Introduction to Methods of Applied Mathematicsor

Advanced Mathematical Methods for Scientists and Engineers

Sean Mauch

April 26, 2001

Contents

1 Anti-Copyright 22

2 Preface 232.1 Advice to Teachers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3 Warnings and Disclaimers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.4 Suggested Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.5 About the Title . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

I Algebra 1

3 Sets and Functions 23.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.2 Single Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.3 Inverses and Multi-Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.4 Transforming Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Vectors 114.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1

4.1.1 Scalars and Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.1.2 The Kronecker Delta and Einstein Summation Convention . . . . . . . . . . . . . . . . . . 144.1.3 The Dot and Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2 Sets of Vectors in n Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.4 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

II Calculus 36

5 Differential Calculus 375.1 Limits of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.2 Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.3 The Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.4 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.5 Maxima and Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.6 Mean Value Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.6.1 Application: Using Taylor’s Theorem to Approximate Functions. . . . . . . . . . . . . . . . 575.6.2 Application: Finite Difference Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.7 L’Hospital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.10 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6 Integral Calculus 1006.1 The Indefinite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006.2 The Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066.2.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

2

6.3 The Fundamental Theorem of Integral Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.4 Techniques of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.4.1 Partial Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.5 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1186.7 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

7 Vector Calculus 1347.1 Vector Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1347.2 Gradient, Divergence and Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1357.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1427.4 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1447.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

III Functions of a Complex Variable 150

8 Complex Numbers 1518.1 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1518.2 The Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1548.3 Polar Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1588.4 Arithmetic and Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1638.5 Integer Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1648.6 Rational Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1668.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1708.8 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1768.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

3

9 Functions of a Complex Variable 2029.1 Curves and Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2029.2 Cartesian and Modulus-Argument Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2069.3 Graphing Functions of a Complex Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2089.4 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2129.5 Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2179.6 Branch Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2269.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2439.8 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2539.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

10 Analytic Functions 30310.1 Complex Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30310.2 Cauchy-Riemann Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31010.3 Harmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31510.4 Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320

10.4.1 Categorization of Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32110.4.2 Isolated and Non-Isolated Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

10.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32710.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33210.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

11 Analytic Continuation 35611.1 Analytic Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35611.2 Analytic Continuation of Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35911.3 Analytic Functions Defined in Terms of Real Variables . . . . . . . . . . . . . . . . . . . . . . . . 360

11.3.1 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36611.3.2 Analytic Functions Defined in Terms of Their Real or Imaginary Parts . . . . . . . . . . . 369

11.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37311.5 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

4

11.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376

12 Contour Integration and Cauchy’s Theorem 38112.1 Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38112.2 Under Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38612.3 Cauchy’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38812.4 Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39012.5 Contour Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39112.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39512.7 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39712.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

13 Cauchy’s Integral Formula 40313.1 Cauchy’s Integral Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40413.2 The Argument Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41113.3 Rouche’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41313.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41513.5 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41713.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418

14 Series and Convergence 42214.1 Series of Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422

14.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42214.1.2 Special Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42514.1.3 Convergence Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426

14.2 Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43214.2.1 Tests for Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43314.2.2 Uniform Convergence and Continuous Functions. . . . . . . . . . . . . . . . . . . . . . . . 435

14.3 Uniformly Convergent Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43614.4 Integration and Differentiation of Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443

5

14.5 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44614.5.1 Newton’s Binomial Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

14.6 Laurent Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45214.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45514.8 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46214.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465

15 The Residue Theorem 49015.1 The Residue Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49015.2 Cauchy Principal Value for Real Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498

15.2.1 The Cauchy Principal Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49815.3 Cauchy Principal Value for Contour Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50315.4 Integrals on the Real Axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50715.5 Fourier Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51215.6 Fourier Cosine and Sine Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51515.7 Contour Integration and Branch Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51715.8 Exploiting Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

15.8.1 Wedge Contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52115.8.2 Box Contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524

15.9 Definite Integrals Involving Sine and Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52515.10Infinite Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52815.11Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53215.12Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54615.13Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553

IV Ordinary Differential Equations 634

16 First Order Differential Equations 63516.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635

6

16.2 One Parameter Families of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63716.3 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639

16.3.1 Separable Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64316.3.2 Homogeneous Coefficient Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645

16.4 The First Order, Linear Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64916.4.1 Homogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64916.4.2 Inhomogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65016.4.3 Variation of Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651

16.5 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65316.5.1 Piecewise Continuous Coefficients and Inhomogeneities . . . . . . . . . . . . . . . . . . . . 654

16.6 Well-Posed Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65916.7 Equations in the Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661

16.7.1 Ordinary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66116.7.2 Regular Singular Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66416.7.3 Irregular Singular Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66916.7.4 The Point at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671

16.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67416.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68016.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683

17 First Order Systems of Differential Equations 70517.1 Matrices and Jordan Canonical Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70517.2 Systems of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71317.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71917.4 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72517.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727

18 Theory of Linear Ordinary Differential Equations 75718.1 Nature of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75818.2 Transformation to a First Order System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761

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18.3 The Wronskian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76218.3.1 Derivative of a Determinant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76218.3.2 The Wronskian of a Set of Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76318.3.3 The Wronskian of the Solutions to a Differential Equation . . . . . . . . . . . . . . . . . . 765

18.4 Well-Posed Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76818.5 The Fundamental Set of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77018.6 Adjoint Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77318.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77618.8 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77818.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 780

19 Techniques for Linear Differential Equations 78619.1 Constant Coefficient Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786

19.1.1 Second Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78719.1.2 Higher Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79119.1.3 Real-Valued Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793

19.2 Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79519.2.1 Real-Valued Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798

19.3 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80119.4 Equations Without Explicit Dependence on y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80219.5 Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80319.6 *Reduction of Order and the Adjoint Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80419.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80719.8 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81419.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817

20 Techniques for Nonlinear Differential Equations 84220.1 Bernoulli Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84220.2 Riccati Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84420.3 Exchanging the Dependent and Independent Variables . . . . . . . . . . . . . . . . . . . . . . . . 848

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20.4 Autonomous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85020.5 *Equidimensional-in-x Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85420.6 *Equidimensional-in-y Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85620.7 *Scale-Invariant Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85920.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86020.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86420.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866

21 Transformations and Canonical Forms 87821.1 The Constant Coefficient Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87821.2 Normal Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 881

21.2.1 Second Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88121.2.2 Higher Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883

21.3 Transformations of the Independent Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88521.3.1 Transformation to the form u” + a(x) u = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 88521.3.2 Transformation to a Constant Coefficient Equation . . . . . . . . . . . . . . . . . . . . . . 886

21.4 Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88821.4.1 Initial Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88921.4.2 Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 891

21.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89421.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89621.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897

22 The Dirac Delta Function 90422.1 Derivative of the Heaviside Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90422.2 The Delta Function as a Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90622.3 Higher Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90822.4 Non-Rectangular Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90822.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91022.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 911

9

22.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 912

23 Inhomogeneous Differential Equations 91523.1 Particular Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91523.2 Method of Undetermined Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91723.3 Variation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 921

23.3.1 Second Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92123.3.2 Higher Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925

23.4 Piecewise Continuous Coefficients and Inhomogeneities . . . . . . . . . . . . . . . . . . . . . . . . 92723.5 Inhomogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 931

23.5.1 Eliminating Inhomogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 93123.5.2 Separating Inhomogeneous Equations and Inhomogeneous Boundary Conditions . . . . . . 93323.5.3 Existence of Solutions of Problems with Inhomogeneous Boundary Conditions . . . . . . . 934

23.6 Green Functions for First Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93623.7 Green Functions for Second Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 939

23.7.1 Green Functions for Sturm-Liouville Problems . . . . . . . . . . . . . . . . . . . . . . . . . 94923.7.2 Initial Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95323.7.3 Problems with Unmixed Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 95623.7.4 Problems with Mixed Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 958

23.8 Green Functions for Higher Order Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96223.9 Fredholm Alternative Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96823.10Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97523.11Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98223.12Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986

24 Difference Equations 102724.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102724.2 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102924.3 Homogeneous First Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103024.4 Inhomogeneous First Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1032

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24.5 Homogeneous Constant Coefficient Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103524.6 Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103824.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104024.8 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104124.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1042

25 Series Solutions of Differential Equations 104625.1 Ordinary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046

25.1.1 Taylor Series Expansion for a Second Order Differential Equation . . . . . . . . . . . . . . 105125.2 Regular Singular Points of Second Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 1060

25.2.1 Indicial Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106325.2.2 The Case: Double Root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106525.2.3 The Case: Roots Differ by an Integer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069

25.3 Irregular Singular Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107925.4 The Point at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107925.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108225.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108725.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1089

26 Asymptotic Expansions 111326.1 Asymptotic Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111326.2 Leading Order Behavior of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 111726.3 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112626.4 Asymptotic Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113326.5 Asymptotic Expansions of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135

26.5.1 The Parabolic Cylinder Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135

27 Hilbert Spaces 114127.1 Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114127.2 Inner Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1143

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27.3 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114427.4 Linear Independence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114727.5 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114727.6 Gramm-Schmidt Orthogonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114727.7 Orthonormal Function Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115127.8 Sets Of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115227.9 Least Squares Fit to a Function and Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . 115827.10Closure Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116127.11Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116627.12Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116727.13Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116827.14Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1169

28 Self Adjoint Linear Operators 117128.1 Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117128.2 Self-Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117228.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117528.4 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117628.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1177

29 Self-Adjoint Boundary Value Problems 117829.1 Summary of Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117829.2 Formally Self-Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117929.3 Self-Adjoint Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118229.4 Self-Adjoint Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118329.5 Inhomogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118829.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119129.7 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119229.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1193

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30 Fourier Series 119530.1 An Eigenvalue Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119530.2 Fourier Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119830.3 Least Squares Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120430.4 Fourier Series for Functions Defined on Arbitrary Ranges . . . . . . . . . . . . . . . . . . . . . . . 120730.5 Fourier Cosine Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121030.6 Fourier Sine Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121130.7 Complex Fourier Series and Parseval’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121230.8 Behavior of Fourier Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121530.9 Gibb’s Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122430.10Integrating and Differentiating Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122430.11Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122930.12Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123830.13Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1241

31 Regular Sturm-Liouville Problems 129131.1 Derivation of the Sturm-Liouville Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129131.2 Properties of Regular Sturm-Liouville Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129431.3 Solving Differential Equations With Eigenfunction Expansions . . . . . . . . . . . . . . . . . . . . 130531.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131131.5 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131531.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1317

32 Integrals and Convergence 134232.1 Uniform Convergence of Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134232.2 The Riemann-Lebesgue Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134332.3 Cauchy Principal Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1345

32.3.1 Integrals on an Infinite Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134532.3.2 Singular Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1345

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33 The Laplace Transform 134733.1 The Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134733.2 The Inverse Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1349

33.2.1 F(s) with Poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135233.2.2 f̂(s) with Branch Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135733.2.3 Asymptotic Behavior of F(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1360

33.3 Properties of the Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136233.4 Constant Coefficient Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136633.5 Systems of Constant Coefficient Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . 136833.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137033.7 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137833.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1382

34 The Fourier Transform 141534.1 Derivation from a Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141534.2 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1417

34.2.1 A Word of Caution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142034.3 Evaluating Fourier Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1421

34.3.1 Integrals that Converge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142134.3.2 Cauchy Principal Value and Integrals that are Not Absolutely Convergent. . . . . . . . . . 142434.3.3 Analytic Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1426

34.4 Properties of the Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142834.4.1 Closure Relation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142934.4.2 Fourier Transform of a Derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143034.4.3 Fourier Convolution Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143134.4.4 Parseval’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143534.4.5 Shift Property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143634.4.6 Fourier Transform of x f(x). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1437

34.5 Solving Differential Equations with the Fourier Transform . . . . . . . . . . . . . . . . . . . . . . 143734.6 The Fourier Cosine and Sine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1440

14

34.6.1 The Fourier Cosine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144034.6.2 The Fourier Sine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1441

34.7 Properties of the Fourier Cosine and Sine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 144234.7.1 Transforms of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144234.7.2 Convolution Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144434.7.3 Cosine and Sine Transform in Terms of the Fourier Transform . . . . . . . . . . . . . . . . 1446

34.8 Solving Differential Equations with the Fourier Cosine and Sine Transforms . . . . . . . . . . . . . 144734.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144934.10Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145534.11Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1458

35 The Gamma Function 148435.1 Euler’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148435.2 Hankel’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148635.3 Gauss’ Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148835.4 Weierstrass’ Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149035.5 Stirling’s Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149235.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149735.7 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149835.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1499

36 Bessel Functions 150136.1 Bessel’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150136.2 Frobeneius Series Solution about z = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1502

36.2.1 Behavior at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150536.3 Bessel Functions of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1507

36.3.1 The Bessel Function Satisfies Bessel’s Equation . . . . . . . . . . . . . . . . . . . . . . . . 150836.3.2 Series Expansion of the Bessel Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150936.3.3 Bessel Functions of Non-Integral Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151236.3.4 Recursion Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1515

15

36.3.5 Bessel Functions of Half-Integral Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151836.4 Neumann Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151936.5 Bessel Functions of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152336.6 Hankel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152536.7 The Modified Bessel Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152536.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152936.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153436.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1536

V Partial Differential Equations 1559

37 Transforming Equations 156037.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156137.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156237.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1563

38 Classification of Partial Differential Equations 156438.1 Classification of Second Order Quasi-Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . 1564

38.1.1 Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156538.1.2 Parabolic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157038.1.3 Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1570

38.2 Equilibrium Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157238.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157438.4 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157538.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1576

39 Separation of Variables 158039.1 Eigensolutions of Homogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158039.2 Homogeneous Equations with Homogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . 1580

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39.3 Time-Independent Sources and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 158239.4 Inhomogeneous Equations with Homogeneous Boundary Conditions . . . . . . . . . . . . . . . . . 158539.5 Inhomogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158739.6 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158939.7 General Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159339.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159439.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160839.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1613

40 Finite Transforms 169040.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169440.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169540.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1696

41 Waves 170141.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170241.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170841.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1710

42 The Diffusion Equation 172742.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172842.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173042.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1731

43 Similarity Methods 173443.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173943.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174043.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1741

17

44 Method of Characteristics 174344.1 The Method of Characteristics and the Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . 174344.2 The Method of Characteristics for an Infinite Domain . . . . . . . . . . . . . . . . . . . . . . . . . 174544.3 The Method of Characteristics for a Semi-Infinite Domain . . . . . . . . . . . . . . . . . . . . . . 174644.4 Envelopes of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174744.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175044.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175244.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1753

45 Transform Methods 175945.1 Fourier Transform for Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 175945.2 The Fourier Sine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176145.3 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176245.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176345.5 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176845.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1771

46 Green Functions 179446.1 Inhomogeneous Equations and Homogeneous Boundary Conditions . . . . . . . . . . . . . . . . . 179446.2 Homogeneous Equations and Inhomogeneous Boundary Conditions . . . . . . . . . . . . . . . . . 179546.3 Eigenfunction Expansions for Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179746.4 The Method of Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180246.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180346.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180846.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1810

47 Conformal Mapping 185147.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185247.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185547.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1856

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48 Non-Cartesian Coordinates 186448.1 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186448.2 Laplace’s Equation in a Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186548.3 Laplace’s Equation in an Annulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1868

VI Calculus of Variations 1872

49 Calculus of Variations 187349.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187449.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189149.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1897

VII Nonlinear Differential Equations 1990

50 Nonlinear Ordinary Differential Equations 199150.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199250.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199750.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1999

51 Nonlinear Partial Differential Equations 202151.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202251.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202551.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2026

VIII Appendices 2044

A Greek Letters 2045

19

B Notation 2047

C Formulas from Complex Variables 2049

D Table of Derivatives 2052

E Table of Integrals 2056

F Definite Integrals 2060

G Table of Sums 2063

H Table of Taylor Series 2066

I Table of Laplace Transforms 2069

J Table of Fourier Transforms 2074

K Table of Fourier Transforms in n Dimensions 2077

L Table of Fourier Cosine Transforms 2078

M Table of Fourier Sine Transforms 2080

N Table of Wronskians 2082

O Sturm-Liouville Eigenvalue Problems 2084

P Green Functions for Ordinary Differential Equations 2086

Q Trigonometric Identities 2089Q.1 Circular Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2089

20

Q.2 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2091

R Bessel Functions 2094R.1 Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2094

S Formulas from Linear Algebra 2095

T Vector Analysis 2097

U Partial Fractions 2099

V Finite Math 2103

W Probability 2104W.1 Independent Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2104W.2 Playing the Odds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2105

X Economics 2106

Y Glossary 2107

21

Chapter 1

Anti-Copyright

Anti-Copyright @ 1995-2000 by Mauch Publishing Company, un-Incorporated.

No rights reserved. Any part of this publication by be reproduced, stored in a retrieval system, transmitted ordesecrated without permission.

22

Chapter 2

Preface

During the summer before my final undergraduate year at Caltech I set out to write a math text unlike anyother, namely, one written by me. In that respect I have succeeded beautifully. Unfortunately, the text is neithercomplete nor polished. I have a “Warnings and Disclaimers” section below that is a little amusing, and anappendix on probability that I feel concisesly captures the essence of the subject. However, all the material inbetween is in some stage of development. I am currently working to improve and expand this text.

This text is freely available from my web set. Currently I’m at http://www.its.caltech.edu/˜sean. I postnew versions a couple of times a year.

2.1 Advice to Teachers

If you have something worth saying, write it down.

2.2 Acknowledgments

I would like to thank Professor Saffman for advising me on this project and the Caltech SURF program forproviding the funding for me to write the first edition of this book.

23

http://www.its.caltech.edu/~sean

2.3 Warnings and Disclaimers

• This book is a work in progress. It contains quite a few mistakes and typos. I would greatly appreciateyour constructive criticism. You can reach me at ‘sean@its.caltech.edu’.

• Reading this book impairs your ability to drive a car or operate machinery.

• This book has been found to cause drowsiness in laboratory animals.

• This book contains twenty-three times the US RDA of fiber.

• Caution: FLAMMABLE - Do not read while smoking or near a fire.

• If infection, rash, or irritation develops, discontinue use and consult a physician.

• Warning: For external use only. Use only as directed. Intentional misuse by deliberately concentratingcontents can be harmful or fatal. KEEP OUT OF REACH OF CHILDREN.

• In the unlikely event of a water landing do not use this book as a flotation device.

• The material in this text is fiction; any resemblance to real theorems, living or dead, is purely coincidental.

• This is by far the most amusing section of this book.

• Finding the typos and mistakes in this book is left as an exercise for the reader. (Eye ewes a spellingchequer from thyme too thyme, sew their should knot bee two many misspellings. Though I ain’t so surethe grammar’s too good.)

• The theorems and methods in this text are subject to change without notice.

• This is a chain book. If you do not make seven copies and distribute them to your friends within ten daysof obtaining this text you will suffer great misfortune and other nastiness.

• The surgeon general has determined that excessive studying is detrimental to your social life.

24

• This text has been buffered for your protection and ribbed for your pleasure.

• Stop reading this rubbish and get back to work!

2.4 Suggested Use

This text is well suited to the student, professional or lay-person. It makes a superb gift. This text has a boquetthat is light and fruity, with some earthy undertones. It is ideal with dinner or as an apertif. Bon apetit!

2.5 About the Title

The title is only making light of naming conventions in the sciences and is not an insult to engineers. If you want tofind a good math text to learn a subject, look for books with “Introduction” and “Elementary” in the title. If it isan “Intermediate” text it will be incomprehensible. If it is “Advanced” then not only will it be incomprehensible,it will have low production qualities, i.e. a crappy typewriter font, no graphics and no examples. There is anexception to this rule when the title also contains the word “Scientists” or “Engineers”. Then an advanced bookmay be quite suitable for actually learning the material.

25

Part I

Algebra

1

Chapter 3

Sets and Functions

3.1 Sets

Definition. A set is a collection of objects. We call the objects, elements. A set is denoted by listing theelements between braces. For example: {e, i, π, 1}. We use ellipses to indicate patterns. The set of positiveintegers is {1, 2, 3, . . . }. We also denote a sets with the notation {x|conditions on x} for sets that are more easilydescribed than enumerated. This is read as “the set of elements x such that x satisfies . . . ”. x ∈ S is the notationfor “x is an element of the set S.” To express the opposite we have x 6∈ S for “x is not an element of the set S.”

Examples. We have notations for denoting some of the commonly encountered sets.

• ∅ = {} is the empty set, the set containing no elements.

• Z = {. . . ,−1, 0, 1 . . . } is the set of integers. (Z is for “Zahlen”, the German word for “number”.)

• Q = {p/q|p, q ∈ Z, q 6= 0} is the set of rational numbers. (Q is for quotient.)

• R = {x|x = a1a2 · · · an.b1b2 · · · } is the set of real numbers, i.e. the set of numbers with decimal expansions.

2

• C = {a + ib|a, b ∈ R, i2 = −1} is the set of complex numbers. i is the square root of −1. (If you haven’tseen complex numbers before, don’t dismay. We’ll cover them later.)

• Z+, Q+ and R+ are the sets of positive integers, rationals and reals, respectively. For example, Z+ ={1, 2, 3, . . . }.

• Z0+, Q0+ and R0+ are the sets of non-negative integers, rationals and reals, respectively. For example,Z0+ = {0, 1, 2, . . . }.

• (a . . . b) denotes an open interval on the real axis. (a . . . b) ≡ {x|x ∈ R, a < x < b}

• We use brackets to denote the closed interval. [a . . . b] ≡ {x|x ∈ R, a ≤ x ≤ b}

The cardinality or order of a set S is denoted |S|. For finite sets, the cardinality is the number of elementsin the set. The Cartesian product of two sets is the set of ordered pairs:

X × Y ≡ {(x, y)|x ∈ X, y ∈ Y }.

The Cartesian product of n sets is the set of ordered n-tuples:

X1 ×X2 × · · · ×Xn ≡ {(x1, x2, . . . , xn)|x1 ∈ X1, x2 ∈ X2, . . . , xn ∈ Xn}.

Equality. Two sets S and T are equal if each element of S is an element of T and vice versa. This is denoted,S = T . Inequality is S 6= T , of course. S is a subset of T , S ⊆ T , if every element of S is an element of T . S isa proper subset of T , S ⊂ T , if S ⊆ T and S 6= T . For example: The empty set is a subset of every set, ∅ ⊆ S.The rational numbers are a proper subset of the real numbers, Q ⊂ R.

Operations. The union of two sets, S ∪ T , is the set whose elements are in either of the two sets. The unionof n sets,

∪nj=1Sj ≡ S1 ∪ S2 ∪ · · · ∪ Sn

3

is the set whose elements are in any of the sets Sj. The intersection of two sets, S ∩ T , is the set whose elementsare in both of the two sets. In other words, the intersection of two sets in the set of elements that the two setshave in common. The intersection of n sets,

∩nj=1Sj ≡ S1 ∩ S2 ∩ · · · ∩ Sn

is the set whose elements are in all of the sets Sj. If two sets have no elements in common, S ∩ T = ∅, then thesets are disjoint. If T ⊆ S, then the difference between S and T , S \ T , is the set of elements in S which are notin T .

S \ T ≡ {x|x ∈ S, x 6∈ T}

The difference of sets is also denoted S − T .

Properties. The following properties are easily verified from the above definitions.

• S ∪ ∅ = S, S ∩ ∅ = ∅, S \ ∅ = S, S \ S = ∅.

• Commutative. S ∪ T = T ∪ S, S ∩ T = T ∩ S.

• Associative. (S ∪ T ) ∪ U = S ∪ (T ∪ U) = S ∪ T ∪ U , (S ∩ T ) ∩ U = S ∩ (T ∩ U) = S ∩ T ∩ U .

• Distributive. S ∪ (T ∩ U) = (S ∪ T ) ∩ (S ∪ U), S ∩ (T ∪ U) = (S ∩ T ) ∪ (S ∩ U).

3.2 Single Valued Functions

Single-Valued Functions. A single-valued function or single-valued mapping is a mapping of the elements

x ∈ X into elements y ∈ Y . This is expressed notationally as f : X → Y or X f→ Y . If such a function iswell-defined, then for each x ∈ X there exists a unique element of y such that f(x) = y. The set X is the domainof the function, Y is the codomain, (not to be confused with the range, which we introduce shortly). To denotethe value of a function on a particular element we can use any of the notations: f(x) = y, f : x 7→ y or simplyx 7→ y. f is the identity map on X if f(x) = x for all x ∈ X.

4

Let f : X → Y . The range or image of f is

f(X) = {y|y = f(x) for some x ∈ X}.

The range is a subset of the codomain. For each Z ⊆ Y , the inverse image of Z is defined:

f−1(Z) ≡ {x ∈ X|f(x) = z for some z ∈ Z}.

Examples.

• Finite polynomials and the exponential function are examples of single valued functions which map realnumbers to real numbers.

• The greatest integer function, b·c, is a mapping from R to Z. bxc in the greatest integer less than or equalto x. Likewise, the least integer function, dxe, is the least integer greater than or equal to x.

The -jectives. A function is injective if for each x1 6= x2, f(x1) 6= f(x2). In other words, for each x in thedomain there is a unique y = f(x) in the range. f is surjective if for each y in the codomain, there is an x suchthat y = f(x). If a function is both injective and surjective, then it is bijective. A bijective function is also calleda one-to-one mapping.

Examples.

• The exponential function y = ex is a bijective function, (one-to-one mapping), that maps R to R+. (R isthe set of real numbers; R+ is the set of positive real numbers.)

• f(x) = x2 is a bijection from R+ to R+. f is not injective from R to R+. For each positive y in the range,there are two values of x such that y = x2.

• f(x) = sinx is not injective from R to [−1..1]. For each y ∈ [−1, 1] there exists an infinite number of valuesof x such that y = sinx.

5

Injective Surjective Bijective

Figure 3.1: Depictions of Injective, Surjective and Bijective Functions

3.3 Inverses and Multi-Valued Functions

If y = f(x), then we can write x = f−1(y) where f−1 is the inverse of f . If y = f(x) is a one-to-one function,then f−1(y) is also a one-to-one function. In this case, x = f−1(f(x)) = f(f−1(x)) for values of x where bothf(x) and f−1(x) are defined. For example log x, which maps R+ to R is the inverse of ex. x = elog x = log( ex)for all x ∈ R+. (Note the x ∈ R+ ensures that log x is defined.)

If y = f(x) is a many-to-one function, then x = f−1(y) is a one-to-many function. f−1(y) is a multi-valuedfunction. We have x = f(f−1(x)) for values of x where f−1(x) is defined, however x 6= f−1(f(x)). There arediagrams showing one-to-one, many-to-one and one-to-many functions in Figure 3.2.

Example 3.3.1 y = x2, a many-to-one function has the inverse x = y1/2. For each positive y, there are twovalues of x such that x = y1/2. y = x2 and y = x1/2 are graphed in Figure 3.3.

We say that there are two branches of y = x1/2: the positive and the negative branch. We denote the positivebranch as y =

√x; the negative branch is y = −

√x. We call

√x the principal branch of x1/2. Note that

√x

is a one-to-one function. Finally, x = (x1/2)2 since (±√x)2 = x, but x 6= (x2)1/2 since (x2)1/2 = ±x. y =

√x is

6

rangedomain rangedomain rangedomain

one-to-one many-to-one one-to-many

Figure 3.2: Diagrams of One-To-One, Many-To-One and One-To-Many Functions

Figure 3.3: y = x2 and y = x1/2

graphed in Figure 3.4.

Figure 3.4: y =√x

7

Now consider the many-to-one function y = sinx. The inverse is x = arcsin y. For each y ∈ [−1, 1] there arean infinite number of values x such that x = arcsin y. In Figure 3.5 is a graph of y = sinx and a graph of a fewbranches of y = arcsinx.

Figure 3.5: y = sinx and y = arcsinx

Example 3.3.2 arcsinx has an infinite number of branches. We will denote the principal branch by Arcsinxwhich maps [−1, 1] to

[−π

2, π

2

]. Note that x = sin(arcsinx), but x 6= arcsin(sinx). y = Arcsinx in Figure 3.6.

Figure 3.6: y = Arcsinx

Example 3.3.3 Consider 11/3. Since x3 is a one-to-one function, x1/3 is a single-valued function. (See Figure 3.7.)11/3 = 1.

8

Figure 3.7: y = x3 and y = x1/3

Example 3.3.4 Consider arccos(1/2). cosx and a few branches of arccos x are graphed in Figure 3.8. cos x = 1/2

Figure 3.8: y = cos x and y = arccos x

has the two solutions x = ±π/3 in the range x ∈ [−π, π]. Since cos(x+ π) = − cos x,

arccos(1/2) = {±π/3 + nπ}.

3.4 Transforming Equations

We must take care in applying functions to equations. It is always safe to apply a one-to-one function to anequation, (provided it is defined for that domain). For example, we can apply y = x3 or y = ex to the equationx = 1. The equations x3 = 1 and ex = e have the unique solution x = 1.

9

If we apply a many-to-one function to an equation, we may introduce spurious solutions. Applying y = x2 andy = sinx to the equation x = π

2results in x2 = π

2

4and sinx = 1. The former equation has the two solutions

x = ±π2; the latter has the infinite number of solutions x = π

2+ 2nπ, n ∈ Z.

We do not generally apply a one-to-many function to both sides of an equation as this rarely is useful. Considerthe equation

sin2 x = 1.

Applying the function f(x) = x1/2 to the equation would not get us anywhere

(sin2 x)1/2 = 11/2.

Since (sin2 x)1/2 6= sinx, we cannot simplify the left side of the equation. Instead we could use the definition off(x) = x1/2 as the inverse of the x2 function to obtain

sinx = 11/2 = ±1.

Then we could use the definition of arcsin as the inverse of sin to get

x = arcsin(±1).

x = arcsin(1) has the solutions x = π/2 + 2nπ and x = arcsin(−1) has the solutions x = −π/2 + 2nπ. Thus

x =π

2+ nπ, n ∈ Z.

Note that we cannot just apply arcsin to both sides of the equation as arcsin(sinx) 6= x.

10

Chapter 4

Vectors

4.1 Vectors

4.1.1 Scalars and Vectors

A vector is a quantity having both a magnitude and a direction. Examples of vector quantities are velocity, forceand position. One can represent a vector in n-dimensional space with an arrow whose initial point is at the origin,(Figure 4.1). The magnitude is the length of the vector. Typographically, variables representing vectors are oftenwritten in capital letters, bold face or with a vector over-line, A, a,~a. The magnitude of a vector is denoted |a|.

A scalar has only a magnitude. Examples of scalar quantities are mass, time and speed.

Vector Algebra. Two vectors are equal if they have the same magnitude and direction. The negative of avector, denoted −a, is a vector of the same magnitude as a but in the opposite direction. We add two vectors aand b by placing the tail of b at the head of a and defining a + b to be the vector with tail at the origin andhead at the head of b. (See Figure 4.2.)

The difference, a − b, is defined as the sum of a and the negative of b, a + (−b). The result of multiplyinga by a scalar α is a vector of magnitude |α| |a| with the same/opposite direction if α is positive/negative. (SeeFigure 4.2.)

11

x

z

y

Figure 4.1: Graphical Representation of a Vector in Three Dimensions

a+b

a

b

-a

a2a

Figure 4.2: Vector Arithmetic

Here are the properties of adding vectors and multiplying them by a scalar. They are evident from geometricconsiderations.

a + b = b + a αa = aα commutative laws

(a + b) + c = a + (b + c) α(βa) = (αβ)a associative laws

α(a + b) = αa + αb (α + β)a = αa + βa distributive laws

12

Zero and Unit Vectors. The additive identity element for vectors is the zero vector or null vector. This is avector of magnitude zero which is denoted as 0. A unit vector is a vector of magnitude one. If a is nonzero thena/|a| is a unit vector in the direction of a. Unit vectors are often denoted with a caret over-line, n̂.

Rectangular Unit Vectors. In n dimensional Cartesian space, Rn, the unit vectors in the directions of thecoordinates axes are e1, . . . en. These are called the rectangular unit vectors. To cut down on subscripts, the unitvectors in three dimensional space are often denoted with i, j and k. (Figure 4.3).

x

z

yj

k

i

Figure 4.3: Rectangular Unit Vectors

Components of a Vector. Consider a vector a with tail at the origin and head having the Cartesian coordinates(a1, . . . , an). We can represent this vector as the sum of n rectangular component vectors, a = a1e1 + · · ·+ anen.(See Figure 4.4.) Another notation for the vector a is 〈a1, . . . , an〉. By the Pythagorean theorem, the magnitudeof the vector a is |a| =

√a21 + · · ·+ a2n.

13

x

z

y

a

a

a

1

3

i

k

ja2

Figure 4.4: Components of a Vector

4.1.2 The Kronecker Delta and Einstein Summation Convention

The Kronecker Delta tensor is defined

δij =

{1 if i = j,

0 if i 6= j.

This notation will be useful in our work with vectors.

Consider writing a vector in terms of its rectangular components. Instead of using ellipses: a = a1e1+· · ·+anen,we could write the expression as a sum: a =

∑ni=1 aiei. We can shorten this notation by leaving out the sum:

a = aiei, where it is understood that whenever an index is repeated in a term we sum over that index from 1 ton. This is the Einstein summation convention. A repeated index is called a summation index or a dummy index.Other indices can take any value from 1 to n and are called free indices.

14

Example 4.1.1 Consider the matrix equation: A · x = b. We can write out the matrix and vectors explicitly.a11 · · · a1n... . . . ...an1 · · · ann

x1...xn

=b1...bn

This takes much less space when we use the summation convention.

aijxj = bi

Here j is a summation index and i is a free index.

4.1.3 The Dot and Cross Product

Dot Product. The dot product or scalar product of two vectors is defined,

a · b ≡ |a||b| cos θ,

where θ is the angle from a to b. From this definition one can derive the following properties:

• a · b = b · a, commutative.

• α(a · b) = (αa) · b = a · (αb), associativity of scalar multiplication.

• a · (b + c) = a · b + a · c, distributive.

• eiej = δij. In three dimension, this is

i · i = j · j = k · k = 1, i · j = j · k = k · i = 0.

• a · b = aibi ≡ a1b1 + · · ·+ anbn, dot product in terms of rectangular components.

• If a · b = 0 then either a and b are orthogonal, (perpendicular), or one of a and b are zero.

15

The Angle Between Two Vectors. We can use the dot product to find the angle between two vectors, a andb. From the definition of the dot product,

a · b = |a||b| cos θ.

If the vectors are nonzero, then

θ = arccos

(a · b|a||b|

).

Example 4.1.2 What is the angle between i and i + j?

θ = arccos

(i · (i + j)|i||i + j|

)= arccos

(1√2

)=π

4.

Parametric Equation of a Line. Consider a line that passes through the point a and is parallel to the vectort, (tangent). A parametric equation of the line is

x = a + ut, u ∈ R.

Implicit Equation of a Line. Consider a line that passes through the point a and is normal, (orthogonal,perpendicular), to the vector n. All the lines that are normal to n have the property that x · n is a constant,where x is any point on the line. (See Figure 4.5.) x · n = 0 is the line that is normal to n and passes throughthe origin. The line that is normal to n and passes through the point a is

x · n = a · n.

16

=0

=1 = a n

n a

=-1

x n

x n

x n

x n

Figure 4.5: Equation for a Line

The normal to a line determines an orientation of the line. The normal points in the direction that is abovethe line. A point b is (above/on/below) the line if (b− a) · n is (positive/zero/negative). The signed distance ofa point b from the line x · n = a · n is

(b− a) · n|n|

.

Implicit Equation of a Hyperplane. A hyperplane in Rn is an n − 1 dimensional “sheet” which passesthrough a given point and is normal to a given direction. In R3 we call this a plane. Consider a hyperplane thatpasses through the point a and is normal to the vector n. All the hyperplanes that are normal to n have theproperty that x · n is a constant, where x is any point in the hyperplane. x · n = 0 is the hyperplane that isnormal to n and passes through the origin. The hyperplane that is normal to n and passes through the point a is

x · n = a · n.

The normal determines an orientation of the hyperplane. The normal points in the direction that is above thehyperplane. A point b is (above/on/below) the hyperplane if (b− a) · n is (positive/zero/negative). The signed

17

distance of a point b from the hyperplane x · n = a · n is

(b− a) · n|n|

.

Right and Left-Handed Coordinate Systems. Consider a rectangular coordinate system in two dimensions.Angles are measured from the positive x axis in the direction of the positive y axis. There are two ways of labelingthe axes. (See Figure 4.6.) In one the angle increases in the counterclockwise direction and in the other the angleincreases in the clockwise direction. The former is the familiar Cartesian coordinate system.

x y

xy

θθ

Figure 4.6: There are Two Ways of Labeling the Axes in Two Dimensions.

There are also two ways of labeling the axes in a three-dimensional rectangular coordinate system. These arecalled right-handed and left-handed coordinate systems. See Figure 4.7. Any other labelling of the axes could berotated into one of these configurations. The right-handed system is the one that is used by default. If you putyour right thumb in the direction of the z axis in a right-handed coordinate system, then your fingers curl in thedirection from the x axis to the y axis.

Cross Product. The cross product or vector product is defined,

a× b = |a||b| sin θ n,

where θ is the angle from a to b and n is a unit vector that is orthogonal to a and b and in the direction suchthat a, b and n form a right-handed system.

18

x

z

yj

i

k

z

k

j

i

y

x

Figure 4.7: Right and Left Handed Coordinate Systems

You can visualize the direction of a × b by applying the right hand rule. Curl the fingers of your right handin the direction from a to b. Your thumb points in the direction of a× b. Warning: Unless you are a lefty, getin the habit of putting down your pencil before applying the right hand rule.

The dot and cross products behave a little differently. First note that unlike the dot product, the cross productis not commutative. The magnitudes of a × b and b × a are the same, but their directions are opposite. (SeeFigure 4.8.)

Let

a× b = |a||b| sin θ n and b× a = |b||a| sinφ m.

The angle from a to b is the same as the angle from b to a. Since {a,b,n} and {b, a,m} are right-handed systems,m points in the opposite direction as n. Since a×b = −b×a we say that the cross product is anti-commutative.

Next we note that since

|a× b| = |a||b| sin θ,

the magnitude of a× b is the area of the parallelogram defined by the two vectors. (See Figure 4.9.) The area ofthe triangle defined by two vectors is then 1

2|a× b|.

19

a

b

b a

a b

Figure 4.8: The Cross Product is Anti-Commutative.

bsin

bb

a

θ

a

Figure 4.9: The Parallelogram and the Triangle Defined by Two Vectors

From the definition of the cross product, one can derive the following properties:

• a× b = −b× a, anti-commutative.

• α(a× b) = (αa)× b = a× (αb), associativity of scalar multiplication.

• a× (b + c) = a× b + a× c, distributive.

• (a× b)× c 6= a× (b× c). The cross product is not associative.

• i× i = j× j = k× k = 0.

20

• i× j = k, j× k = i, k× i = j.

•

a× b = (a2b3 − a3b2)i + (a3b1 − a1b3)j + (a1b2 − a2b1)k =

∣∣∣∣∣∣i j ka1 a2 a3b1 b2 b3

∣∣∣∣∣∣ ,cross product in terms of rectangular components.

• If a · b = 0 then either a and b are parallel or one of a or b is zero.

Scalar Triple Product. Consider the volume of the parallelopiped defined by three vectors. (See Figure 4.10.)The area of the base is ||b||c| sin θ|, where θ is the angle between b and c. The height is |a| cosφ, where φ is theangle between b× c and a. Thus the volume of the parallelopiped is |a||b||c| sin θ cosφ.

φ

θ

b ca

b

c

Figure 4.10: The Parallelopiped Defined by Three Vectors

Note that

|a · (b× c)| = |a · (|b||c| sin θ n)|= ||a||b||c| sin θ cosφ| .

21

Thus |a · (b× c)| is the volume of the parallelopiped. a · (b × c) is the volume or the negative of the volumedepending on whether {a,b, c} is a right or left-handed system.

Note that parentheses are unnecessary in a ·b×c. There is only one way to interpret the expression. If you didthe dot product first then you would be left with the cross product of a scalar and a vector which is meaningless.a · b× c is called the scalar triple product.

Plane Defined by Three Points. Three points which are not collinear define a plane. Consider a plane thatpasses through the three points a, b and c. One way of expressing that the point x lies in the plane is that thevectors x−a, b−a and c−a are coplanar. (See Figure 4.11.) If the vectors are coplanar, then the parallelopipeddefined by these three vectors will have zero volume. We can express this in an equation using the scalar tripleproduct,

(x− a) · (b− a)× (c− a) = 0.

b

c

x

a

Figure 4.11: Three Points Define a Plane.

22

4.2 Sets of Vectors in n Dimensions

Orthogonality. Consider two n-dimensional vectors

x = (x1, x2, . . . , xn), y = (y1, y2, . . . , yn).

The inner product of these vectors can be defined

〈x|y〉 ≡ x · y =n∑i=1

xiyi.

The vectors are orthogonal if x·y = 0. The norm of a vector is the length of the vector generalized to n dimensions.

‖x‖ =√

x · x

Consider a set of vectors

{x1,x2, . . . ,xm}.

If each pair of vectors in the set is orthogonal, then the set is orthogonal.

xi · xj = 0 if i 6= j

If in addition each vector in the set has norm 1, then the set is orthonormal.

xi · xj = δij =

{1 if i = j

0 if i 6= j

Here δij is known as the Kronecker delta function.

23

Completeness. A set of n, n-dimensional vectors

{x1,x2, . . . ,xn}

is complete if any n-dimensional vector can be written as a linear combination of the vectors in the set. That is,any vector y can be written

y =n∑i=1

cixi.

Taking the inner product of each side of this equation with xm,

y · xm =

(n∑i=1

cixi

)· xm

=n∑i=1

cixi · xm

= cmxm · xmcm =

y · xm‖xm‖2

Thus y has the expansion

y =n∑i=1

y · xi‖xi‖2

xi.

If in addition the set is orthonormal, then

y =n∑i=1

(y · xi)xi.

24

4.3 Exercises

The Dot and Cross Product

Exercise 4.1Prove the distributive law for the dot product,

a · (b + c) = a · b + a · c.

Exercise 4.2Prove that

a · b = aibi ≡ a1b1 + · · ·+ anbn.

Exercise 4.3What is the angle between the vectors i + j and i + 3j?

Exercise 4.4Prove the distributive law for the cross product,

a× (b + c) = a× b + a× b.

Exercise 4.5Show that

a× b =

∣∣∣∣∣∣i j ka1 a2 a3b1 b2 b3

∣∣∣∣∣∣25

Exercise 4.6What is the area of the quadrilateral with vertices at (1, 1), (4, 2), (3, 7) and (2, 3)?

Exercise 4.7What is the volume of the tetrahedron with vertices at (1, 1, 0), (3, 2, 1), (2, 4, 1) and (1, 2, 5)?

Exercise 4.8What is the equation of the plane that passes through the points (1, 2, 3), (2, 3, 1) and (3, 1, 2)? What is thedistance from the point (2, 3, 5) to the plane?

26

4.4 Hints

The Dot and Cross Product

Hint 4.1First prove the distributive law when the first vector is of unit length,

n · (b + c) = n · b + n · c.Then all the quantities in the equation are projections onto the unit vector n and you can use geometry.

Hint 4.2First prove that the dot product of a rectangular unit vector with itself is one and the dot product of two distinctrectangular unit vectors is zero. Then write a and b in rectangular components and use the distributive law.

Hint 4.3Use a · b = |a||b| cos θ.

Hint 4.4First consider the case that both b and c are orthogonal to a. Prove the distributive law in this case fromgeometric considerations.

Next consider two arbitrary vectors a and b. We can write b = b⊥ + b‖ where b⊥ is orthogonal to a and b‖is parallel to a. Show that

a× b = a× b⊥.Finally prove the distributive law for arbitrary b and c.

Hint 4.5Write the vectors in their rectangular components and use,

i× j = k, j× k = i, k× i = j,

27

and,

i× i = j× j = k× k = 0.

Hint 4.6The quadrilateral is composed of two triangles. The area of a triangle defined by the two vectors a and b is12|a · b|.

Hint 4.7Justify that the area of a tetrahedron determined by three vectors is one sixth the area of the parallelogramdetermined by those three vectors. The area of a parallelogram determined by three vectors is the magnitude ofthe scalar triple product of the vectors: a · b× c.

Hint 4.8The equation of a line that is orthogonal to a and passes through the point b is a · x = a · b. The distance of apoint c from the plane is ∣∣∣∣(c− b) · a|a|

∣∣∣∣

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4.5 Solutions

The Dot and Cross Product

Solution 4.1First we prove the distributive law when the first vector is of unit length, i.e.,

n · (b + c) = n · b + n · c. (4.1)

From Figure 4.12 we see that the projection of the vector b + c onto n is equal to the sum of the projections b ·nand c · n.

b

c

n b

n c

b+cn

n (b+c)

Figure 4.12: The Distributive Law for the Dot Product

Now we extend the result to the case when the first vector has arbitrary length. We define a = |a|n andmultiply Equation 4.1 by the scalar, |a|.

|a|n · (b + c) = |a|n · b + |a|n · c

a · (b + c) = a · b + a · c.

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Solution 4.2First note that

ei · ei = |ei||ei| cos(0) = 1.

Then note that that dot product of any two distinct rectangular unit vectors is zero because they are orthogonal.Now we write a and b in terms of their rectangular components and use the distributive law.

a · b = aiei · bjej= aibjei · ej= aibjδij

= aibi

Solution 4.3Since a · b = |a||b| cos θ, we have

θ = arccos

(a · b|a||b|

)when a and b are nonzero.

θ = arccos

((i + j) · (i + 3j)|i + j||i + 3j|

)= arccos

(4√

2√

10

)= arccos

(2√

5

5

)≈ 0.463648

Solution 4.4First consider the case that both b and c are orthogonal to a. b + c is the diagonal of the parallelogram definedby b and c, (see Figure 4.13). Since a is orthogonal to each of these vectors, taking the cross product of a withthese vectors has the effect of rotating the vectors through π/2 radians about a and multiplying their length by|a|. Note that a× (b + c) is the diagonal of the parallelogram defined by a× b and a× c. Thus we see that thedistributive law holds when a is orthogonal to both b and c,

a× (b + c) = a× b + a× c.

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b

cb+c

a c

a

a b

a (b+c)

Figure 4.13: The Distributive Law for the Cross Product

Now consider two arbitrary vectors a and b. We can write b = b⊥ + b‖ where b⊥ is orthogonal to a and b‖is parallel to a, (see Figure 4.14).

a

bb

θ

b

Figure 4.14: The Vector b Written as a Sum of Components Orthogonal and Parallel to a

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By the definition of the cross product,

a× b = |a||b| sin θ n.

Note that

|b⊥| = |b| sin θ,

and that a× b⊥ is a vector in the same direction as a× b. Thus we see that

a× b = |a||b| sin θ n= |a|(sin θ|b|)n= |a||b⊥|n = |a||b⊥| sin(π/2)n

a× b = a× b⊥.

Now we are prepared to prove the distributive law for arbitrary b and c.

a× (b + c) = a× (b⊥ + b‖ + c⊥ + c‖)= a× ((b + c)⊥ + (b + c)‖)= a× ((b + c)⊥)= a× b⊥ + a× c⊥= a× b + a× c

a× (b + c) = a× b + a× c

Solution 4.5We know that

i× j = k, j× k = i, k× i = j,

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and that

i× i = j× j = k× k = 0.

Now we write a and b in terms of their rectangular components and use the distributive law to expand the crossproduct.

a× b = (a1i + a2j + a3k)× (b1i + b2j + b3k)= a1i× (b1i + b2j + b3k) + a2j× (b1i + b2j + b3k) + a3k× (b1i + b2j + b3k)= a1b2k + a1b3(−j) + a2b1(−k) + a2b3i + a3b1j + a3b2(−i)= (a2b3 − a3b2)i− (a1b3 − a3b1)j + (a1b2 − a2b1)k

Next we evaluate the determinant.∣∣∣∣∣∣i j ka1 a2 a3b1 b2 b3

∣∣∣∣∣∣ = i∣∣∣∣a2 a3b2 b3

∣∣∣∣− j ∣∣∣∣a1 a3b1 b3∣∣∣∣+ k ∣∣∣∣a1 a2b1 b2

∣∣∣∣= (a2b3 − a3b2)i− (a1b3 − a3b1)j + (a1b2 − a2b1)k

Thus we see that,

a× b =

∣∣∣∣∣∣i j ka1 a2 a3b1 b2 b3