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Derivations of applied mathematics PDF

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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

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