Guide to Essential Math (eBook)

Front Cover 1
Guide to Essential Math 4
Copyright Page 5
To the Reader 8
Table of Contents 10
Chapter 1. Mathematical Thinking 16
1.1 The NCAA March Madness Problem 17
1.2 Gauss and the Arithmetic Series 17
1.3 The Pythagorean Theorem 18
1.4 Torus Area and Volume 19
1.5 Einstein’s Velocity Addition Law 20
1.6 The Birthday Problem 21
1.7 Fibonacci Numbers and the Golden Ratio 22
1.8 vp in the Gaussian Integral 23
1.9 Function Equal to Its Derivative 24
1.10 Log of N Factorial for Large N 26
1.11 Potential and Kinetic Energies 26
1.12 Riemann Zeta Function and Prime Numbers 29
1.13 How to Solve It 30
1.14 A Note on Mathematical Rigor 32
Chapter 2. Numbers 34
2.1 Integers 34
2.2 Primes 34
2.3 Divisibility 36
2.4 Rational Numbers 37
2.5 Exponential Notation 38
2.6 Powers of 10 39
2.7 Binary Number System 40
2.8 Infinity 42
Chapter 3. Algebra 46
3.1 Symbolic Variables 46
3.2 Legal and Illegal Algebraic Manipulations 47
3.3 Factor-Label Method 50
3.4 Powers and Roots 51
3.5 Logarithms 53
3.6 The Quadratic Formula 55
3.7 Imagining i 57
3.8 Factorials, Permutations, and Combinations 61
3.9 The Binomial Theorem 63
3.10 e Is for Euler 64
Chapter 4. Trigonometry 69
4.1 What Use is Trigonometry? 69
4.2 The Pythagorean Theorem 69
4.3 p in the Sky 72
4.4 Sine and Cosine 75
4.5 Tangent and Secant 79
4.6 Trigonometry in the Complex Plane 80
4.7 de Moivre’s Theorem 82
4.8 Euler’s Theorem 83
4.9 Hyperbolic Functions 85
Chapter 5. Analytic Geometry 88
5.1 Functions and Graphs 88
5.2 Linear Functions 89
5.3 Conic Sections 92
5.4 Conic Sections in Polar Coordinates 97
Chapter 6. Calculus 100
6.1 A Little Road Trip 101
6.2 A Speedboat Ride 103
6.3 Differential and Integral Calculus 104
6.4 Basic Formulas of Differential Calculus 108
6.5 More on Derivatives 110
6.6 Indefinite Integrals 112
6.7 Techniques of Integration 114
6.8 Curvature, Maxima, and Minima 115
6.9 The Gamma Function 117
6.10 Gaussian and Error Functions 119
Chapter 7. Series and Integrals 123
7.1 Some Elementary Series 123
7.2 Power Series 125
7.3 Convergence of Series 127
7.4 Taylor Series 129
7.5 L’Hôpital’s Rule 131
7.6 Fourier Series 132
7.7 Dirac Deltafunction 139
7.8 Fourier Integrals 142
7.9 Generalized Fourier Expansions 145
7.10 Asymptotic Series 145
Chapter 8. Differential Equations 149
8.1 First-Order Differential Equations 150
8.2 AC Circuits 152
8.3 Second-Order Differential Equations 156
8.4 Some Examples from Physics 158
8.5 Boundary Conditions 164
8.6 Series Solutions 167
8.7 Bessel Functions 169
8.8 Second Solution 172
Chapter 9. Matrix Algebra 175
9.1 Matrix Multiplication 176
9.2 Further Properties of Matrices 178
9.3 Determinants 179
9.4 Matrix Inverse 182
9.5 Wronskian Determinant 184
9.6 Special Matrices 184
9.7 Similarity Transformations 186
9.8 Eigenvalue Problems 187
9.9 Group Theory 190
9.10 Minkowski Spacetime 194
Chapter 10. Multivariable Calculus 198
10.1 Partial Derivatives 198
10.2 Multiple Integration 202
10.3 Polar Coordinates 204
10.4 Cylindrical Coordinates 206
10.5 Spherical Polar Coordinates 207
10.6 Differential Expressions 209
10.7 Line Integrals 213
10.8 Green’s Theorem 215
Chapter 11. Vector Analysis 218
11.1 Scalars and Vectors 218
11.2 Scalar or Dot Product 221
11.3 Vector or Cross Product 222
11.4 Triple Products of Vectors 226
11.5 Vector Velocity and Acceleration 227
11.6 Circular Motion 228
11.7 Angular Momentum 230
11.8 Gradient of a Scalar Field 232
11.9 Divergence of a Vector Field 234
11.10 Curl of a Vector Field 236
11.11 Maxwell’s Equations 239
11.12 Covariant Electrodynamics 243
11.13 Curvilinear Coordinates 246
11.14 Vector Identities 249
Chapter 12. Partial Differential Equations and Special Functions 250
12.1 Partial Differential Equations 250
12.2 Separation of Variables 252
12.3 Special Functions 254
12.4 Leibniz’s Formula 255
12.5 Vibration of a Circular Membrane 256
12.6 Bessel Functions 258
12.7 Laplace’s Equation in Spherical Coordinates 261
12.8 Legendre Polynomials 262
12.9 Spherical Harmonics 264
12.10 Spherical Bessel Functions 267
12.11 Hermite Polynomials 269
12.12 Laguerre Polynomials 271
Chapter 13. Complex Variables 275
13.1 Analytic Functions 275
13.2 Derivative of an Analytic Function 279
13.3 Contour Integrals 279
13.4 Cauchy’s Theorem 280
13.5 Cauchy’s Integral Formula 281
13.6 Taylor Series 282
13.7 Laurent Expansions 284
13.8 Calculus of Residues 286
13.9 Multivalued Functions 290
13.10 Integral Representations for Special Functions 293
About the Author 295
Index 296