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om C Zo ne Derivations of Applied Mathematics Thaddeus H Black Si nh Vi en Revised 14 December 2006 SinhVienZone.com https://fb.com/sinhvienzonevn Si nh Vi en Zo ne C om ii Derivations of Applied Mathematics 14 December 2006 Copyright c 1983–2006 by Thaddeus H Black derivations@b-tk.org 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 SinhVienZone.com https://fb.com/sinhvienzonevn om Contents xiii 1 2 5 Classical algebra and geometry 2.1 Basic arithmetic relationships 2.1.1 Commutivity, associativity, distributivity 2.1.2 Negative numbers 2.1.3 Inequality 2.1.4 The change of variable 2.2 Quadratics 2.3 Notation for series sums and products 2.4 The arithmetic series 2.5 Powers and roots 2.5.1 Notation and integral powers 2.5.2 Roots 2.5.3 Powers of products and powers of powers 2.5.4 Sums of powers 2.5.5 Summary and remarks 2.6 Multiplying and dividing power series 2.6.1 Multiplying power series 2.6.2 Dividing power series 2.6.3 Common quotients and the geometric series 7 10 11 11 13 15 15 15 17 19 19 20 20 21 21 26 Si nh Vi en Zo ne Introduction 1.1 Applied mathematics 1.2 Rigor 1.2.1 Axiom and definition 1.2.2 Mathematical extension 1.3 Complex numbers and complex variables 1.4 On the text C Preface iii SinhVienZone.com https://fb.com/sinhvienzonevn iv CONTENTS ne C om 2.6.4 Variations on the geometric series 2.7 Constants and variables 2.8 Exponentials and logarithms 2.8.1 The logarithm 2.8.2 Properties of the logarithm 2.9 Triangles and other polygons: simple facts 2.9.1 Triangle area 2.9.2 The triangle inequalities 2.9.3 The sum of interior angles 2.10 The Pythagorean theorem 2.11 Functions 2.12 Complex numbers (introduction) 2.12.1 Rectangular complex multiplication 2.12.2 Complex conjugation 2.12.3 Power series and analytic functions (preview) 26 27 29 29 30 30 31 31 32 33 35 36 38 38 40 43 43 45 45 49 51 52 53 53 57 58 60 62 63 The derivative 4.1 Infinitesimals and limits 4.1.1 The infinitesimal 4.1.2 Limits 4.2 Combinatorics 4.2.1 Combinations and permutations 4.2.2 Pascal’s triangle 4.3 The binomial theorem 4.3.1 Expanding the binomial 65 65 66 67 68 68 70 70 70 Si nh Vi en Zo Trigonometry 3.1 Definitions 3.2 Simple properties 3.3 Scalars, vectors, and vector notation 3.4 Rotation 3.5 Trigonometric sums and differences 3.5.1 Variations on the sums and differences 3.5.2 Trigonometric functions of double and half angles 3.6 Trigonometrics of the hour angles 3.7 The laws of sines and cosines 3.8 Summary of properties 3.9 Cylindrical and spherical coordinates 3.10 The complex triangle inequalities 3.11 De Moivre’s theorem SinhVienZone.com https://fb.com/sinhvienzonevn CONTENTS 71 72 73 73 74 76 77 77 78 78 79 80 82 83 om C ne Si 5.9 complex exponential The real exponential The natural logarithm Fast and slow functions Euler’s formula Complex exponentials and de Moivre Complex trigonometrics Summary of properties Derivatives of complex exponentials 5.8.1 Derivatives of sine and cosine 5.8.2 Derivatives of the trigonometrics 5.8.3 Derivatives of the inverse trigonometrics The actuality of complex quantities Zo The 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 en 4.6 4.7 4.8 Vi 4.5 4.3.2 Powers of numbers near unity 4.3.3 Complex powers of numbers near unity The derivative 4.4.1 The derivative of the power series 4.4.2 The Leibnitz notation 4.4.3 The derivative of a function of a complex variable 4.4.4 The derivative of z a 4.4.5 The logarithmic derivative Basic manipulation of the derivative 4.5.1 The derivative chain rule 4.5.2 The derivative product rule Extrema and higher derivatives L’Hˆopital’s rule The Newton-Raphson iteration nh 4.4 v Primes, roots and averages 6.1 Prime numbers 6.1.1 The infinite supply of primes 6.1.2 Compositional uniqueness 6.1.3 Rational and irrational numbers 6.2 The existence and number of roots 6.2.1 Polynomial roots 6.2.2 The fundamental theorem of algebra 6.3 Addition and averages 6.3.1 Serial and parallel addition 6.3.2 Averages SinhVienZone.com 87 87 90 91 92 96 96 97 97 97 100 100 102 105 105 105 106 109 110 110 111 112 112 115 https://fb.com/sinhvienzonevn vi CONTENTS en Zo ne C om The integral 7.1 The concept of the integral 7.1.1 An introductory example 7.1.2 Generalizing the introductory example 7.1.3 The balanced definition and the trapezoid rule 7.2 The antiderivative 7.3 Operators, linearity and multiple integrals 7.3.1 Operators 7.3.2 A formalism 7.3.3 Linearity 7.3.4 Summational and integrodifferential transitivity 7.3.5 Multiple integrals 7.4 Areas and volumes 7.4.1 The area of a circle 7.4.2 The volume of a cone 7.4.3 The surface area and volume of a sphere 7.5 Checking integrations 7.6 Contour integration 7.7 Discontinuities 7.8 Remarks (and exercises) Si nh Vi The Taylor series 8.1 The power series expansion of 1/(1 − z) n+1 8.1.1 The formula 8.1.2 The proof by induction 8.1.3 Convergence 8.1.4 General remarks on mathematical induction 8.2 Shifting a power series’ expansion point 8.3 Expanding functions in Taylor series 8.4 Analytic continuation 8.5 Branch points 8.6 Cauchy’s integral formula 8.6.1 The meaning of the symbol dz 8.6.2 Integrating along the contour 8.6.3 The formula 8.7 Taylor series for specific functions 8.8 Bounds 8.9 Calculating 2π 8.10 The multidimensional Taylor series SinhVienZone.com https://fb.com/sinhvienzonevn 119 119 120 123 123 124 126 126 127 128 129 130 131 131 132 133 136 137 138 141 143 143 144 145 146 148 149 151 152 154 155 156 156 160 161 164 165 166 CONTENTS vii 185 186 186 189 191 193 195 om 169 169 170 171 173 176 178 178 180 182 184 199 en 11 The matrix (to be written) ne Zo 10 Cubics and quartics 10.1 Vieta’s transform 10.2 Cubics 10.3 Superfluous roots 10.4 Edge cases 10.5 Quartics 10.6 Guessing the roots C Integration techniques 9.1 Integration by antiderivative 9.2 Integration by substitution 9.3 Integration by parts 9.4 Integration by unknown coefficients 9.5 Integration by closed contour 9.6 Integration by partial-fraction expansion 9.6.1 Partial-fraction expansion 9.6.2 Multiple poles 9.6.3 Integrating rational functions 9.7 Integration by Taylor series Vi A Hex and other notational matters 203 A.1 Hexadecimal numerals 204 A.2 Avoiding notational clutter 205 207 C Manuscript history 211 Si nh B The Greek alphabet SinhVienZone.com https://fb.com/sinhvienzonevn CONTENTS Si nh Vi en Zo ne C om viii SinhVienZone.com https://fb.com/sinhvienzonevn om List of Figures Two triangles 2.1 2.2 2.3 2.4 2.5 Multiplicative commutivity The sum of a triangle’s inner angles: turning at A right triangle The Pythagorean theorem The complex (or Argand) plane 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 The sine and the cosine The sine function ˆx + y ˆ y A two-dimensional vector u = x ˆx + y ˆ y + zˆz A three-dimensional vector v = x Vector basis rotation The 0x18 hours in a circle Calculating the hour trigonometrics The laws of sines and cosines A point on a sphere 4.1 4.2 4.3 4.4 4.5 The plan for Pascal’s triangle Pascal’s triangle A local extremum A level inflection The Newton-Raphson iteration 5.1 5.2 5.3 5.4 The The The The 7.1 Areas representing discrete sums 120 C 1.1 32 34 34 37 44 45 47 47 50 55 55 57 61 70 71 80 81 84 natural exponential natural logarithm complex exponential and Euler’s formula derivatives of the sine and cosine functions 90 91 94 99 Si nh Vi en Zo ne the corner ix SinhVienZone.com https://fb.com/sinhvienzonevn x LIST OF FIGURES An area representing an infinite sum of infinitesimals Integration by the trapezoid rule The area of a circle The volume of a cone A sphere An element of a sphere’s surface A contour of integration The Heaviside unit step u(t) The Dirac delta δ(t) 8.1 8.2 A complex contour of integration in two parts 157 A Cauchy contour integral 161 9.1 Integration by closed contour 177 C om 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 122 124 132 133 134 134 138 139 139 Si nh Vi en Zo ne 10.1 Vieta’s transform, plotted logarithmically 187 SinhVienZone.com https://fb.com/sinhvienzonevn 216 APPENDIX C MANUSCRIPT HISTORY ’em Si nh Vi en Zo ne C om THB SinhVienZone.com https://fb.com/sinhvienzonevn om Bibliography C [1] http://encyclopedia.laborlawtalk.com/Applied_mathematics As retrieved Sept 2005 ne [2] http://www-history.mcs.st-andrews.ac.uk/ As retrieved 12 Oct 2005 through Nov 2005 [3] Wikipedia http://en.wikipedia.org/ Zo [4] Kristie H Black Private conversation en [5] Thaddeus H Black Derivations of Applied Mathematics The Debian Project, http://www.debian.org/, 14 December 2006 Vi [6] R Courant and D Hilbert Methods of Mathematical Physics Interscience (Wiley), New York, first English edition, 1953 nh [7] The Debian Project http://www.debian.org/ [8] The Debian Project Debian Free Software Guidelines, version 1.1 http://www.debian.org/social_contract#guidelines Si [9] G Doetsch Guide to the Applications of the Laplace and z-Transforms Van Nostrand Reinhold, London, 1971 Referenced indirectly by way of [19] [10] Richard P Feynman, Robert B Leighton, and Matthew Sands The Feynman Lectures on Physics Addison-Wesley, Reading, Mass., 1963– 65 Three volumes [11] The Free Software Foundation GNU General Public License, version /usr/share/common-licenses/GPL-2 on a Debian system The Debian Project: http://www.debian.org/ The Free Software Foundation: 51 Franklin St., Fifth Floor, Boston, Mass 02110-1301, USA 217 SinhVienZone.com https://fb.com/sinhvienzonevn 218 BIBLIOGRAPHY [12] Edward Gibbon The History of the Decline and Fall of the Roman Empire 1788 [13] William Goldman The Princess Bride Ballantine, New York, 1973 [14] Richard W Hamming Methods of Mathematics Applied to Calculus, Probability, and Statistics Books on Mathematics Dover, Mineola, N.Y., 1985 om [15] Francis B Hildebrand Advanced Calculus for Applications PrenticeHall, Englewood Cliffs, N.J., 2nd edition, 1976 .C [16] N.N Lebedev Special Functions and Their Applications Books on Mathematics Dover, Mineola, N.Y., revised English edition, 1965 ne [17] David McMahon Quantum Mechanics Demystified Demystified Series McGraw-Hill, New York, 2006 Zo [18] Ali H Nayfeh and Balakumar Balachandran Applied Nonlinear Dynamics: Analytical, Computational and Experimental Methods Series in Nonlinear Science Wiley, New York, 1995 en [19] Charles L Phillips and John M Parr Signals, Systems and Transforms Prentice-Hall, Englewood Cliffs, N.J., 1995 Vi [20] Carl Sagan Cosmos Random House, New York, 1980 nh [21] Adel S Sedra and Kenneth C Smith Microelectronic Circuits Series in Electrical Engineering Oxford University Press, New York, 3rd edition, 1991 Si [22] Al Shenk Calculus and Analytic Geometry Scott, Foresman & Co., Glenview, Ill., 3rd edition, 1984 [23] William L Shirer The Rise and Fall of the Third Reich Simon & Schuster, New York, 1960 [24] Murray R Spiegel Complex Variables: with an Introduction to Conformal Mapping and Its Applications Schaum’s Outline Series McGrawHill, New York, 1964 [25] Susan Stepney “Euclid’s proof that there are an infinite number of primes” http://www-users.cs.york.ac.uk/susan/cyc/p/ primeprf.htm As retrieved 28 April 2006 SinhVienZone.com https://fb.com/sinhvienzonevn BIBLIOGRAPHY 219 [26] James Stewart, Lothar Redlin, and Saleem Watson Precalculus: Mathematics for Calculus Brooks/Cole, Pacific Grove, Calif., 3rd edition, 1993 [27] Eric W Weisstein Mathworld http://mathworld.wolfram.com/ As retrieved 29 May 2006 http://www.xplora.org/downloads/ Si nh Vi en Zo ne C [29] Xplora Xplora Knoppix Knoppix/ om [28] Eric W Weisstein CRC Concise Encyclopedia of Mathematics Chapman & Hall/CRC, Boca Raton, Fla., 2nd edition, 2003 SinhVienZone.com https://fb.com/sinhvienzonevn Index C om altitude, 35 amortization, 174 analytic continuation, 152 analytic function, 40, 152 angle, 32, 43, 53 double, 53 half, 53 hour, 53 interior, 32 of a polygon, 33 of a triangle, 32 of rotation, 51 right, 43 square, 43 sum of, 32 antiderivative, 124, 169 and the natural logarithm, 170 guessing, 173 antiquity, 185 applied mathematics, 1, 139, 148 arc, 43 arccosine, 43 derivative of, 103 arcsine, 43 derivative of, 103 arctangent, 43 derivative of, 103 area, 7, 33, 131 surface, 131 arg, 37 Argand domain and range planes, 153 Argand plane, 37 Argand, Jean-Robert, 37 arithmetic, arithmetic mean, 116 arithmetic series, 15 Si nh Vi en Zo ne , 49 (zero), dividing by, 66 (one), 7, 43 2π, 32, 43, 203 calculating, 165 δ, 66 , 66 ≡, 11 ←, 11 and , 67 0x, 204 π, 203 d , 137 dz, 156 e, 87 i, 36 nth root calculation by Newton-Raphson, 85 nth-order expression, 185 reductio ad absurdum, 106 16th century, 185 absolute value, 37 accretion, 124 addition parallel, 112, 186 serial, 112 series, 112 algebra classical, fundamental theorem of, 111 higher-order, 185 linear, 199 alternating signs, 164 220 SinhVienZone.com https://fb.com/sinhvienzonevn INDEX 221 C ne Zo baroquity, 195 binomial theorem, 70 blackbody radiation, 26 bond, 77 borrower, 174 boundary condition, 174 bounds on a power series, 164 box, branch point, 36, 154 strategy to avoid, 155 businessperson, 115 Si nh Vi en C and C++, 9, 11 calculus, 65, 119 fundamental theorem of, 124 the two complementary questions of, 65, 119, 125 Cardano, Girolamo (also known as Cardanus or Cardan), 185 Cauchy’s integral formula, 155, 176 Cauchy, Augustine Louis, 155, 176 chain rule, derivative, 78 change, 65 change of variable, 11 change, rate of, 65 checking division, 136 checking integrations, 136 choosing blocks, 68 circle, 43, 53, 92 area of, 131 unit, 43 cis, 97 citation, classical algebra, cleverness, 142, 193 SinhVienZone.com clock, 53 closed analytic form, 184 closed contour integration, 137 closed form, 184 closed surface integration, 135 clutter, notational, 205 coefficient inscrutable, 188 unknown, 173 combination, 68 properties of, 69 combinatorics, 68 commutivity, completing the square, 12 complex conjugation, 38 complex exponent, 94 complex exponential, 87 and de Moivre’s theorem, 96 derivative of, 97 inverse, derivative of, 97 properties, 98 complex number, 5, 36, 63 actuality of, 102 conjugating, 38 imaginary part of, 37 magnitude of, 37 multiplication and division, 38, 63 phase of, 37 real part of, 37 complex plane, 37 complex power, 72 complex trigonometrics, 96 complex variable, 5, 76 composite number, 105 compositional uniqueness of, 106 concert hall, 28 cone volume of, 132 conjugate, 36, 38, 103 conjugation, 38 constant, 27 constant expression, 11 constant, indeterminate, 27 constraint, 194 om arm, 93 assignment, 11 associativity, average, 115 axes, 49 changing, 49 rotation of, 49 axiom, axis, 60 https://fb.com/sinhvienzonevn 222 INDEX C om density, 131 dependent variable, 74 derivation, derivative, 65 balanced form, 73 chain rule for, 78 cross-, 166 definition of, 73 higher, 80 Leibnitz notation for, 74 logarithmic, 77 manipulation of, 78 Newton notation for, 73 of z a /a, 170 of a complex exponential, 97 of a function of a complex variable, 76 of a trigonometric, 101 of an inverse trigonometric, 103 of arcsine, arccosine and arctangent, 103 of sine and cosine, 97 of sine, cosine and tangent, 101 of the natural exponential, 88, 101 of the natural logarithm, 91, 103 of z a , 77 product rule for, 79, 171 second, 80 unbalanced form, 73 DFSG (Debian Free Software Guidelines), diagonal, 33, 43 three-dimensional, 35 differentiability, 76 differential equation, 174 solving by unknown coefficients, 174 differentiation analytical versus numeric, 137 dimensionality, 49 dimensionlessness, 43 Dirac delta function, 139 sifting property of, 139 Dirac, Paul, 139 direction, 45 Si nh Vi en Zo ne contour, 138, 154 complex, 156, 161, 177 contour integration, 137 closed, 137 closed complex, 155, 176 complex, 156 of a vector quantity, 138 contract, 115 contradiction, proof by, 106 convention, 203 convergence, 62, 129, 146 coordinate rotation, 60 coordinates, 60 cylindrical, 60 rectangular, 43, 60 relations among, 61 spherical, 60 corner case, 192 corner value, 186 cosine, 43 derivative of, 97, 101 in complex exponential form, 96 law of cosines, 58 Courant, Richard, cross-derivative, 166 cross-term, 166 cryptography, 105 cubic expression, 11, 112, 185, 186 roots of, 189 cubic formula, 189 cubing, 190 cylindrical coordinates, 60 day, 53 de Moivre’s theorem, 63 and the complex exponential, 96 de Moivre, Abraham, 63 Debian, Debian Free Software Guidelines, definite integral, 136 definition, 2, 139 definition notation, 11 delta function, Dirac, 139 sifting property of, 139 denominator, 21, 178 SinhVienZone.com https://fb.com/sinhvienzonevn INDEX 223 extension, extremum, 80 factorial, 13, 172 factoring, 11 factorization, prime, 106 fast function, 91 Ferrari, Lodovico, 185, 193 flaw, logical, 108 forbidden point, 150, 153 formalism, 128 fourfold integral, 131 Fourier transform spatial, 131 fraction, 178 function, 35, 139 analytic, 40, 152 extremum of, 80 fast, 91 fitting of, 143 linear, 128 nonanalytic, 40 nonlinear, 128 of a complex variable, 76 rational, 179 single- and multiple-valued, 154 single- vs multiple-valued, 153 slow, 91 fundamental theorem of algebra, 111 fundamental theorem of calculus, 124 Si nh Vi en Zo east, 43 edge case, 5, 191 elf, 36 equation solving a set of simultaneously, 51 equator, 133 Euclid, 33, 106 Euler’s formula, 92 curious consequences of, 95 Euler, Leonhard, 92, 94 evaluation, 80 exercises, 141 expansion of 1/(1 − z)n+1 , 143 expansion point shifting, 149 exponent, 15, 29 complex, 94 exponential, 29, 94 general, 90 exponential, complex, 87 and de Moivre’s theorem, 96 exponential, natural, 87 compared to xa , 91 derivative of, 88, 101 existence of, 87 exponential, real, 87 ne C om discontinuity, 138 distributivity, divergence to infinity, 35 dividend, 21 dividing by zero, 66 division, 38, 63 checking, 136 divisor, 21 domain, 35 domain contour, 154 domain neighborhood, 153 double angle, 53 double integral, 131 double pole, 180 double root, 191 down, 43 dummy variable, 13, 127, 176 SinhVienZone.com Gamma function, 172 general exponential, 90 General Public License, GNU, geometric mean, 116 geometric series, 26 geometric series, variations on, 26 geometrical arguments, geometry, 30 GNU General Public License, GNU GPL, Goldman, William, 139 GPL, grapes, 103 Greek alphabet, 207 https://fb.com/sinhvienzonevn 224 INDEX C ne half angle, 53 Hamming, Richard W., 119, 148 handwriting, reflected, 103 harmonic mean, 116 Heaviside unit step function, 138 Heaviside, Oliver, 3, 102, 138 hexadecimal, 204 higher-order algebra, 185 Hilbert, David, hour, 53 hour angle, 53 hyperbolic functions, 96 hypotenuse, 33 Si nh Vi en Zo identity, arithmetic, iff, 113, 128, 146 imaginary number, 36 imaginary part, 37 imaginary unit, 36 indefinite integral, 136 independent infinitesimal variable, 124 independent variable, 74 indeterminate form, 82 index of summation, 13 induction, 38, 145 inequality, 10 infinite differentiability, 152 infinitesimal, 66 and the Leibnitz notation, 74 dropping when negligible, 156 independent, variable, 124 practical size of, 66 second- and higher-order, 67 infinity, 66 inflection, 80 integer, 15 composite, 105 compositional uniqueness of, 106 prime, 105 integral, 119 SinhVienZone.com as accretion or area, 120 as antiderivative, 124 as shortcut to a sum, 121 balanced form, 123 closed complex contour, 155, 176 closed contour, 137 closed surface, 135 complex contour, 156 concept of, 119 contour, 137 definite, 136 double, 131 fourfold, 131 indefinite, 136 multiple, 130 sixfold, 131 surface, 131, 133 triple, 131 vector contour, 138 volume, 131 integral swapping, 130 integrand, 169 integration analytical versus numeric, 137 by antiderivative, 169 by closed contour, 176 by partial-fraction expansion, 178 by parts, 171 by substitution, 170 by Taylor series, 184 by unknown coefficients, 173 checking, 136 integration techniques, 169 interest, 77, 174 inverse complex exponential derivative of, 97 inverse trigonometric family of functions, 97 inversion, arithmetic, irrational number, 109 irreducibility, iteration, 83 om Greenwich, 53 guessing roots, 195 guessing the form of a solution, 173 l’Hˆ opital’s rule, 82 l’Hˆ opital, Guillaume de, 82 https://fb.com/sinhvienzonevn INDEX 225 om mean, 115 arithmetic, 116 geometric, 116 harmonic, 116 minimum, 80 mirror, 103 model, 2, 108 modulus, 37 multiple pole, 35, 180 multiple-valued function, 153, 154 multiplication, 13, 38, 63 nh Si C ne magnitude, 37, 45, 62 majorization, 147, 164 mapping, 35 mason, 112 mass density, 131 mathematician applied, professional, 2, 108 mathematics applied, 1, 139, 148 professional or pure, 2, 108, 139 matrix, 199 maximum, 80 SinhVienZone.com natural exponential, 87 compared to xa , 91 complex, 87 derivative of, 88, 101 existence of, 87 real, 87 natural exponential family of functions, 97 natural logarithm, 90 and the antiderivative, 170 compared to xa , 91 derivative of, 91, 103 of a complex number, 95 natural logarithmic family of functions, 97 neighborhood, 153 Newton, Sir Isaac, 65, 73, 83 Newton-Raphson iteration, 83, 185 nobleman, 139 nonanalytic function, 40 nonanalytic point, 153, 159 normal vector or line, 132 north, 43 number, 36 complex, 5, 36, 63 complex, actuality of, 102 imaginary, 36 irrational, 109 rational, 109 real, 36 very large or very small, 66 number theory, 105 numerator, 21, 178 Zo Vi en law of cosines, 58 law of sines, 57 leg, 33 Leibnitz notation, 74, 124 Leibnitz, Gottfried Wilhelm, 65, 74 length curved, 43 limit, 67 line, 49 linear algebra, 199 linear combination, 128 linear expression, 11, 128, 185 linear operator, 128 linear superposition, 160 linearity, 128 of a function, 128 of an operator, 128 loan, 174 locus, 111 logarithm, 29 properties of, 30 logarithm, natural, 90 and the antiderivative, 170 compared to xa , 91 derivative of, 91, 103 logarithmic derivative, 77 logical flaw, 108 long division, 21 by z − α, 110 procedure for, 24, 25 loop counter, 13 https://fb.com/sinhvienzonevn 226 INDEX Si nh Vi en C ne Zo parallel addition, 112, 186 parallel subtraction, 115 Parseval’s principle, 180 Parseval, Marc-Antoine, 180 partial sum, 164 partial-fraction expansion, 178 Pascal’s triangle, 70 neighbors in, 69 Pascal, Blaise, 70 path integration, 137 payment rate, 174 permutation, 68 Pfufnik, Gorbag, 154 phase, 37 physicist, Planck, Max, 26 plane, 49 plausible assumption, 106 point, 49, 60 in vector notation, 49 pole, 35, 153, 154, 159 circle of, 180 double, 180 multiple, 35, 180 polygon, 30 polynomial, 20, 110 has at least one root, 111 of order N has N roots, 111 power, 15 SinhVienZone.com complex, 72 fractional, 17 integral, 15 notation for, 15 of a power, 19 of a product, 19 properties, 16 real, 18 sum of, 19 power series, 20, 40 bounds on, 164 common quotients of, 26 derivative of, 73 dividing, 21 extending the technique, 26 multiplying, 21 shifting the expansion point of, 149 prime factorization, 106 prime mark ( ), 49 prime number, 105 infinite supply of, 105 relative, 197 product, 13 product rule, derivative, 79, 171 productivity, 115 professional mathematician, 108 professional mathematics, 2, 139 proof, by contradiction, 106 by induction, 38, 145 by sketch, 2, 31 proving backward, 117 pure mathematics, 2, 108 pyramid volume of, 132 Pythagoras, 33 Pythagorean theorem, 33 and the hyperbolic functions, 96 and the sine and cosine functions, 44 in three dimensions, 35 om Observatory, Old Royal, 53 Occam’s razor abusing, 104 Old Royal Observatory, 53 one, 7, 43 operator, 126 + and − as, 127 linear, 128 nonlinear, 128 using a variable up, 127 order, 11 orientation, 49 origin, 43, 49 orthonormal vectors, 60 quadrant, 43 quadratic expression, 11, 112, 185, 188 https://fb.com/sinhvienzonevn INDEX 227 om C scalar, 45 complex, 48 screw, 49 second derivative, 80 selecting blocks, 68 serial addition, 112 series, 13 arithmetic, 15 convergence of, 62 geometric, 26 geometric, variations on, 26 multiplication order of, 14 notation for, 13 product of, 13 sum of, 13 series addition, 112 shape area of, 131 sifting property, 139 sign alternating, 164 Simpson’s rule, 123 sine, 43 derivative of, 97, 101 in complex exponential form, 96 law of sines, 57 single-valued function, 153, 154 singularity, 35, 82 sinusoid, 45 sixfold integral, 131 Si nh Vi en Zo radian, 43, 53 range, 35 range contour, 154 Raphson, Joseph, 83 rate, 65 relative, 77 rate of change, 65 rate of change, instantaneous, 65 ratio, 18, 178 fully reduced, 109 rational function, 179 rational number, 109 rational root, 197 real exponential, 87 real number, 36 approximating as a ratio of integers, 18 real part, 37 rectangle, splitting down the diagonal, 31 rectangular coordinates, 43, 60 regular part, 160 relative primeness, 197 relative rate, 77 remainder, 21 after division by z − α, 110 zero, 110 residue, 160, 179 resolvent cubic, 194 revolution, 43 right triangle, 31, 43 right-hand rule, 49 rigor, 2, 148 rise, 43 Roman alphabet, 207 root, 11, 17, 35, 82, 110, 185 double, 191 finding of numerically, 83 guessing, 195 rational, 197 superfluous, 189 triple, 192 root extraction from a cubic polynomial, 189 from a quadratic polynomial, 12 from a quartic polynomial, 196 rotation, 49 angle of, 51 Royal Observatory, Old, 53 run, 43 ne quadratic formula, 12 quadratics, 11 quartic expression, 11, 112, 185, 193 resolvent cubic of, 194 roots of, 196 quartic formula, 196 quintic expression, 11, 112, 197 quotient, 21, 178 SinhVienZone.com https://fb.com/sinhvienzonevn 228 INDEX C om tapered strip, 133 Tartaglia, Niccol` o Fontana, 185 Taylor expansion, first-order, 72 Taylor series, 143, 151 converting a power series to, 149 for specific functions, 161 integration by, 184 multidimensional, 166 transposing to a different expansion point, 152 Taylor, Brook, 143, 151 term cross-, 166 finite number of, 164 time and space, 131 transitivity summational and integrodifferential, 129 trapezoid rule, 123 triangle, 30, 57 area of, 31 equilateral, 54 right, 31, 43 triangle inequalities, 31 complex, 62 vector, 62 trigonometric family of functions, 97 trigonometric function, 43 derivative of, 101 inverse, 43 inverse, derivative of, 103 of a double or half angle, 53 of a hour angle, 53 of a sum or difference of angles, 51 trigonometrics, complex, 96 trigonometry, 43 properties, 46, 59 triple integral, 131 triple root, 192 Si nh Vi en Zo ne sketch, proof by, 2, 31 slope, 43, 80 slow function, 91 solid surface area of, 133 volume of, 131 solution guessing the form of, 173 sound, 27 south, 43 space, 49 space and time, 131 sphere, 61 surface area of, 133 volume of, 135 spherical coordinates, 60 square, 54 tilted, 33 square root, 17, 36 calculation by Newton-Raphson, 85 square, completing the, 12 squares, sum or difference of, 11 squaring, 190 strip, tapered, 133 style, 3, 108, 141 subtraction parallel, 115 sum, 13 partial, 164 summation, 13 convergence of, 62 superfluous root, 189 superposition, 103, 160 surface, 130 surface area, 133 surface integral, 131 surface integration, 133 closed, 135 symmetry, appeal to, 49 tangent, 43 derivative of, 101 in complex exponential form, 96 tangent line, 83, 88 SinhVienZone.com unit, 36, 43, 45 imaginary, 36 real, 36 unit basis vector, 45 https://fb.com/sinhvienzonevn INDEX 229 cylindrical, 60 spherical, 60 variable, 60 unit circle, 43 unit step function, Heaviside, 138 unity, 7, 43, 45 unknown coefficient, 173 unsureness, logical, 108 up, 43 utility variable, 57 complex, 103 propagating, 103 west, 43 C ne Si nh Vi en Zo variable, 27 assignment, 11 change of, 11 complex, 5, 76 definition notation for, 11 dependent, 27, 74 independent, 27, 74 utility, 57 variable independent infinitesimal, 124 variable dτ , 124 vector, 45, 166 generalized, 166 integer, 167 nonnegative integer, 167 notation for, 45 orthonormal, 60 point, 49 rotation of, 49 three-dimensional, 47 two-dimensional, 47 unit, 45 unit basis, 45 unit basis, cylindrical, 60 unit basis, spherical, 60 unit basis, variable, 60 vertex, 132 Vieta (Fran¸cois Vi`ete), 185, 186 Vieta’s parallel transform, 186 Vieta’s substitution, 186 Vieta’s transform, 186, 187 volume, 7, 130, 131 volume integral, 131 om zero, 7, 35 dividing by, 66 wave SinhVienZone.com https://fb.com/sinhvienzonevn INDEX Si nh Vi en Zo ne C om 230 SinhVienZone.com https://fb.com/sinhvienzonevn ... introductory matters of general interest Si What is applied mathematics? Applied mathematics is a branch of mathematics that concerns itself with the application of mathematical knowledge to other... physicists have ceased to appreciate the attitudes of mathematicians [6, Preface] Si Although the present book treats “the attitudes of mathematicians” with greater deference than some of the... stylistic error to mix the two indiscriminately, clearly the two have much to with one another However this may be, this book is a book of derivations of applied mathematics The derivations here
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