Introduction to Methods of Applied Mathematicsor
Advanced Mathematical Methods for Scientists and Engineers
Sean Mauchhttp://www.its.caltech.edu/˜sean
January 24, 2004
http://www.its.caltech.edu/~sean
Contents
Anti-Copyright xxiv
Preface xxv0.1 Advice to Teachers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv0.2 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv0.3 Warnings and Disclaimers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvi0.4 Suggested Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvii0.5 About the Title . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvii
I Algebra 1
1 Sets and Functions 21.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Single Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Inverses and Multi-Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Transforming Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
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2 Vectors 222.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.1.1 Scalars and Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.1.2 The Kronecker Delta and Einstein Summation Convention . . . . . . . . . . . . . . . . . . . . 252.1.3 The Dot and Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2 Sets of Vectors in n Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.4 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
II Calculus 47
3 Differential Calculus 483.1 Limits of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.2 Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.3 The Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.4 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.5 Maxima and Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.6 Mean Value Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.6.1 Application: Using Taylor’s Theorem to Approximate Functions. . . . . . . . . . . . . . . . . . 683.6.2 Application: Finite Difference Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.7 L’Hospital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.8.1 Limits of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.8.2 Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.8.3 The Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.8.4 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.8.5 Maxima and Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.8.6 Mean Value Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
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3.8.7 L’Hospital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873.10 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933.11 Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1133.12 Quiz Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4 Integral Calculus 1164.1 The Indefinite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1164.2 The Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1224.2.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.3 The Fundamental Theorem of Integral Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.4 Techniques of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.4.1 Partial Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1274.5 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1304.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.6.1 The Indefinite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1344.6.2 The Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1344.6.3 The Fundamental Theorem of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1364.6.4 Techniques of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1364.6.5 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.7 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1384.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1414.9 Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1504.10 Quiz Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5 Vector Calculus 1545.1 Vector Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1545.2 Gradient, Divergence and Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
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5.4 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1665.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1685.6 Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1775.7 Quiz Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
III Functions of a Complex Variable 179
6 Complex Numbers 1806.1 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1806.2 The Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1846.3 Polar Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1886.4 Arithmetic and Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1936.5 Integer Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1956.6 Rational Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1976.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2016.8 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2086.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
7 Functions of a Complex Variable 2397.1 Curves and Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2397.2 The Point at Infinity and the Stereographic Projection . . . . . . . . . . . . . . . . . . . . . . . . . . 2427.3 A Gentle Introduction to Branch Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2467.4 Cartesian and Modulus-Argument Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2467.5 Graphing Functions of a Complex Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2497.6 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2527.7 Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2597.8 Riemann Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2687.9 Branch Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2707.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
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7.11 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2977.12 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
8 Analytic Functions 3608.1 Complex Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3608.2 Cauchy-Riemann Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3678.3 Harmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3728.4 Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
8.4.1 Categorization of Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3778.4.2 Isolated and Non-Isolated Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
8.5 Application: Potential Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3838.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3888.7 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3968.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
9 Analytic Continuation 4379.1 Analytic Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4379.2 Analytic Continuation of Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4409.3 Analytic Functions Defined in Terms of Real Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 442
9.3.1 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4469.3.2 Analytic Functions Defined in Terms of Their Real or Imaginary Parts . . . . . . . . . . . . . . 450
9.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4549.5 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4569.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
10 Contour Integration and the Cauchy-Goursat Theorem 46210.1 Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46210.2 Contour Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464
10.2.1 Maximum Modulus Integral Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46610.3 The Cauchy-Goursat Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
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10.4 Contour Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46910.5 Morera’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47110.6 Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47310.7 Fundamental Theorem of Calculus via Primitives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474
10.7.1 Line Integrals and Primitives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47410.7.2 Contour Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474
10.8 Fundamental Theorem of Calculus via Complex Calculus . . . . . . . . . . . . . . . . . . . . . . . . . 47510.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47810.10Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48210.11Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483
11 Cauchy’s Integral Formula 49311.1 Cauchy’s Integral Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49411.2 The Argument Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50111.3 Rouche’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50211.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50511.5 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50911.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511
12 Series and Convergence 52512.1 Series of Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525
12.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52512.1.2 Special Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52712.1.3 Convergence Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529
12.2 Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53612.2.1 Tests for Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53712.2.2 Uniform Convergence and Continuous Functions. . . . . . . . . . . . . . . . . . . . . . . . . . 539
12.3 Uniformly Convergent Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53912.4 Integration and Differentiation of Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54712.5 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550
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12.5.1 Newton’s Binomial Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55312.6 Laurent Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55512.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560
12.7.1 Series of Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56012.7.2 Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56612.7.3 Uniformly Convergent Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56612.7.4 Integration and Differentiation of Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . 56812.7.5 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56912.7.6 Laurent Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571
12.8 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57412.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582
13 The Residue Theorem 62613.1 The Residue Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62613.2 Cauchy Principal Value for Real Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634
13.2.1 The Cauchy Principal Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63413.3 Cauchy Principal Value for Contour Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63913.4 Integrals on the Real Axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64313.5 Fourier Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64713.6 Fourier Cosine and Sine Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64913.7 Contour Integration and Branch Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65213.8 Exploiting Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655
13.8.1 Wedge Contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65513.8.2 Box Contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658
13.9 Definite Integrals Involving Sine and Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65913.10Infinite Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66213.11Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66613.12Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68013.13Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686
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IV Ordinary Differential Equations 772
14 First Order Differential Equations 77314.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77314.2 Example Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775
14.2.1 Growth and Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77514.3 One Parameter Families of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77714.4 Integrable Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 779
14.4.1 Separable Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78014.4.2 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78214.4.3 Homogeneous Coefficient Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786
14.5 The First Order, Linear Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79114.5.1 Homogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79114.5.2 Inhomogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79214.5.3 Variation of Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795
14.6 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79614.6.1 Piecewise Continuous Coefficients and Inhomogeneities . . . . . . . . . . . . . . . . . . . . . . 797
14.7 Well-Posed Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80114.8 Equations in the Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803
14.8.1 Ordinary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80314.8.2 Regular Singular Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80614.8.3 Irregular Singular Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81214.8.4 The Point at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814
14.9 Additional Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81614.10Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81914.11Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82214.12Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84314.13Quiz Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844
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15 First Order Linear Systems of Differential Equations 84615.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84615.2 Using Eigenvalues and Eigenvectors to find Homogeneous Solutions . . . . . . . . . . . . . . . . . . . 84715.3 Matrices and Jordan Canonical Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85215.4 Using the Matrix Exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86015.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86515.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87015.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872
16 Theory of Linear Ordinary Differential Equations 90016.1 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90016.2 Nature of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90116.3 Transformation to a First Order System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90516.4 The Wronskian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905
16.4.1 Derivative of a Determinant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90516.4.2 The Wronskian of a Set of Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90616.4.3 The Wronskian of the Solutions to a Differential Equation . . . . . . . . . . . . . . . . . . . . 908
16.5 Well-Posed Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91116.6 The Fundamental Set of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91316.7 Adjoint Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91516.8 Additional Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91916.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92016.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92216.11Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92816.12Quiz Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 929
17 Techniques for Linear Differential Equations 93017.1 Constant Coefficient Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 930
17.1.1 Second Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93117.1.2 Real-Valued Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935
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17.1.3 Higher Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93717.2 Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 940
17.2.1 Real-Valued Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94217.3 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94517.4 Equations Without Explicit Dependence on y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94617.5 Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94717.6 *Reduction of Order and the Adjoint Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94817.7 Additional Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95117.8 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95717.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 960
18 Techniques for Nonlinear Differential Equations 98418.1 Bernoulli Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98418.2 Riccati Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98618.3 Exchanging the Dependent and Independent Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 99018.4 Autonomous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99218.5 *Equidimensional-in-x Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99518.6 *Equidimensional-in-y Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99718.7 *Scale-Invariant Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100018.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100118.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100418.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006
19 Transformations and Canonical Forms 101819.1 The Constant Coefficient Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101819.2 Normal Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1021
19.2.1 Second Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102119.2.2 Higher Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1022
19.3 Transformations of the Independent Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102419.3.1 Transformation to the form u” + a(x) u = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024
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19.3.2 Transformation to a Constant Coefficient Equation . . . . . . . . . . . . . . . . . . . . . . . . 102519.4 Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1027
19.4.1 Initial Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102719.4.2 Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1029
19.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103219.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103419.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035
20 The Dirac Delta Function 104120.1 Derivative of the Heaviside Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104120.2 The Delta Function as a Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104320.3 Higher Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104520.4 Non-Rectangular Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104620.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104820.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105020.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1052
21 Inhomogeneous Differential Equations 105921.1 Particular Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105921.2 Method of Undetermined Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106121.3 Variation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065
21.3.1 Second Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106521.3.2 Higher Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1068
21.4 Piecewise Continuous Coefficients and Inhomogeneities . . . . . . . . . . . . . . . . . . . . . . . . . . 107121.5 Inhomogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074
21.5.1 Eliminating Inhomogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 107421.5.2 Separating Inhomogeneous Equations and Inhomogeneous Boundary Conditions . . . . . . . . . 107621.5.3 Existence of Solutions of Problems with Inhomogeneous Boundary Conditions . . . . . . . . . . 1077
21.6 Green Functions for First Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107921.7 Green Functions for Second Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082
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21.7.1 Green Functions for Sturm-Liouville Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 109221.7.2 Initial Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109521.7.3 Problems with Unmixed Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 109821.7.4 Problems with Mixed Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1100
21.8 Green Functions for Higher Order Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110421.9 Fredholm Alternative Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110921.10Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111721.11Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112321.12Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112621.13Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116421.14Quiz Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165
22 Difference Equations 116622.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116622.2 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116822.3 Homogeneous First Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116922.4 Inhomogeneous First Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117122.5 Homogeneous Constant Coefficient Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117422.6 Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117722.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117922.8 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118022.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1181
23 Series Solutions of Differential Equations 118423.1 Ordinary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1184
23.1.1 Taylor Series Expansion for a Second Order Differential Equation . . . . . . . . . . . . . . . . 118823.2 Regular Singular Points of Second Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1198
23.2.1 Indicial Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120123.2.2 The Case: Double Root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120323.2.3 The Case: Roots Differ by an Integer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1206
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23.3 Irregular Singular Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121623.4 The Point at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121623.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121923.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122423.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122523.8 Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124823.9 Quiz Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1249
24 Asymptotic Expansions 125124.1 Asymptotic Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125124.2 Leading Order Behavior of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125524.3 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126324.4 Asymptotic Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127024.5 Asymptotic Expansions of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1272
24.5.1 The Parabolic Cylinder Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1272
25 Hilbert Spaces 127825.1 Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127825.2 Inner Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128025.3 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128125.4 Linear Independence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128325.5 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128325.6 Gramm-Schmidt Orthogonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128425.7 Orthonormal Function Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128725.8 Sets Of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128825.9 Least Squares Fit to a Function and Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129425.10Closure Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129725.11Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130225.12Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130325.13Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1304
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25.14Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1305
26 Self Adjoint Linear Operators 130726.1 Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130726.2 Self-Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130826.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131126.4 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131226.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1313
27 Self-Adjoint Boundary Value Problems 131427.1 Summary of Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131427.2 Formally Self-Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131527.3 Self-Adjoint Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131827.4 Self-Adjoint Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131827.5 Inhomogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132327.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132627.7 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132727.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1328
28 Fourier Series 133028.1 An Eigenvalue Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133028.2 Fourier Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133328.3 Least Squares Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133728.4 Fourier Series for Functions Defined on Arbitrary Ranges . . . . . . . . . . . . . . . . . . . . . . . . . 134128.5 Fourier Cosine Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134428.6 Fourier Sine Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134528.7 Complex Fourier Series and Parseval’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134628.8 Behavior of Fourier Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134928.9 Gibb’s Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135828.10Integrating and Differentiating Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1358
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28.11Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136328.12Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137128.13Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1373
29 Regular Sturm-Liouville Problems 142029.1 Derivation of the Sturm-Liouville Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142029.2 Properties of Regular Sturm-Liouville Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142229.3 Solving Differential Equations With Eigenfunction Expansions . . . . . . . . . . . . . . . . . . . . . . 143329.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143929.5 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144329.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1445
30 Integrals and Convergence 147030.1 Uniform Convergence of Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147030.2 The Riemann-Lebesgue Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147130.3 Cauchy Principal Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1472
30.3.1 Integrals on an Infinite Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147230.3.2 Singular Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1473
31 The Laplace Transform 147531.1 The Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147531.2 The Inverse Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1477
31.2.1 f̂(s) with Poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148031.2.2 f̂(s) with Branch Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148431.2.3 Asymptotic Behavior of f̂(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1488
31.3 Properties of the Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148931.4 Constant Coefficient Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149231.5 Systems of Constant Coefficient Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 149531.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149731.7 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1504
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31.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1507
32 The Fourier Transform 153932.1 Derivation from a Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153932.2 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1541
32.2.1 A Word of Caution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154432.3 Evaluating Fourier Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1545
32.3.1 Integrals that Converge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154532.3.2 Cauchy Principal Value and Integrals that are Not Absolutely Convergent. . . . . . . . . . . . . 154832.3.3 Analytic Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1550
32.4 Properties of the Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155232.4.1 Closure Relation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155232.4.2 Fourier Transform of a Derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155332.4.3 Fourier Convolution Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155432.4.4 Parseval’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155732.4.5 Shift Property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155932.4.6 Fourier Transform of x f(x). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1559
32.5 Solving Differential Equations with the Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . 156032.6 The Fourier Cosine and Sine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1562
32.6.1 The Fourier Cosine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156232.6.2 The Fourier Sine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1563
32.7 Properties of the Fourier Cosine and Sine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 156432.7.1 Transforms of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156432.7.2 Convolution Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156632.7.3 Cosine and Sine Transform in Terms of the Fourier Transform . . . . . . . . . . . . . . . . . . 1568
32.8 Solving Differential Equations with the Fourier Cosine and Sine Transforms . . . . . . . . . . . . . . . 156932.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157132.10Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157832.11Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1581
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33 The Gamma Function 160533.1 Euler’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160533.2 Hankel’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160733.3 Gauss’ Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160933.4 Weierstrass’ Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161133.5 Stirling’s Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161333.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161833.7 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161933.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1620
34 Bessel Functions 162234.1 Bessel’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162234.2 Frobeneius Series Solution about z = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1623
34.2.1 Behavior at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162634.3 Bessel Functions of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1628
34.3.1 The Bessel Function Satisfies Bessel’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 162934.3.2 Series Expansion of the Bessel Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163034.3.3 Bessel Functions of Non-Integer Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163334.3.4 Recursion Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163634.3.5 Bessel Functions of Half-Integer Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1639
34.4 Neumann Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164034.5 Bessel Functions of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164434.6 Hankel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164634.7 The Modified Bessel Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164634.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165034.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165534.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1657
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V Partial Differential Equations 1680
35 Transforming Equations 168135.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168235.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168335.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1684
36 Classification of Partial Differential Equations 168536.1 Classification of Second Order Quasi-Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 1685
36.1.1 Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168636.1.2 Parabolic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169136.1.3 Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1692
36.2 Equilibrium Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169436.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169636.4 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169736.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1698
37 Separation of Variables 170437.1 Eigensolutions of Homogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170437.2 Homogeneous Equations with Homogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . . 170437.3 Time-Independent Sources and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 170637.4 Inhomogeneous Equations with Homogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . 170937.5 Inhomogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171037.6 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171337.7 General Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171637.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171837.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173437.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1739
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38 Finite Transforms 182138.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182538.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182638.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1827
39 The Diffusion Equation 183139.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183239.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183439.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1835
40 Laplace’s Equation 184140.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184140.2 Fundamental Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1841
40.2.1 Two Dimensional Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184240.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184340.4 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184640.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1847
41 Waves 185941.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186041.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186641.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1868
42 Similarity Methods 188842.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189242.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189342.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1894
43 Method of Characteristics 189743.1 First Order Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189743.2 First Order Quasi-Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1898
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43.3 The Method of Characteristics and the Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 190043.4 The Wave Equation for an Infinite Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190143.5 The Wave Equation for a Semi-Infinite Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190243.6 The Wave Equation for a Finite Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190443.7 Envelopes of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190543.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190843.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191043.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1911
44 Transform Methods 191844.1 Fourier Transform for Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191844.2 The Fourier Sine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192044.3 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192044.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192244.5 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192644.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1928
45 Green Functions 195045.1 Inhomogeneous Equations and Homogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . 195045.2 Homogeneous Equations and Inhomogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . 195145.3 Eigenfunction Expansions for Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195345.4 The Method of Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195845.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196045.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197145.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1974
46 Conformal Mapping 203446.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203546.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203846.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2039
xx
47 Non-Cartesian Coordinates 205147.1 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205147.2 Laplace’s Equation in a Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205247.3 Laplace’s Equation in an Annulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2055
VI Calculus of Variations 2059
48 Calculus of Variations 206048.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206148.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207548.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2079
VII Nonlinear Differential Equations 2166
49 Nonlinear Ordinary Differential Equations 216749.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216849.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217349.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2174
50 Nonlinear Partial Differential Equations 219650.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219750.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220050.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2201
VIII Appendices 2220
A Greek Letters 2221
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B Notation 2223
C Formulas from Complex Variables 2225
D Table of Derivatives 2228
E Table of Integrals 2232
F Definite Integrals 2236
G Table of Sums 2238
H Table of Taylor Series 2241
I Continuous Transforms 2244I.1 Properties of Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2244I.2 Table of Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2247I.3 Table of Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2250I.4 Table of Fourier Transforms in n Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2253I.5 Table of Fourier Cosine Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2254I.6 Table of Fourier Sine Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2255
J Table of Wronskians 2257
K Sturm-Liouville Eigenvalue Problems 2259
L Green Functions for Ordinary Differential Equations 2261
M Trigonometric Identities 2264M.1 Circular Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2264M.2 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2266
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N Bessel Functions 2269N.1 Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2269
O Formulas from Linear Algebra 2270
P Vector Analysis 2271
Q Partial Fractions 2273
R Finite Math 2276
S Physics 2277
T Probability 2278T.1 Independent Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2278T.2 Playing the Odds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2279
U Economics 2280
V Glossary 2281
W whoami 2283
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Anti-Copyright
Anti-Copyright @ 1995-2001 by Mauch Publishing Company, un-Incorporated.
No rights reserved. Any part of this publication may be reproduced, stored in a retrieval system, transmitted ordesecrated without permission.
xxiv
Preface
During the summer before my final undergraduate year at Caltech I set out to write a math text unlike any other,namely, one written by me. In that respect I have succeeded beautifully. Unfortunately, the text is neither complete norpolished. I have a “Warnings and Disclaimers” section below that is a little amusing, and an appendix on probabilitythat I feel concisesly captures the essence of the subject. However, all the material in between is in some stage ofdevelopment. 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 post newversions a couple of times a year.
0.1 Advice to Teachers
If you have something worth saying, write it down.
0.2 Acknowledgments
I would like to thank Professor Saffman for advising me on this project and the Caltech SURF program for providingthe funding for me to write the first edition of this book.
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http://www.its.caltech.edu/~sean
0.3 Warnings and Disclaimers
• This book is a work in progress. It contains quite a few mistakes and typos. I would greatly appreciate yourconstructive criticism. You can reach me at ‘[email protected]’.
• 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 concentrating contentscan 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 spelling chequerfrom thyme too thyme, sew their should knot bee two many misspellings. Though I ain’t so sure the grammar’stoo 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 days ofobtaining this text you will suffer great misfortune and other nastiness.
• The surgeon general has determined that excessive studying is detrimental to your social life.
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• This text has been buffered for your protection and ribbed for your pleasure.
• Stop reading this rubbish and get back to work!
0.4 Suggested Use
This text is well suited to the student, professional or lay-person. It makes a superb gift. This text has a boquet thatis light and fruity, with some earthy undertones. It is ideal with dinner or as an apertif. Bon apetit!
0.5 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 tolearn about some mathematical subject, look for books with “Introduction” or “Elementary” in the title. If it is an“Intermediate” text it will be incomprehensible. If it is “Advanced” then not only will it be incomprehensible, it willhave low production qualities, i.e. a crappy typewriter font, no graphics and no examples. There is an exception to thisrule: When the title also contains the word “Scientists” or “Engineers” the advanced book may be quite suitable foractually learning the material.
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Part I
Algebra
1
Chapter 1
Sets and Functions
1.1 Sets
Definition. A set is a collection of objects. We call the objects, elements. A set is denoted by listing the elementsbetween braces. For example: {e, ı, π, 1} is the set of the integer 1, the pure imaginary number ı =
√−1 and the
transcendental numbers e = 2.7182818 . . . and π = 3.1415926 . . .. For elements of a set, we do not count multiplicities.We regard the set {1, 2, 2, 3, 3, 3} as identical to the set {1, 2, 3}. Order is not significant in sets. The set {1, 2, 3} isequivalent to {3, 2, 1}.
In enumerating the elements of a set, we use ellipses to indicate patterns. We denote the set of positive integers as{1, 2, 3, . . .}. We also denote sets with the notation {x|conditions on x} for sets that are more easily described thanenumerated. This is read as “the set of elements x such that . . . ”. x ∈ S is the notation for “x is an element of theset 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 = {. . . ,−3,−2,−1, 0, 1, 2, 3 . . .} is the set of integers. (Z is for “Zahlen”, the German word for “number”.)
2
• Q = {p/q|p, q ∈ Z, q 6= 0} is the set of rational numbers. (Q is for quotient.) 1
• R = {x|x = a1a2 · · · an.b1b2 · · · } is the set of real numbers, i.e. the set of numbers with decimal expansions. 2
• C = {a + ıb|a, b ∈ R, ı2 = −1} is the set of complex numbers. ı is the square root of −1. (If you haven’t seencomplex 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, . . .}.We use a − superscript to denote the sets of negative numbers.
• 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 elements in theset. 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 is aproper subset of T , S ⊂ T , if S ⊆ T and S 6= T . For example: The empty set is a subset of every set, ∅ ⊆ S. Therational numbers are a proper subset of the real numbers, Q ⊂ R.
1 Note that with this description, we enumerate each rational number an infinite number of times. For example: 1/2 = 2/4 =3/6 = (−1)/(−2) = · · · . This does not pose a problem as we do not count multiplicities.
2Guess what R is for.
3
Operations. The union of two sets, S ∪ T , is the set whose elements are in either of the two sets. The union of nsets,
∪nj=1Sj ≡ S1 ∪ S2 ∪ · · · ∪ Snis the set whose elements are in any of the sets Sj. The intersection of two sets, S ∩ T , is the set whose elements arein both of the two sets. In other words, the intersection of two sets in the set of elements that the two sets have incommon. The intersection of n sets,
∩nj=1Sj ≡ S1 ∩ S2 ∩ · · · ∩ Snis the set whose elements are in all of the sets Sj. If two sets have no elements in common, S ∩ T = ∅, then the setsare disjoint. If T ⊆ S, then the difference between S and T , S \ T , is the set of elements in S which are not in 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).
1.2 Single Valued Functions
Single-Valued Functions. A single-valued function or single-valued mapping is a mapping of the elements x ∈ Xinto elements y ∈ Y . This is expressed as f : X → Y or X f→ Y . If such a function is well-defined, then for eachx ∈ X there exists a unique element of y such that f(x) = y. The set X is the domain of the function, Y is thecodomain, (not to be confused with the range, which we introduce shortly). To denote the value of a function on a
4
particular element we can use any of the notations: f(x) = y, f : x 7→ y or simply x 7→ y. f is the identity map onX if f(x) = x for all x ∈ X.
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, f(x) =∑n
k=0 akxk, ak ∈ R, and the exponential function, f(x) = ex, are examples of single
valued functions which map real numbers to real numbers.
• The greatest integer function, f(x) = bxc, is a mapping from R to Z. bxc is defined as the greatest integer lessthan or equal to x. Likewise, the least integer function, f(x) = dxe, is the least integer greater than or equal tox.
The -jectives. A function is injective if for each x1 6= x2, f(x1) 6= f(x2). In other words, distinct elements aremapped to distinct elements. f is surjective if for each y in the codomain, there is an x such that y = f(x). If afunction is both injective and surjective, then it is bijective. A bijective function is also called a one-to-one mapping.
Examples.
• The exponential function f(x) = ex, considered as a mapping from R to R+, is bijective, (a one-to-one mapping).
• 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, thereare 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 values ofx such that y = sinx.
5
Injective Surjective Bijective
Figure 1.1: Depictions of Injective, Surjective and Bijective Functions
1.3 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, thenf−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 both f(x) andf−1(x) are defined. For example lnx, which maps R+ to R is the inverse of ex. x = elnx = ln(ex) for all x ∈ R+.(Note the x ∈ R+ ensures that lnx 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-valued function.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 are diagrams showingone-to-one, many-to-one and one-to-many functions in Figure 1.2.
Example 1.3.1 y = x2, a many-to-one function has the inverse x = y1/2. For each positive y, there are two values ofx such that x = y1/2. y = x2 and y = x1/2 are graphed in Figure 1.3.
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 graphed
in Figure 1.4.
6
rangedomain rangedomain rangedomain
one-to-one many-to-one one-to-many
Figure 1.2: Diagrams of One-To-One, Many-To-One and One-To-Many Functions
Figure 1.3: y = x2 and y = x1/2
Figure 1.4: y =√x
7
Now consider the many-to-one function y = sin x. The inverse is x = arcsin y. For each y ∈ [−1..1] there are aninfinite number of values x such that x = arcsin y. In Figure 1.5 is a graph of y = sinx and a graph of a few branchesof y = arcsin x.
Figure 1.5: y = sinx and y = arcsin x
Example 1.3.2 arcsinx has an infinite number of branches. We will denote the principal branch by Arcsin x whichmaps [−1..1] to
[−π
2..π
2
]. Note that x = sin(arcsinx), but x 6= arcsin(sinx). y = Arcsinx in Figure 1.6.
Figure 1.6: y = Arcsinx
Example 1.3.3 Consider 11/3. Since x3 is a one-to-one function, x1/3 is a single-valued function. (See Figure 1.7.)11/3 = 1.
Example 1.3.4 Consider arccos(1/2). cosx and a portion of arccos x are graphed in Figure 1.8. The equationcosx = 1/2 has the two solutions x = ±π/3 in the range x ∈ (−π..π]. We use the periodicity of the cosine,
8
Figure 1.7: y = x3 and y = x1/3
cos(x+ 2π) = cosx, to find the remaining solutions.
arccos(1/2) = {±π/3 + 2nπ}, n ∈ Z.
Figure 1.8: y = cos x and y = arccosx
1.4 Transforming Equations
Consider the equation g(x) = h(x) and the single-valued function f(x). A particular value of x is a solution of theequation if substituting that value into the equation results in an identity. In determining the solutions of an equation,we often apply functions to each side of the equation in order to simplify its form. We apply the function to obtaina second equation, f(g(x)) = f(h(x)). If x = ξ is a solution of the former equation, (let ψ = g(ξ) = h(ξ)), then it
9
is necessarily a solution of latter. This is because f(g(ξ)) = f(h(ξ)) reduces to the identity f(ψ) = f(ψ). If f(x) isbijective, then the converse is true: any solution of the latter equation is a solution of the former equation. Supposethat x = ξ is a solution of the latter, f(g(ξ)) = f(h(ξ)). That f(x) is a one-to-one mapping implies that g(ξ) = h(ξ).Thus x = ξ is a solution of the former equation.
It is always safe to apply a one-to-one, (bijective), function to an equation, (provided it is defined for that domain).For example, we can apply f(x) = x3 or f(x) = ex, considered as mappings on R, to the equation x = 1. Theequations x3 = 1 and ex = e each have the unique solution x = 1 for x ∈ R.
In general, we must take care in applying functions to equations. If we apply a many-to-one function, we mayintroduce spurious solutions. Applying f(x) = x2 to the equation x = π
2results in x2 = π
2
4, which has the two solutions,
x = {±π2}. Applying f(x) = sin x results in x2 = π2
4, which has an infinite number of solutions, x = {π
2+2nπ |n ∈ Z}.
We do not generally apply a one-to-many, (multi-valued), function to both sides of an equation as this rarely is useful.Rather, we typically use the definition of the inverse function. Consider the 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= sin x, 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
sin x = 11/2 = ±1.Now note that we should not just apply arcsin to both sides of the equation as arcsin(sinx) 6= x. Instead we use thedefinition of arcsin as the inverse of sin.
x = arcsin(±1)x = arcsin(1) has the solutions x = π/2+2nπ and x = arcsin(−1) has the solutions x = −π/2+2nπ. We enumeratethe solutions.
x ={π
2+ nπ | n ∈ Z
}
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1.5 ExercisesExercise 1.1The area of a circle is directly proportional to the square of its diameter. What is the constant of proportionality?Hint, Solution
Exercise 1.2Consider the equation
x+ 1
y − 2=x2 − 1y2 − 4
.
1. Why might one think that this is the equation of a line?
2. Graph the solutions of the equation to demonstrate that it is not the equation of a line.
Hint, Solution
Exercise 1.3Consider the function of a real variable,
f(x) =1
x2 + 2.
What is the domain and range of the function?Hint, Solution
Exercise 1.4The temperature measured in degrees Celsius 3 is linearly related to the temperature measured in degrees Fahrenheit 4.Water freezes at 0◦ C = 32◦ F and boils at 100◦ C = 212◦ F . Write the temperature in degrees Celsius as a functionof degrees Fahrenheit.
3 Originally, it was called degrees Centigrade. centi because there are 100 degrees between the two calibration points. It is nowcalled degrees Celsius in honor of the inventor.
4 The Fahrenheit scale, named for Daniel Fahrenheit, was originally calibrated with the freezing point of salt-saturated water tobe 0◦. Later, the calibration points became the freezing point of water, 32◦, and body temperature, 96◦. With this method, there are64 divisions between the calibration points. Finally, the upper calibration point was changed to the boiling point of water at 212◦.This gave 180 divisions, (the number of degrees in a half circle), between the two calibration points.
11
Hint, Solution
Exercise 1.5Consider the function graphed in Figure 1.9. Sketch graphs of f(−x), f(x+ 3), f(3− x) + 2, and f−1(x). You mayuse the blank grids in Figure 1.10.
Figure 1.9: Graph of the function.
Hint, Solution
Exercise 1.6A culture of bacteria grows at the rate of 10% per minute. At 6:00 pm there are 1 billion bacteria. How many bacteriaare there at 7:00 pm? How many were there at 3:00 pm?
Hint, Solution
Exercise 1.7The graph in Figure 1.11 shows an even function f(x) = p(x)/q(x) where p(x) and q(x) are rational quadraticpolynomials. Give possible formulas for p(x) and q(x).
Hint, Solution
12
Figure 1.10: Blank grids.
Exercise 1.8Find a polynomial of degree 100 which is zero only at x = −2, 1, π and is non-negative.Hint, Solution
13
1 2
1
2
2 4 6 8 10
1
2
Figure 1.11: Plots of f(x) = p(x)/q(x).
1.6 Hints
Hint 1.1area = constant× diameter2.
Hint 1.2A pair (x, y) is a solution of the equation if it make the equation an identity.
Hint 1.3The domain is the subset of R on which the function is defined.
Hint 1.4Find the slope and x-intercept of the line.
Hint 1.5The inverse of the function is the reflection of the function across the line y = x.
Hint 1.6The formula for geometric growth/decay is x(t) = x0r
t, where r is the rate.
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Hint 1.7Note that p(x) and q(x) appear as a ratio, they are determined only up to a multiplicative constant. We may take theleading coefficient of q(x) to be unity.
f(x) =p(x)
q(x)=ax2 + bx+ c
x2 + βx+ χ
Use the properties of the function to solve for the unknown parameters.
Hint 1.8Write the polynomial in factored form.
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1.7 SolutionsSolution 1.1
area = π × radius2
area =π
4× diameter2
The constant of proportionality is π4.
Solution 1.21. If we multiply the equation by y2 − 4 and divide by x+ 1, we obtain the equation of a line.
y + 2 = x− 1
2. We factor the quadratics on the right side of the equation.
x+ 1
y − 2=
(x+ 1)(x− 1)(y − 2)(y + 2)
.
We note that one or both sides of the equation are undefined at y = ±2 because of division by zero. There areno solutions for these two values of y and we assume from this point that y 6= ±2. We multiply by (y−2)(y+2).
(x+ 1)(y + 2) = (x+ 1)(x− 1)
For x = −1, the equation becomes the identity 0 = 0. Now we consider x 6= −1. We divide by x+ 1 to obtainthe equation of a line.
y + 2 = x− 1y = x− 3
Now we collect the solutions we have found.
{(−1, y) : y 6= ±2} ∪ {(x, x− 3) : x 6= 1, 5}
The solutions are depicted in Figure /reffig not a line.
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-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
Figure 1.12: The solutions of x+1y−2 =
x2−1y2−4 .
Solution 1.3The denominator is nonzero for all x ∈ R. Since we don’t have any division by zero problems, the domain of thefunction is R. For x ∈ R,
0 <1
x2 + 2≤ 2.
Consider
y =1
x2 + 2. (1.1)
For any y ∈ (0 . . . 1/2], there is at least one value of x that satisfies Equation 1.1.
x2 + 2 =1
y
x = ±√
1
y− 2
Thus the range of the function is (0 . . . 1/2]
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Solution 1.4Let c denote degrees Celsius and f denote degrees Fahrenheit. The line passes through the points (f, c) = (32, 0) and(f, c) = (212, 100). The x-intercept is f = 32. We calculate the slope of the line.
slope =100− 0212− 32
=100
180=
5
9
The relationship between fahrenheit and celcius is
c =5
9(f − 32).
Solution 1.5We plot the various transformations of f(x).
Solution 1.6The formula for geometric growth/decay is x(t) = x0r
t, where r is the rate. Let t = 0 coincide with 6:00 pm. Wedetermine x0.
x(0) = 109 = x0
(11
10
)0= x0
x0 = 109
At 7:00 pm the number of bacteria is
109(
11
10
)60=
1160
1051≈ 3.04× 1011
At 3:00 pm the number of bacteria was
109(
11
10
)−180=
10189
11180≈ 35.4
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Figure 1.13: Graphs of f(−x), f(x+ 3), f(3− x) + 2, and f−1(x).
Solution 1.7We write p(x) and q(x) as general quadratic polynomials.
f(x) =p(x)
q(x)=
ax2 + bx+ c
αx2 + βx+ χ
We will use the properties of the function to solve for the unknown parameters.
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Note that p(x) and q(x) appear as a ratio, they are determined only up to a multiplicative constant. We may takethe leading coefficient of q(x) to be unity.
f(x) =p(x)
q(x)=ax2 + bx+ c
x2 + βx+ χ
f(x) has a second order zero at x = 0. This means that p(x) has a second order zero there and that χ 6= 0.
f(x) =ax2
x2 + βx+ χ
We note that f(x) → 2 as x→∞. This determines the parameter a.
limx→∞
f(x) = limx→∞
ax2
x2 + βx+ χ
= limx→∞
2ax
2x+ β
= limx→∞
2a
2= a
f(x) =2x2
x2 + βx+ χ
Now we use the fact that f(x) is even to conclude that q(x) is even and thus β = 0.
f(x) =2x2
x2 + χ
Finally, we use that f(1) = 1 to determine χ.
f(x) =2x2
x2 + 1
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Solution 1.8Consider the polynomial
p(x) = (x+ 2)40(x− 1)30(x− π)30.
It is of degree 100. Since the factors only vanish at x = −2, 1, π, p(x) only vanishes there. Since factors are non-negative, the polynomial is non-negative.
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Chapter 2
Vectors
2.1 Vectors
2.1.1 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 2.1). 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 a vector,denoted −a, is a vector of the same magnitude as a but in the opposite direction. We add two vectors a and b byplacing the tail of b at the head of a and defining a + b to be the vector with tail at the origin and head at the headof b. (See Figure 2.2.)
The difference, a− b, is defined as the sum of a and the negative of b, a + (−b). The result of multiplying a bya scalar α is a vector of magnitude |α| |a| with the same/opposite direction if α is positive/negative. (See Figure 2.2.)
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x
z
y
Figure 2.1: Graphical representation of a vector in three dimensions.
a+b
a
b
-a
a2a
Figure 2.2: Vector arithmetic.
Here are the properties of adding vectors and multiplying them by a scalar. They are evident from geometric
23
considerations.
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
Zero and Unit Vectors. The additive identity element for vectors is the zero vector or null vector. This is a vectorof magnitude zero which is denoted as 0. A unit vector is a vector of magnitude one. If a is nonzero then a/|a| is aunit 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 2.3).
x
z
yj
k
i
Figure 2.3: Rectangular unit vectors.
24
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 2.4.) Another notation for the vector a is 〈a1, . . . , an〉. By the Pythagorean theorem, the magnitude ofthe vector a is |a| =
√a21 + · · ·+ a2n.
x
z
y
a
a
a
1
3
i
k
ja2
Figure 2.4: Components of a vector.
2.1.2 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 = a1