Derivations of applied mathematics

246 Pages·2006·0.831 MB·English

by  Black T.H.

Derivations of Applied Mathematics
Thaddeus H. Black
Revised 14 December 2006
ii
Derivations of Applied Mathematics.
14 December 2006.
Copyright c 1983{2006 by Thaddeus H. Black [email protected] .
(cid:13) h i
Published by the Debian Project [7].
This book is free software. You can redistribute and/or modify it under the
terms of the GNU General Public License [11], version 2.
Contents
Preface xiii
1 Introduction 1
1.1 Applied mathematics . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Rigor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Axiom and de(cid:12)nition . . . . . . . . . . . . . . . . . . . 2
1.2.2 Mathematical extension . . . . . . . . . . . . . . . . . 4
1.3 Complex numbers and complex variables . . . . . . . . . . . . 5
1.4 On the text . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Classical algebra and geometry 7
2.1 Basic arithmetic relationships . . . . . . . . . . . . . . . . . . 7
2.1.1 Commutivity, associativity, distributivity . . . . . . . 7
2.1.2 Negative numbers . . . . . . . . . . . . . . . . . . . . 9
2.1.3 Inequality . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.4 The change of variable . . . . . . . . . . . . . . . . . . 11
2.2 Quadratics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Notation for series sums and products . . . . . . . . . . . . . 13
2.4 The arithmetic series . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Powers and roots . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5.1 Notation and integral powers . . . . . . . . . . . . . . 15
2.5.2 Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5.3 Powers of products and powers of powers . . . . . . . 19
2.5.4 Sums of powers . . . . . . . . . . . . . . . . . . . . . . 19
2.5.5 Summary and remarks . . . . . . . . . . . . . . . . . . 20
2.6 Multiplying and dividing power series . . . . . . . . . . . . . 20
2.6.1 Multiplying power series . . . . . . . . . . . . . . . . . 21
2.6.2 Dividing power series . . . . . . . . . . . . . . . . . . 21
2.6.3 Common quotients and the geometric series . . . . . . 26
iii
iv CONTENTS
2.6.4 Variations on the geometric series . . . . . . . . . . . 26
2.7 Constants and variables . . . . . . . . . . . . . . . . . . . . . 27
2.8 Exponentials and logarithms . . . . . . . . . . . . . . . . . . 29
2.8.1 The logarithm . . . . . . . . . . . . . . . . . . . . . . 29
2.8.2 Properties of the logarithm . . . . . . . . . . . . . . . 30
2.9 Triangles and other polygons: simple facts . . . . . . . . . . . 30
2.9.1 Triangle area . . . . . . . . . . . . . . . . . . . . . . . 31
2.9.2 The triangle inequalities . . . . . . . . . . . . . . . . . 31
2.9.3 The sum of interior angles . . . . . . . . . . . . . . . . 32
2.10 The Pythagorean theorem . . . . . . . . . . . . . . . . . . . . 33
2.11 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.12 Complex numbers (introduction) . . . . . . . . . . . . . . . . 36
2.12.1 Rectangular complex multiplication . . . . . . . . . . 38
2.12.2 Complex conjugation . . . . . . . . . . . . . . . . . . . 38
2.12.3 Power series and analytic functions (preview) . . . . . 40
3 Trigonometry 43
3.1 De(cid:12)nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 Simple properties . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3 Scalars, vectors, and vector notation . . . . . . . . . . . . . . 45
3.4 Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.5 Trigonometric sums and di(cid:11)erences . . . . . . . . . . . . . . . 51
3.5.1 Variations on the sums and di(cid:11)erences . . . . . . . . . 52
3.5.2 Trigonometric functions of double and half angles . . . 53
3.6 Trigonometrics of the hour angles . . . . . . . . . . . . . . . . 53
3.7 The laws of sines and cosines . . . . . . . . . . . . . . . . . . 57
3.8 Summary of properties . . . . . . . . . . . . . . . . . . . . . . 58
3.9 Cylindrical and spherical coordinates . . . . . . . . . . . . . . 60
3.10 The complex triangle inequalities . . . . . . . . . . . . . . . . 62
3.11 De Moivre’s theorem . . . . . . . . . . . . . . . . . . . . . . . 63
4 The derivative 65
4.1 In(cid:12)nitesimals and limits . . . . . . . . . . . . . . . . . . . . . 65
4.1.1 The in(cid:12)nitesimal . . . . . . . . . . . . . . . . . . . . . 66
4.1.2 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2 Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2.1 Combinations and permutations . . . . . . . . . . . . 68
4.2.2 Pascal’s triangle . . . . . . . . . . . . . . . . . . . . . 70
4.3 The binomial theorem . . . . . . . . . . . . . . . . . . . . . . 70
4.3.1 Expanding the binomial . . . . . . . . . . . . . . . . . 70
CONTENTS v
4.3.2 Powers of numbers near unity . . . . . . . . . . . . . . 71
4.3.3 Complex powers of numbers near unity . . . . . . . . 72
4.4 The derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.4.1 The derivative of the power series. . . . . . . . . . . . 73
4.4.2 The Leibnitz notation . . . . . . . . . . . . . . . . . . 74
4.4.3 The derivative of a function of a complex variable . . 76
4.4.4 The derivative of za . . . . . . . . . . . . . . . . . . . 77
4.4.5 The logarithmic derivative . . . . . . . . . . . . . . . . 77
4.5 Basic manipulation of the derivative . . . . . . . . . . . . . . 78
4.5.1 The derivative chain rule . . . . . . . . . . . . . . . . 78
4.5.2 The derivative product rule . . . . . . . . . . . . . . . 79
4.6 Extrema and higher derivatives . . . . . . . . . . . . . . . . . 80
4.7 L’Ho^pital’s rule . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.8 The Newton-Raphson iteration . . . . . . . . . . . . . . . . . 83
5 The complex exponential 87
5.1 The real exponential . . . . . . . . . . . . . . . . . . . . . . . 87
5.2 The natural logarithm . . . . . . . . . . . . . . . . . . . . . . 90
5.3 Fast and slow functions . . . . . . . . . . . . . . . . . . . . . 91
5.4 Euler’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.5 Complex exponentials and de Moivre . . . . . . . . . . . . . . 96
5.6 Complex trigonometrics . . . . . . . . . . . . . . . . . . . . . 96
5.7 Summary of properties . . . . . . . . . . . . . . . . . . . . . . 97
5.8 Derivatives of complex exponentials . . . . . . . . . . . . . . 97
5.8.1 Derivatives of sine and cosine . . . . . . . . . . . . . . 97
5.8.2 Derivatives of the trigonometrics . . . . . . . . . . . . 100
5.8.3 Derivatives of the inverse trigonometrics . . . . . . . . 100
5.9 The actuality of complex quantities . . . . . . . . . . . . . . . 102
6 Primes, roots and averages 105
6.1 Prime numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.1.1 The in(cid:12)nite supply of primes . . . . . . . . . . . . . . 105
6.1.2 Compositional uniqueness . . . . . . . . . . . . . . . . 106
6.1.3 Rational and irrational numbers . . . . . . . . . . . . 109
6.2 The existence and number of roots . . . . . . . . . . . . . . . 110
6.2.1 Polynomial roots . . . . . . . . . . . . . . . . . . . . . 110
6.2.2 The fundamental theorem of algebra . . . . . . . . . . 111
6.3 Addition and averages . . . . . . . . . . . . . . . . . . . . . . 112
6.3.1 Serial and parallel addition . . . . . . . . . . . . . . . 112
6.3.2 Averages . . . . . . . . . . . . . . . . . . . . . . . . . 115
vi CONTENTS
7 The integral 119
7.1 The concept of the integral . . . . . . . . . . . . . . . . . . . 119
7.1.1 An introductory example . . . . . . . . . . . . . . . . 120
7.1.2 Generalizing the introductory example . . . . . . . . . 123
7.1.3 The balanced de(cid:12)nition and the trapezoid rule . . . . 123
7.2 The antiderivative . . . . . . . . . . . . . . . . . . . . . . . . 124
7.3 Operators, linearity and multiple integrals . . . . . . . . . . . 126
7.3.1 Operators . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.3.2 A formalism . . . . . . . . . . . . . . . . . . . . . . . . 127
7.3.3 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . 128
7.3.4 Summational and integrodi(cid:11)erential transitivity. . . . 129
7.3.5 Multiple integrals . . . . . . . . . . . . . . . . . . . . . 130
7.4 Areas and volumes . . . . . . . . . . . . . . . . . . . . . . . . 131
7.4.1 The area of a circle . . . . . . . . . . . . . . . . . . . . 131
7.4.2 The volume of a cone . . . . . . . . . . . . . . . . . . 132
7.4.3 The surface area and volume of a sphere . . . . . . . . 133
7.5 Checking integrations . . . . . . . . . . . . . . . . . . . . . . 136
7.6 Contour integration . . . . . . . . . . . . . . . . . . . . . . . 137
7.7 Discontinuities . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.8 Remarks (and exercises) . . . . . . . . . . . . . . . . . . . . . 141
8 The Taylor series 143
8.1 The power series expansion of 1=(1 z)n+1 . . . . . . . . . . 143
(cid:0)
8.1.1 The formula . . . . . . . . . . . . . . . . . . . . . . . . 144
8.1.2 The proof by induction . . . . . . . . . . . . . . . . . 145
8.1.3 Convergence . . . . . . . . . . . . . . . . . . . . . . . 146
8.1.4 General remarks on mathematical induction . . . . . . 148
8.2 Shifting a power series’ expansion point . . . . . . . . . . . . 149
8.3 Expanding functions in Taylor series . . . . . . . . . . . . . . 151
8.4 Analytic continuation . . . . . . . . . . . . . . . . . . . . . . 152
8.5 Branch points . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
8.6 Cauchy’s integral formula . . . . . . . . . . . . . . . . . . . . 155
8.6.1 The meaning of the symbol dz . . . . . . . . . . . . . 156
8.6.2 Integrating along the contour . . . . . . . . . . . . . . 156
8.6.3 The formula . . . . . . . . . . . . . . . . . . . . . . . . 160
8.7 Taylor series for speci(cid:12)c functions . . . . . . . . . . . . . . . . 161
8.8 Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
8.9 Calculating 2(cid:25) . . . . . . . . . . . . . . . . . . . . . . . . . . 165
8.10 The multidimensional Taylor series . . . . . . . . . . . . . . . 166
CONTENTS vii
9 Integration techniques 169
9.1 Integration by antiderivative . . . . . . . . . . . . . . . . . . . 169
9.2 Integration by substitution . . . . . . . . . . . . . . . . . . . 170
9.3 Integration by parts . . . . . . . . . . . . . . . . . . . . . . . 171
9.4 Integration by unknown coe(cid:14)cients . . . . . . . . . . . . . . . 173
9.5 Integration by closed contour . . . . . . . . . . . . . . . . . . 176
9.6 Integration by partial-fraction expansion . . . . . . . . . . . . 178
9.6.1 Partial-fraction expansion . . . . . . . . . . . . . . . . 178
9.6.2 Multiple poles . . . . . . . . . . . . . . . . . . . . . . 180
9.6.3 Integrating rational functions . . . . . . . . . . . . . . 182
9.7 Integration by Taylor series . . . . . . . . . . . . . . . . . . . 184
10 Cubics and quartics 185
10.1 Vieta’s transform . . . . . . . . . . . . . . . . . . . . . . . . . 186
10.2 Cubics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
10.3 Super(cid:13)uous roots . . . . . . . . . . . . . . . . . . . . . . . . . 189
10.4 Edge cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
10.5 Quartics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
10.6 Guessing the roots . . . . . . . . . . . . . . . . . . . . . . . . 195
11 The matrix (to be written) 199
A Hex and other notational matters 203
A.1 Hexadecimal numerals . . . . . . . . . . . . . . . . . . . . . . 204
A.2 Avoiding notational clutter . . . . . . . . . . . . . . . . . . . 205
B The Greek alphabet 207
C Manuscript history 211
viii CONTENTS
List of Figures
1.1 Two triangles. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1 Multiplicative commutivity. . . . . . . . . . . . . . . . . . . . 8
2.2 The sum of a triangle’s inner angles: turning at the corner. . 32
2.3 A right triangle. . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4 The Pythagorean theorem. . . . . . . . . . . . . . . . . . . . 34
2.5 The complex (or Argand) plane. . . . . . . . . . . . . . . . . 37
3.1 The sine and the cosine. . . . . . . . . . . . . . . . . . . . . . 44
3.2 The sine function. . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3 A two-dimensional vector u= x^x+y^y. . . . . . . . . . . . . 47
3.4 A three-dimensional vector v = x^x+y^y+z^z. . . . . . . . . . 47
3.5 Vector basis rotation.. . . . . . . . . . . . . . . . . . . . . . . 50
3.6 The 0x18 hours in a circle. . . . . . . . . . . . . . . . . . . . . 55
3.7 Calculating the hour trigonometrics. . . . . . . . . . . . . . . 55
3.8 The laws of sines and cosines. . . . . . . . . . . . . . . . . . . 57
3.9 A point on a sphere. . . . . . . . . . . . . . . . . . . . . . . . 61
4.1 The plan for Pascal’s triangle. . . . . . . . . . . . . . . . . . . 70
4.2 Pascal’s triangle. . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.3 A local extremum. . . . . . . . . . . . . . . . . . . . . . . . . 80
4.4 A level in(cid:13)ection. . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.5 The Newton-Raphson iteration. . . . . . . . . . . . . . . . . . 84
5.1 The natural exponential. . . . . . . . . . . . . . . . . . . . . . 90
5.2 The natural logarithm. . . . . . . . . . . . . . . . . . . . . . . 91
5.3 The complex exponential and Euler’s formula. . . . . . . . . . 94
5.4 The derivatives of the sine and cosine functions. . . . . . . . . 99
7.1 Areas representing discrete sums. . . . . . . . . . . . . . . . . 120
ix
x LIST OF FIGURES
7.2 An area representing an in(cid:12)nite sum of in(cid:12)nitesimals. . . . . 122
7.3 Integration by the trapezoid rule. . . . . . . . . . . . . . . . . 124
7.4 The area of a circle. . . . . . . . . . . . . . . . . . . . . . . . 132
7.5 The volume of a cone. . . . . . . . . . . . . . . . . . . . . . . 133
7.6 A sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7.7 An element of a sphere’s surface. . . . . . . . . . . . . . . . . 134
7.8 A contour of integration. . . . . . . . . . . . . . . . . . . . . . 138
7.9 The Heaviside unit step u(t). . . . . . . . . . . . . . . . . . . 139
7.10 The Dirac delta (cid:14)(t). . . . . . . . . . . . . . . . . . . . . . . . 139
8.1 A complex contour of integration in two parts. . . . . . . . . 157
8.2 A Cauchy contour integral. . . . . . . . . . . . . . . . . . . . 161
9.1 Integration by closed contour. . . . . . . . . . . . . . . . . . . 177
10.1 Vieta’s transform, plotted logarithmically. . . . . . . . . . . . 187