• C. Binomial Series 28
• D. Substitution of a Polynomial or a Series for the Variable in Another Series 29
• E. Combination of Methods 30
• F. Taylor Series Using the Basic Maclaurin Series 30
• G. Using a Computer 31

14. Accuracy of Series Approximations 33
15. Some Uses of Series 36
16. Miscellaneous Problems 44

2 COMPLEX NUMBERS 46

1. Introduction 46
2. Real and Imaginary Parts of a Complex Number 47
3. The Complex Plane 47
4. Terminology and Notation 49
5. Complex Algebra 51

• A. Simplifying to x+iy form 51
• B. Complex Conjugate of a Complex Expression 52
• C. Finding the Absolute Value of z 53
• D. Complex Equations 54
• E. Graphs 54
• F. Physical Applications 55

6. Complex Infinite Series 56
7. Complex Power Series; Disk of Convergence 58
8. Elementary Functions of Complex Numbers 60
9. Euler’s Formula 61
10. Powers and Roots of Complex Numbers 64
11. The Exponential and Trigonometric Functions 67
12. Hyperbolic Functions 70
13. Logarithms 72
14. Complex Roots and Powers 73
15. Inverse Trigonometric and Hyperbolic Functions 74
16. Some Applications 76
17. Miscellaneous Problems 80

3 LINEAR ALGEBRA 82

1. Introduction 82
2. Matrices; Row Reduction 83
3. Determinants; Cramer’s Rule 89
4. Vectors 96
5. Lines and Planes 106
6. Matrix Operations 114
7. Linear Combinations, Linear Functions, Linear Operators 124
8. Linear Dependence and Independence 132
9. Special Matrices and Formulas 137
10. Linear Vector Spaces 142
11. Eigenvalues and Eigenvectors; Diagonalizing Matrices 148
12. Applications of Diagonalization 162
13. A Brief Introduction to Groups 172
14. General Vector Spaces 179
15. Miscellaneous Problems 184

4 PARTIAL DIFFERENTIATION 188

1. Introduction and Notation 188
2. Power Series in Two Variables 191
3. Total Differentials 193
4. Approximations using Differentials 196
5. Chain Rule or Differentiating a Function of a Function 199
6. Implicit Differentiation 202
7. More Chain Rule 203
8. Application of Partial Differentiation to Maximum and Minimum Problems 211
9. Maximum and Minimum Problems with Constraints; Lagrange Multipliers 214
10. Endpoint or Boundary Point Problems 223
11. Change of Variables 228
12. Differentiation of Integrals; Leibniz’ Rule 233
13. Miscellaneous problems 238

5 MULTIPLE INTEGRALS 241

1. Introduction 241
2. Double and Triple Integrals 242
3. Applications of Integration; Single and Multiple Integrals 249
4. Change of Variables in Integrals; Jacobians 258
5. Surface Integrals 270
6. Miscellaneous Problems 273

6 VECTOR ANALYSIS 276

1. Introduction 276
2. Applications of Vector Multiplication 276
3. Triple Products 278
4. Differentiation of Vectors 285
5. Fields 289
6. Directional Derivative; Gradient 290
7. Some Other Expressions Involving ∇ 296
8. Line Integrals 299
9. Green’s Theorem in the Plane 309
10. The Divergence and the Divergence Theorem 314
11. The Curl and Stokes’ Theorem 324
12. Miscellaneous Problems 336

7 FOURIER SERIES AND TRANSFORMS 340

1. Introduction 340
2. Simple Harmonic Motion and Wave Motion; Periodic Functions 340
3. Applications of Fourier Series 345
4. Average Value of a Function 347
5. Fourier Coefficients 350
6. Dirichlet Conditions 355
7. Complex Form of Fourier Series 358
8. Other Intervals 360
9. Even and Odd Functions 364
10. An Application to Sound 372
11. Parseval’s Theorem 375
12. Fourier Transforms 378
13. Miscellaneous Problems 386

8 ORDINARY DIFFERENTIAL EQUATIONS 390

1. Introduction 390
2. Separable Equations 395
3. Linear First-Order Equations 401
4. Other Methods for First-Order Equations 404
5. Second-Order Linear Equations with Constant Coefficients and Zero Right-Hand Side 408
6. Second-Order Linear Equations with Constant Coefficients and Right-Hand Side Not Zero 417
7. Other Second-Order Equations 430
8. The Laplace Transform 437
9. Solution of Differential Equations by Laplace Transforms 440
10. Convolution 444
11. The Dirac Delta Function 449
12. A Brief Introduction to Green Functions 461
13. Miscellaneous Problems 466

9 CALCULUS OF VARIATIONS 472

1. Introduction 472
2. The Euler Equation 474
3. Using the Euler Equation 478
4. The Brachistochrone Problem; Cycloids 482
5. Several Dependent Variables; Lagrange’s Equations 485
6. Isoperimetric Problems 491
7. Variational Notation 493
8. Miscellaneous Problems 494

10 TENSOR ANALYSIS 496

1. Introduction 496
2. Cartesian Tensors 498
3. Tensor Notation and Operations 502
4. Inertia Tensor 505
5. Kronecker Delta and Levi-Civita Symbol 508
6. Pseudovectors and Pseudotensors 514
7. More About Applications 518
8. Curvilinear Coordinates 521
9. Vector Operators in Orthogonal Curvilinear Coordinates 525
10. Non-Cartesian Tensors 529
11. Miscellaneous Problems 535

11 SPECIAL FUNCTIONS 537

1. Introduction 537
2. The Factorial Function 538
3. Definition of the Gamma Function; Recursion Relation 538
4. The Gamma Function of Negative Numbers 540
5. Some Important Formulas Involving Gamma Functions 541
6. Beta Functions 542
7. Beta Functions in Terms of Gamma Functions 543
8. The Simple Pendulum 545
9. The Error Function 547
10. Asymptotic Series 549
11. Stirling’s Formula 552
12. Elliptic Integrals and Functions 554
13. Miscellaneous Problems 560

12 SERIES SOLUTIONS OF DIFFERENTIAL EQUATIONS; LEGENDRE, BESSEL, HERMITE, AND LAGUERRE FUNCTIONS 562

1. Introduction 562
2. Legendre’s Equation 564
3. Leibniz’ Rule for Differentiating Products 567
4. Rodrigues’ Formula 568
5. Generating Function for Legendre Polynomials 569
6. Complete Sets of Orthogonal Functions 575
7. Orthogonality of the Legendre Polynomials 577
8. Normalization of the Legendre Polynomials 578
9. Legendre Series 580
10. The Associated Legendre Functions 583
11. Generalized Power Series or the Method of Frobenius 585
12. Bessel’s Equation 587
13. The Second Solution of Bessel’s Equation 590
14. Graphs and Zeros of Bessel Functions 591
15. Recursion Relations 592
16. Differential Equations with Bessel Function Solutions 593
17. Other Kinds of Bessel Functions 595
18. The Lengthening Pendulum 598
19. Orthogonality of Bessel Functions 601
20. Approximate Formulas for Bessel Functions 604
21. Series Solutions; Fuchs’s Theorem 605
22. Hermite Functions; Laguerre Functions; Ladder Operators 607
23. Miscellaneous Problems 615

13 PARTIAL DIFFERENTIAL EQUATIONS 619

1. Introduction 619
2. Laplace’s Equation; Steady-State Temperature in a Rectangular Plate 621
3. The Diffusion or Heat Flow Equation; the Schr ̈odinger Equation 628
4. The Wave Equation; the Vibrating String 633
5. Steady-state Temperature in a Cylinder 638
6. Vibration of a Circular Membrane 644
7. Steady-state Temperature in a Sphere 647
8. Poisson’s Equation 652
9. Integral Transform Solutions of Partial Differential Equations 659
10. Miscellaneous Problems 663

14 FUNCTIONS OF A COMPLEX VARIABLE 666

1. Introduction 666
2. Analytic Functions 667
3. Contour Integrals 674
4. Laurent Series 678
5. The Residue Theorem 682
6. Methods of Finding Residues 683
7. Evaluation of Definite Integrals by Use of the Residue Theorem 687
8. The Point at Infinity; Residues at Infinity 702
9. Mapping 705
10. Some Applications of Conformal Mapping 710
11. Miscellaneous Problems 718

15 PROBABILITY AND STATISTICS 722

1. Introduction 722
2. Sample Space 724
3. Probability Theorems 729
4. Methods of Counting 736
5. Random Variables 744
6. Continuous Distributions 750
7. Binomial Distribution 756
8. The Normal or Gaussian Distribution 761
9. The Poisson Distribution 767
10. Statistics and Experimental Measurements 770
11. Miscellaneous Problems 776

REFERENCES 779

ANSWERS TO SELECTED PROBLEMS 781

INDEX 811