Mathematics Formulary     By ir. J.C.A. Wevers
c 1999, 2008 J.C.A. Wevers                                                                    Version: May 4, 2008Dear rea...
ContentsContents                                                                                                          ...
II                                                                              Mathematics Formulary door J.C.A. Wevers4 ...
Mathematics Formulary by J.C.A. Wevers                                                                                    ...
IV   Mathematics Formulary door J.C.A. Wevers
Chapter 1Basics1.1     Goniometric functionsFor the goniometric ratios for a point p on the unit circle holds:            ...
2                                                                   Mathematics Formulary by ir. J.C.A. Wevers1.3     Calc...
Chapter 1: Basics                                                                                                         ...
4                                                                  Mathematics Formulary by ir. J.C.A. Wevers1.6     Geome...
Chapter 1: Basics                                                                                                         ...
6                                                                                 Mathematics Formulary by ir. J.C.A. Weve...
Chapter 1: Basics                                                                                                   71.9  ...
8                                                             Mathematics Formulary by ir. J.C.A. WeversIf π(x) is the num...
Chapter 2Probability and statistics2.1     CombinationsThe number of possible combinations of k elements from n elements i...
10                                                                     Mathematics Formulary by ir. J.C.A. WeversThe covar...
Chapter 2: Probability and statistics                                                                              11   6....
Chapter 3Calculus3.1     Integrals3.1.1 Arithmetic rulesThe primitive function F (x) of f (x) obeys the rule F (x) = f (x)...
Chapter 3: Calculus                                                                                              13The vol...
14                                                                            Mathematics Formulary by ir. J.C.A. WeversTh...
Chapter 3: Calculus                                                                                                     15...
16                                                                        Mathematics Formulary by ir. J.C.A. Wevers3.2.4 ...
Chapter 3: Calculus                                                                                                17     ...
18                                                                                      Mathematics Formulary by ir. J.C.A...
Chapter 3: Calculus                                                                          19A Fourier series can also b...
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  1. 1. Mathematics Formulary By ir. J.C.A. Wevers
  2. 2. c 1999, 2008 J.C.A. Wevers Version: May 4, 2008Dear reader,This document contains 66 pages with mathematical equations intended for physicists and engineers. It is intendedto be a short reference for anyone who often needs to look up mathematical equations.This document can also be obtained from the author, Johan Wevers (johanw@vulcan.xs4all.nl).It can also be found on the WWW on http://www.xs4all.nl/˜johanw/index.html.This document is Copyright by J.C.A. Wevers. All rights reserved. Permission to use, copy and distribute thisunmodified document by any means and for any purpose except profit purposes is hereby granted. Reproducing thisdocument by any means, included, but not limited to, printing, copying existing prints, publishing by electronic orother means, implies full agreement to the above non-profit-use clause, unless upon explicit prior written permissionof the author.The C code for the rootfinding via Newtons method and the FFT in chapter 8 are from “Numerical Recipes in C ”,2nd Edition, ISBN 0-521-43108-5.The Mathematics Formulary is made with teTEX and LTEX version 2.09. AIf you prefer the notation in which vectors are typefaced in boldface, uncomment the redefinition of the veccommand and recompile the file.If you find any errors or have any comments, please let me know. I am always open for suggestions and possiblecorrections to the mathematics formulary.Johan Wevers
  3. 3. ContentsContents I1 Basics 1 1.1 Goniometric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.4 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.5 Complex numbers and quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.5.1 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.5.2 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.6 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.6.1 Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.6.2 Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.7 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.8 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.8.1 Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.8.2 Convergence and divergence of series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.8.3 Convergence and divergence of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.9 Products and quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.10 Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.11 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.12 Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Probability and statistics 9 2.1 Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Probability theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3.2 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 Regression analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Calculus 12 3.1 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.1.1 Arithmetic rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.1.2 Arc lengts, surfaces and volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.1.3 Separation of quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.1.4 Special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.1.5 Goniometric integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 Functions with more variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2.1 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2.2 Taylor series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2.3 Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2.4 The -operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2.5 Integral theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2.6 Multiple integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2.7 Coordinate transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.3 Orthogonality of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.4 Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 I
  4. 4. II Mathematics Formulary door J.C.A. Wevers4 Differential equations 20 4.1 Linear differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.1.1 First order linear DE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.1.2 Second order linear DE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.1.3 The Wronskian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.1.4 Power series substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.2 Some special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.2.1 Frobenius’ method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.2.2 Euler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.2.3 Legendre’s DE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.2.4 The associated Legendre equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.2.5 Solutions for Bessel’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.2.6 Properties of Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2.7 Laguerre’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2.8 The associated Laguerre equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.2.9 Hermite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.2.10 Chebyshev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.2.11 Weber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.3 Non-linear differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.4 Sturm-Liouville equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.5 Linear partial differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.5.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.5.2 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.5.3 Potential theory and Green’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 Linear algebra 29 5.1 Vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5.2 Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5.3 Matrix calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5.3.1 Basic operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5.3.2 Matrix equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.4 Linear transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.5 Plane and line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.6 Coordinate transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.7 Eigen values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.8 Transformation types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.9 Homogeneous coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.10 Inner product spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.11 The Laplace transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.12 The convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.13 Systems of linear differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.14 Quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5.14.1 Quadratic forms in I 2 . . . . . R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5.14.2 Quadratic surfaces in I 3 . . . . R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 Complex function theory 39 6.1 Functions of complex variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 6.2 Complex integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 6.2.1 Cauchy’s integral formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 6.2.2 Residue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6.3 Analytical functions definied by series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 6.4 Laurent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 6.5 Jordan’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
  5. 5. Mathematics Formulary by J.C.A. Wevers III7 Tensor calculus 43 7.1 Vectors and covectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 7.2 Tensor algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 7.3 Inner product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 7.4 Tensor product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 7.5 Symmetric and antisymmetric tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 7.6 Outer product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 7.7 The Hodge star operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 7.8 Differential operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 7.8.1 The directional derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 7.8.2 The Lie-derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 7.8.3 Christoffel symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 7.8.4 The covariant derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 7.9 Differential operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 7.10 Differential geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 7.10.1 Space curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 7.10.2 Surfaces in I 3 . . . . . . . . . R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 7.10.3 The first fundamental tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 7.10.4 The second fundamental tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 7.10.5 Geodetic curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 7.11 Riemannian geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 Numerical mathematics 51 8.1 Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 8.2 Floating point representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 8.3 Systems of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 8.3.1 Triangular matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 8.3.2 Gauss elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 8.3.3 Pivot strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 8.4 Roots of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 8.4.1 Successive substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 8.4.2 Local convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 8.4.3 Aitken extrapolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 8.4.4 Newton iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 8.4.5 The secant method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 8.5 Polynomial interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 8.6 Definite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 8.7 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 8.8 Differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 8.9 The fast Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
  6. 6. IV Mathematics Formulary door J.C.A. Wevers
  7. 7. Chapter 1Basics1.1 Goniometric functionsFor the goniometric ratios for a point p on the unit circle holds: yp cos(φ) = xp , sin(φ) = yp , tan(φ) = xpsin2 (x) + cos2 (x) = 1 and cos−2 (x) = 1 + tan2 (x). cos(a ± b) = cos(a) cos(b) sin(a) sin(b) , sin(a ± b) = sin(a) cos(b) ± cos(a) sin(b) tan(a) ± tan(b) tan(a ± b) = 1 tan(a) tan(b)The sum formulas are: sin(p) + sin(q) = 2 sin( 1 (p + q)) cos( 1 (p − q)) 2 2 sin(p) − sin(q) = 2 cos( 1 (p + q)) sin( 1 (p − q)) 2 2 cos(p) + cos(q) = 2 cos( 2 (p + q)) cos( 1 (p − q)) 1 2 cos(p) − cos(q) = −2 sin( 1 (p + q)) sin( 1 (p − q)) 2 2From these equations can be derived that 2 cos2 (x) = 1 + cos(2x) , 2 sin2 (x) = 1 − cos(2x) sin(π − x) = sin(x) , cos(π − x) = − cos(x) 1 sin( 2 π − x) = cos(x) , cos( 1 π − x) = sin(x) 2Conclusions from equalities: sin(x) = sin(a) ⇒ x = a ± 2kπ or x = (π − a) ± 2kπ, k ∈ IN cos(x) = cos(a) ⇒ x = a ± 2kπ or x = −a ± 2kπ π tan(x) = tan(a) ⇒ x = a ± kπ and x = ± kπ 2The following relations exist between the inverse goniometric functions: x 1 arctan(x) = arcsin √ = arccos √ , sin(arccos(x)) = 1 − x2 x2 +1 x2 +11.2 Hyperbolic functionsThe hyperbolic functions are defined by: ex − e−x ex + e−x sinh(x) sinh(x) = , cosh(x) = , tanh(x) = 2 2 cosh(x)From this follows that cosh2 (x) − sinh2 (x) = 1. Further holds: arsinh(x) = ln |x + x2 + 1| , arcosh(x) = arsinh( x2 − 1) 1
  8. 8. 2 Mathematics Formulary by ir. J.C.A. Wevers1.3 CalculusThe derivative of a function is defined as: df f (x + h) − f (x) = lim dx h→0 hDerivatives obey the following algebraic rules: x ydx − xdy d(x ± y) = dx ± dy , d(xy) = xdy + ydx , d = y y2For the derivative of the inverse function f inv (y), defined by f inv (f (x)) = x, holds at point P = (x, f (x)): df inv (y) df (x) · =1 dy P dx PChain rule: if f = f (g(x)), then holds df df dg = dx dg dxFurther, for the derivatives of products of functions holds: n (n) n (n−k) (k) (f · g) = f ·g k k=0For the primitive function F (x) holds: F (x) = f (x). An overview of derivatives and primitives is: y = f (x) dy/dx = f (x) f (x)dx axn anxn−1 a(n + 1)−1 xn+1 1/x −x−2 ln |x| a 0 ax ax ax ln(a) ax / ln(a) ex ex ex a log(x) (x ln(a))−1 (x ln(x) − x)/ ln(a) ln(x) 1/x x ln(x) − x sin(x) cos(x) − cos(x) cos(x) − sin(x) sin(x) tan(x) cos−2 (x) − ln | cos(x)| sin−1 (x) − sin−2 (x) cos(x) ln | tan( 1 x)| 2 sinh(x) cosh(x) cosh(x) cosh(x) sinh(x) √ sinh(x) √ arcsin(x) 1/ √ − x2 1 x arcsin(x) + √1 − x2 arccos(x) −1/ 1 − x2 x arccos(x) − 1 − x2 arctan(x) (1 + x2 )−1 x arctan(x) − 1 ln(1 + x2 ) 2 √ (a + x2 )−1/2 −x(a + x2 )−3/2 ln |x + a + x2 | 1 (a2 − x2 )−1 2x(a2 + x2 )−2 ln |(a + x)/(a − x)| 2a (1 + (y )2 )3/2The curvature ρ of a curve is given by: ρ = |y | f (x) f (x)The theorem of De ’l Hˆ pital: if f (a) = 0 and g(a) = 0, then is lim o = lim x→a g(x) x→a g (x)
  9. 9. Chapter 1: Basics 31.4 Limits sin(x) ex − 1 tan(x) n x lim = 1 , lim = 1 , lim = 1 , lim (1 + k)1/k = e , lim 1+ = en x→0 x x→0 x x→0 x k→0 x→∞ x lnp (x) ln(x + a) xp lim xa ln(x) = 0 , lim a = 0 , lim =a , lim = 0 als |a| > 1. x↓0 x→∞ x x→0 x x→∞ ax arcsin(x) √ lim a1/x − 1 = ln(a) , lim =1 , lim x x=1 x→0 x→0 x x→∞1.5 Complex numbers and quaternions1.5.1 Complex numbers √The complex number z = a + bi with a and b ∈ I a is the real part, b the imaginary part of z. |z| = a2 + b2 . R.By definition holds: i2 = −1. Every complex number can be written as z = |z| exp(iϕ), with tan(ϕ) = b/a. Thecomplex conjugate of z is defined as z = z ∗ := a − bi. Further holds: (a + bi)(c + di) = (ac − bd) + i(ad + bc) (a + bi) + (c + di) = a + c + i(b + d) a + bi (ac + bd) + i(bc − ad) = c + di c2 + d2Goniometric functions can be written as complex exponents: 1 ix sin(x) = (e − e−ix ) 2i 1 ix cos(x) = (e + e−ix ) 2From this follows that cos(ix) = cosh(x) and sin(ix) = i sinh(x). Further follows from this thate±ix = cos(x) ± i sin(x), so eiz = 0∀z. Also the theorem of De Moivre follows from this:(cos(ϕ) + i sin(ϕ))n = cos(nϕ) + i sin(nϕ).Products and quotients of complex numbers can be written as: z1 · z2 = |z1 | · |z2 |(cos(ϕ1 + ϕ2 ) + i sin(ϕ1 + ϕ2 )) z1 |z1 | = (cos(ϕ1 − ϕ2 ) + i sin(ϕ1 − ϕ2 )) z2 |z2 |The following can be derived: |z1 + z2 | ≤ |z1 | + |z2 | , |z1 − z2 | ≥ | |z1 | − |z2 | |And from z = r exp(iθ) follows: ln(z) = ln(r) + iθ, ln(z) = ln(z) ± 2nπi.1.5.2 QuaternionsQuaternions are defined as: z = a + bi + cj + dk, with a, b, c, d ∈ I and i2 = j 2 = k 2 = −1. The products of Ri, j, k with each other are given by ij = −ji = k, jk = −kj = i and ki = −ik = j.
  10. 10. 4 Mathematics Formulary by ir. J.C.A. Wevers1.6 Geometry1.6.1 TrianglesThe sine rule is: a b c = = sin(α) sin(β) sin(γ)Here, α is the angle opposite to a, β is opposite to b and γ opposite to c. The cosine rule is: a2 = b2 +c2 −2bc cos(α).For each triangle holds: α + β + γ = 180◦ .Further holds: 1 tan( 2 (α + β)) a+b 1 = tan( 2 (α − β)) a−bThe surface of a triangle is given by 2 ab sin(γ) = 1 aha = 1 2 s(s − a)(s − b)(s − c) with ha the perpendicular on 1a and s = 2 (a + b + c).1.6.2 CurvesCycloid: if a circle with radius a rolls along a straight line, the trajectory of a point on this circle has the followingparameter equation: x = a(t + sin(t)) , y = a(1 + cos(t))Epicycloid: if a small circle with radius a rolls along a big circle with radius R, the trajectory of a point on the smallcircle has the following parameter equation: R+a R+a x = a sin t + (R + a) sin(t) , y = a cos t + (R + a) cos(t) a aHypocycloid: if a small circle with radius a rolls inside a big circle with radius R, the trajectory of a point on thesmall circle has the following parameter equation: R−a R−a x = a sin t + (R − a) sin(t) , y = −a cos t + (R − a) cos(t) a aA hypocycloid with a = R is called a cardioid. It has the following parameterequation in polar coordinates:r = 2a[1 − cos(ϕ)].1.7 VectorsThe inner product is defined by: a · b = ai bi = |a | · |b | cos(ϕ) iwhere ϕ is the angle between a and b. The external product is in I 3 defined by: R   ay bz − az by ex ey ez a × b =  az bx − ax bz  = ax ay az ax by − ay bx bx by bzFurther holds: |a × b | = |a | · |b | sin(ϕ), and a × (b × c ) = (a · c )b − (a · b )c.
  11. 11. Chapter 1: Basics 51.8 Series1.8.1 ExpansionThe Binomium of Newton is: n n n−k k (a + b)n = a b k k=0 n n!where := . k k!(n − k)! n nBy subtracting the series rk and r rk one finds: k=0 k=0 n 1 − rn+1 rk = 1−r k=0 ∞ 1and for |r| < 1 this gives the geometric series: rk = . 1−r k=0 N 1The arithmetic series is given by: (a + nV ) = a(N + 1) + 2 N (N + 1)V . n=0The expansion of a function around the point a is given by the Taylor series: (x − a)2 (x − a)n (n) f (x) = f (a) + (x − a)f (a) + f (a) + · · · + f (a) + R 2 n!where the remainder is given by: hn (n+1) Rn (h) = (1 − θ)n f (θh) n!and is subject to: mhn+1 M hn+1 ≤ Rn (h) ≤ (n + 1)! (n + 1)!From this one can deduce that ∞ α n (1 − x)α = x n=0 nOne can derive that: ∞ ∞ ∞ 1 π2 1 π4 1 π6 = , = , = n=1 n2 6 n=1 n4 90 n=1 n6 945 n ∞ ∞ (−1)n+1 π2 (−1)n+1 k 2 = 1 n(n + 1)(2n + 1) , 6 = , = ln(2) n=1 n2 12 n=1 n k=1 ∞ ∞ ∞ ∞ 1 1 1 π2 1 π4 (−1)n+1 π3 2−1 = 2 , = , = , = n=1 4n n=1 (2n − 1)2 8 n=1 (2n − 1)4 96 n=1 (2n − 1)3 321.8.2 Convergence and divergence of seriesIf |un | converges, un also converges. n nIf lim un = 0 then un is divergent. n→∞ nAn alternating series of which the absolute values of the terms drop monotonously to 0 is convergent (Leibniz).
  12. 12. 6 Mathematics Formulary by ir. J.C.A. Wevers ∞If p f (x)dx < ∞, then fn is convergent. nIf un > 0 ∀n then is un convergent if ln(un + 1) is convergent. n n 1 cn+1If un = cn xn the radius of convergence ρ of un is given by: = lim n |cn | = lim . n ρ n→∞ n→∞ cn ∞ 1The series is convergent if p > 1 and divergent if p ≤ 1. n=1 np unIf: lim = p, than the following is true: if p > 0 than un and vn are both divergent or both convergent, if n→∞vn n np = 0 holds: if vn is convergent, than un is also convergent. n n n un+1If L is defined by: L = lim |nn |, or by: L = lim , then is un divergent if L > 1 and convergent if n→∞ n→∞ un nL < 1.1.8.3 Convergence and divergence of functionsf (x) is continuous in x = a only if the upper - and lower limit are equal: lim f (x) = lim f (x). This is written as: x↑a x↓af (a− ) = f (a+ ).If f (x) is continuous in a and: lim f (x) = lim f (x), than f (x) is differentiable in x = a. x↑a x↓aWe define: f W := sup(|f (x)| |x ∈ W ), and lim fn (x) = f (x). Than holds: {fn } is uniform convergent if x→∞ lim fn − f = 0, or: ∀(ε > 0)∃(N )∀(n ≥ N ) fn − f < ε.n→∞Weierstrass’ test: if un W is convergent, than un is uniform convergent. ∞ bWe define S(x) = un (x) and F (y) = f (x, y)dx := F . Than it can be proved that: n=N a Theorem For Demands on W Than holds on W rows fn continuous, f is continuous {fn } uniform convergent C series S(x) uniform convergent, S is continuous un continuous integral f is continuous F is continuous rows fn can be integrated, fn can be integrated, {fn } uniform convergent f (x)dx = lim fn dx n→∞ I series S(x) is uniform convergent, S can be integrated, Sdx = un dx un can be integrated integral f is continuous F dy = f (x, y)dxdy rows {fn } ∈C−1 ; {fn } unif.conv → φ f = φ(x) D series un ∈C−1 ; un conv; un u.c. S (x) = un (x) integral ∂f /∂y continuous Fy = fy (x, y)dx
  13. 13. Chapter 1: Basics 71.9 Products and quotientsFor a, b, c, d ∈ I holds: RThe distributive property: (a + b)(c + d) = ac + ad + bc + bdThe associative property: a(bc) = b(ac) = c(ab) and a(b + c) = ab + acThe commutative property: a + b = b + a, ab = ba.Further holds: n a2n − b2n a2n+1 − b2n+1 = a2n−1 ± a2n−2 b + a2n−3 b2 ± · · · ± b2n−1 , = a2n−k b2k a±b a+b k=0 a3 ± b3 (a ± b)(a2 ± ab + b2 ) = a3 ± b3 , (a + b)(a − b) = a2 + b2 , = a2 ba + b2 a+b1.10 LogarithmsDefinition: a log(x) = b ⇔ ab = x. For logarithms with base e one writes ln(x).Rules: log(xn ) = n log(x), log(a) + log(b) = log(ab), log(a) − log(b) = log(a/b).1.11 PolynomialsEquations of the type n ak xk = 0 k=0have n roots which may be equal to each other. Each polynomial p(z) of order n ≥ 1 has at least one root in C . Ifall ak ∈ I holds: when x = p with p ∈ C a root, than p∗ is also a root. Polynomials up to and including order 4 Rhave a general analytical solution, for polynomials with order ≥ 5 there does not exist a general analytical solution.For a, b, c ∈ I and a = 0 holds: the 2nd order equation ax2 + bx + c = 0 has the general solution: R √ −b ± b2 − 4ac x= 2aFor a, b, c, d ∈ I and a = 0 holds: the 3rd order equation ax3 + bx2 + cx + d = 0 has the general analytical Rsolution: 3ac − b2 b x1 = K− − 9a2 K 3a √ K 3ac − b2 b 3 3ac − b2 x2 = x∗ 3 = − + − +i K+ 2 18a2 K 3a 2 9a2 K √ √ 1/3 9abc − 27da2 − 2b3 3 4ac3 − c2 b2 − 18abcd + 27a2 d2 + 4db3with K = + 54a3 18a21.12 PrimesA prime is a number ∈ I that can only be divided by itself and 1. There are an infinite number of primes. Proof: Nsuppose that the collection of primes P would be finite, than construct the number q = 1 + p, than holds p∈Pq = 1(p) and so Q cannot be written as a product of primes from P . This is a contradiction.
  14. 14. 8 Mathematics Formulary by ir. J.C.A. WeversIf π(x) is the number of primes ≤ x, than holds: π(x) π(x) lim = 1 and lim =1 x→∞ x/ ln(x) x→∞ x dt ln(t) 2For each N ≥ 2 there is a prime between N and 2N .The numbers Fk := 2k + 1 with k ∈ I are called Fermat numbers. Many Fermat numbers are prime. NThe numbers Mk := 2k − 1 are called Mersenne numbers. They occur when one searches for perfect numbers,which are numbers n ∈ I which are the sum of their different dividers, for example 6 = 1 + 2 + 3. There Nare 23 Mersenne numbers for k < 12000 which are prime: for k ∈ {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521,607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213}.To check if a given number n is prime one can use a sieve method. The first known sieve method was developed byEratosthenes. A faster method for large numbers are the 4 Fermat tests, who don’t prove that a number is prime butgive a large probability. 1. Take the first 4 primes: b = {2, 3, 5, 7}, 2. Take w(b) = bn−1 mod n, for each b, 3. If w = 1 for each b, then n is probably prime. For each other value of w, n is certainly not prime.
  15. 15. Chapter 2Probability and statistics2.1 CombinationsThe number of possible combinations of k elements from n elements is given by n n! = k k!(n − k)!The number of permutations of p from n is given by n! n = p! (n − p)! pThe number of different ways to classify ni elements in i groups, when the total number of elements is N , is N! ni ! i2.2 Probability theoryThe probability P (A) that an event A occurs is defined by: n(A) P (A) = n(U )where n(A) is the number of events when A occurs and n(U ) the total number of events.The probability P (¬A) that A does not occur is: P (¬A) = 1 − P (A). The probability P (A ∪ B) that A andB both occur is given by: P (A ∪ B) = P (A) + P (B) − P (A ∩ B). If A and B are independent, than holds:P (A ∩ B) = P (A) · P (B).The probability P (A|B) that A occurs, given the fact that B occurs, is: P (A ∩ B) P (A|B) = P (B)2.3 Statistics2.3.1 GeneralThe average or mean value x of a collection of values is: x = i xi /n. The standard deviation σx in thedistribution of x is given by: n (xi − x )2 i=1 σx = n nWhen samples are being used the sample variance s is given by s2 = σ2 . n−1 9
  16. 16. 10 Mathematics Formulary by ir. J.C.A. WeversThe covariance σxy of x and y is given by:: n (xi − x )(yi − y ) i=1 σxy = n−1The correlation coefficient rxy of x and y than becomes: rxy = σxy /σx σy .The standard deviation in a variable f (x, y) resulting from errors in x and y is: 2 2 2 ∂f ∂f ∂f ∂f σf (x,y) = σx + σy + σxy ∂x ∂y ∂x ∂y2.3.2 Distributions 1. The Binomial distribution is the distribution describing a sampling with replacement. The probability for success is p. The probability P for k successes in n trials is then given by: n k P (x = k) = p (1 − p)n−k k The standard deviation is given by σx = np(1 − p) and the expectation value is ε = np. 2. The Hypergeometric distribution is the distribution describing a sampling without replacement in which the order is irrelevant. The probability for k successes in a trial with A possible successes and B possible failures is then given by: A B k n−k P (x = k) = A+B n The expectation value is given by ε = nA/(A + B). 3. The Poisson distribution is a limiting case of the binomial distribution when p → 0, n → ∞ and also np = λ is constant. λx e−λ P (x) = x! ∞ This distribution is normalized to P (x) = 1. x=0 4. The Normal distribution is a limiting case of the binomial distribution for continuous variables: 2 1 1 x− x P (x) = √ exp − σ 2π 2 σ 5. The Uniform distribution occurs when a random number x is taken from the set a ≤ x ≤ b and is given by:  1  P (x) = if a ≤ x ≤ b b−a    P (x) = 0 in all other cases (b − a)2 x = 2 (b − a) and σ 2 = 1 . 12
  17. 17. Chapter 2: Probability and statistics 11 6. The Gamma distribution is given by: xα−1 e−x/β P (x) = if 0 ≤ y ≤ ∞ β α Γ(α) with α > 0 and β > 0. The distribution has the following properties: x = αβ, σ 2 = αβ 2 . 7. The Beta distribution is given by:  α−1  P (x) = x (1 − x)β−1 if 0 ≤ x ≤ 1   β(α, β)   P (x) = 0 everywhere else  α αβ and has the following properties: x = , σ2 = 2 (α + β + 1) . α+β (α + β) For P (χ2 ) holds: α = V /2 and β = 2. 8. The Weibull distribution is given by:  α α  P (x) = xα−1 e−x if 0 ≤ x ≤ ∞ ∧ α ∧ β > 0  β  P (x) = 0 in all other cases  The average is x = β 1/α Γ((α + 1)α) 9. For a two-dimensional distribution holds: P1 (x1 ) = P (x1 , x2 )dx2 , P2 (x2 ) = P (x1 , x2 )dx1 with ε(g(x1 , x2 )) = g(x1 , x2 )P (x1 , x2 )dx1 dx2 = g·P x1 x22.4 Regression analysesWhen there exists a relation between the quantities x and y of the form y = ax + b and there is a measured set xiwith related yi , the following relation holds for a and b with x = (x1 , x2 , ..., xn ) and e = (1, 1, ..., 1): y − ax − be ∈< x, e >⊥From this follows that the inner products are 0: (y, x ) − a(x, x ) − b(e, x ) = 0 (y, e ) − a(x, e ) − b(e, e ) = 0with (x, x ) = x2 , (x, y ) = i xi yi , (x, e ) = xi and (e, e ) = n. a and b follow from this. i i iA similar method works for higher order polynomial fits: for a second order fit holds: y − ax2 − bx − ce ∈< x2 , x, e >⊥with x2 = (x2 , ..., x2 ). 1 nThe correlation coefficient r is a measure for the quality of a fit. In case of linear regression it is given by: n xy − x y r= (n x2 −( x)2 )(n y2 −( y)2 )
  18. 18. Chapter 3Calculus3.1 Integrals3.1.1 Arithmetic rulesThe primitive function F (x) of f (x) obeys the rule F (x) = f (x). With F (x) the primitive of f (x) holds for thedefinite integral b f (x)dx = F (b) − F (a) aIf u = f (x) holds: b f (b) g(f (x))df (x) = g(u)du a f (a)Partial integration: with F and G the primitives of f and g holds: df (x) f (x) · g(x)dx = f (x)G(x) − G(x) dx dxA derivative can be brought under the intergral sign (see section 1.8.3 for the required conditions):   x=h(y) x=h(y) d  ∂f (x, y) dg(y) dh(y) f (x, y)dx = dx − f (g(y), y) + f (h(y), y)  dy ∂y dy dy  x=g(y) x=g(y)3.1.2 Arc lengts, surfaces and volumesThe arc length of a curve y(x) is given by: 2 dy(x) = 1+ dx dxThe arc length of a parameter curve F (x(t)) is: = F ds = ˙ F (x(t))|x(t)|dtwith dx ˙ x(t) t= = , |t | = 1 ds ˙ |x(t)| (v, t)ds = ˙ (v, t(t))dt = (v1 dx + v2 dy + v3 dz)The surface A of a solid of revolution is: 2 dy(x) A = 2π y 1+ dx dx 12
  19. 19. Chapter 3: Calculus 13The volume V of a solid of revolution is: V =π f 2 (x)dx3.1.3 Separation of quotientsEvery rational function P (x)/Q(x) where P and Q are polynomials can be written as a linear combination offunctions of the type (x − a)k with k ∈ Z and of functions of the type Z, px + q ((x − a)2 + b2 )nwith b > 0 and n ∈ I . So: N n n p(x) Ak p(x) Ak x + B = , = (x − a)n (x − a)k ((x − b)2 + c2 )n ((x − b)2 + c2 )k k=1 k=1Recurrent relation: for n = 0 holds: dx 1 x 2n − 1 dx = + (x2 + 1)n+1 2n (x2 + 1)n 2n (x2 + 1)n3.1.4 Special functionsElliptic functionsElliptic functions can be written as a power series as follows: ∞ (2n − 1)!! 1 − k 2 sin2 (x) = 1 − k 2n sin2n (x) n=1 (2n)!!(2n − 1) ∞ 1 (2n − 1)!! 2n 2n =1+ k sin (x) 1 − k 2 sin2 (x) n=1 (2n)!!with n!! = n(n − 2)!!.The Gamma functionThe gamma function Γ(y) is defined by: ∞ Γ(y) = e−x xy−1 dx 0One can derive that Γ(y + 1) = yΓ(y) = y!. This is a way to define faculties for non-integers. Further one canderive that √ ∞ π Γ(n + 2 ) = n (2n − 1)!! and Γ (y) = e−x xy−1 lnn (x)dx 1 (n) 2 0The Beta functionThe betafunction β(p, q) is defined by: 1 β(p, q) = xp−1 (1 − x)q−1 dx 0with p and q > 0. The beta and gamma functions are related by the following equation: Γ(p)Γ(q) β(p, q) = Γ(p + q)
  20. 20. 14 Mathematics Formulary by ir. J.C.A. WeversThe Delta functionThe delta function δ(x) is an infinitely thin peak function with surface 1. It can be defined by: 0 for |x| > ε δ(x) = lim P (ε, x) with P (ε, x) = 1 ε→0 when |x| < ε 2εSome properties are: ∞ ∞ δ(x)dx = 1 , F (x)δ(x)dx = F (0) −∞ −∞3.1.5 Goniometric integralsWhen solving goniometric integrals it can be useful to change variables. The following holds if one defines 1tan( 2 x) := t: 2dt 1 − t2 2t dx = , cos(x) = , sin(x) = 1 + t2 1 + t2 1 + t2 √Each integral of the type R(x, ax2 + bx + c)dx can be converted into one of the types that were treated insection 3.1.3. After this conversion one can substitute in the integrals of the type: dϕ R(x, x2 + 1)dx : x = tan(ϕ) , dx = of x2 + 1 = t + x cos(ϕ) R(x, 1 − x2 )dx : x = sin(ϕ) , dx = cos(ϕ)dϕ of 1 − x2 = 1 − tx 1 sin(ϕ) R(x, x2 − 1)dx : x= , dx = dϕ of x2 − 1 = x − t cos(ϕ) cos2 (ϕ)These definite integrals are easily solved: π/2 (n − 1)!!(m − 1)!! π/2 when m and n are both even cosn (x) sinm (x)dx = · (m + n)!! 1 in all other cases 0Some important integrals are: ∞ ∞ ∞ xdx π2 x2 dx π2 x3 dx π4 ax + 1 = , = , = e 12a2 (e x + 1)2 3 e x+1 15 0 −∞ 03.2 Functions with more variables3.2.1 DerivativesThe partial derivative with respect to x of a function f (x, y) is defined by: ∂f f (x0 + h, y0 ) − f (x0 , y0 ) = lim ∂x x0 h→0 hThe directional derivative in the direction of α is defined by: ∂f f (x0 + r cos(α), y0 + r sin(α)) − f (x0 , y0 ) f ·v = lim = ( f, (sin α, cos α)) = ∂α r↓0 r |v|
  21. 21. Chapter 3: Calculus 15When one changes to coordinates f (x(u, v), y(u, v)) holds: ∂f ∂f ∂x ∂f ∂y = + ∂u ∂x ∂u ∂y ∂uIf x(t) and y(t) depend only on one parameter t holds: ∂f ∂f dx ∂f dy = + ∂t ∂x dt ∂y dtThe total differential df of a function of 3 variables is given by: ∂f ∂f ∂f df = dx + dy + dz ∂x ∂y ∂zSo df ∂f ∂f dy ∂f dz = + + dx ∂x ∂y dx ∂z dxThe tangent in point x0 at the surface f (x, y) = 0 is given by the equation fx (x0 )(x − x0 ) + fy (x0 )(y − y0 ) = 0.The tangent plane in x0 is given by: fx (x0 )(x − x0 ) + fy (x0 )(y − y0 ) = z − f (x0 ).3.2.2 Taylor seriesA function of two variables can be expanded as follows in a Taylor series: n 1 ∂p ∂p f (x0 + h, y0 + k) = h p +k p f (x0 , y0 ) + R(n) p=0 p! ∂x ∂ywith R(n) the residual error and p ∂p ∂p p m p−m ∂ p f (a, b) h p +k p f (a, b) = h k ∂x ∂y m=0 m ∂xm ∂y p−m3.2.3 ExtremaWhen f is continuous on a compact boundary V there exists a global maximum and a global minumum for f onthis boundary. A boundary is called compact if it is limited and closed.Possible extrema of f (x, y) on a boundary V ∈ I 2 are: R 1. Points on V where f (x, y) is not differentiable, 2. Points where f = 0, 3. If the boundary V is given by ϕ(x, y) = 0, than all points where f (x, y) + λ ϕ(x, y) = 0 are possible for extrema. This is the multiplicator method of Lagrange, λ is called a multiplicator.The same as in I 2 holds in I 3 when the area to be searched is constrained by a compact V , and V is defined by R Rϕ1 (x, y, z) = 0 and ϕ2 (x, y, z) = 0 for extrema of f (x, y, z) for points (1) and (2). Point (3) is rewritten as follows:possible extrema are points where f (x, y, z) + λ1 ϕ1 (x, y, z) + λ2 ϕ2 (x, y, z) = 0.
  22. 22. 16 Mathematics Formulary by ir. J.C.A. Wevers3.2.4 The -operatorIn cartesian coordinates (x, y, z) holds: ∂ ∂ ∂ = ex + ey + ez ∂x ∂y ∂z ∂f ∂f ∂f gradf = ex + ey + ez ∂x ∂y ∂z ∂ax ∂ay ∂az div a = + + ∂x ∂y ∂z ∂az ∂ay ∂ax ∂az ∂ay ∂ax curl a = − ex + − ey + − ez ∂y ∂z ∂z ∂x ∂x ∂y 2 2 2 2 ∂ f ∂ f ∂ f f = 2 + 2 + 2 ∂x ∂y ∂zIn cylindrical coordinates (r, ϕ, z) holds: ∂ 1 ∂ ∂ = er + eϕ + ez ∂r r ∂ϕ ∂z ∂f 1 ∂f ∂f gradf = er + eϕ + ez ∂r r ∂ϕ ∂z ∂ar ar 1 ∂aϕ ∂az div a = + + + ∂r r r ∂ϕ ∂z 1 ∂az ∂aϕ ∂ar ∂az ∂aϕ aϕ 1 ∂ar curl a = − er + − eϕ + + − ez r ∂ϕ ∂z ∂z ∂r ∂r r r ∂ϕ 2 2 2 2 ∂ f 1 ∂f 1 ∂ f ∂ f f = + + 2 + 2 ∂r2 r ∂r r ∂ϕ2 ∂zIn spherical coordinates (r, θ, ϕ) holds: ∂ 1 ∂ 1 ∂ = er + eθ + eϕ ∂r r ∂θ r sin θ ∂ϕ ∂f 1 ∂f 1 ∂f gradf = er + eθ + eϕ ∂r r ∂θ r sin θ ∂ϕ ∂ar 2ar 1 ∂aθ aθ 1 ∂aϕ div a = + + + + ∂r r r ∂θ r tan θ r sin θ ∂ϕ 1 ∂aϕ aθ 1 ∂aθ 1 ∂ar ∂aϕ aϕ curl a = + − er + − − eθ + r ∂θ r tan θ r sin θ ∂ϕ r sin θ ∂ϕ ∂r r ∂aθ aθ 1 ∂ar + − eϕ ∂r r r ∂θ 2 ∂2f 2 ∂f 1 ∂2f 1 ∂f 1 ∂2f f = 2 + + 2 2 + 2 + 2 2 ∂r r ∂r r ∂θ r tan θ ∂θ r sin θ ∂ϕ2General orthonormal curvilinear coordinates (u, v, w) can be derived from cartesian coordinates by the transforma-tion x = x(u, v, w). The unit vectors are given by: 1 ∂x 1 ∂x 1 ∂x eu = , ev = , ew = h1 ∂u h2 ∂v h3 ∂wwhere the terms hi give normalization to length 1. The differential operators are than given by: 1 ∂f 1 ∂f 1 ∂f gradf = eu + ev + ew h1 ∂u h2 ∂v h3 ∂w
  23. 23. Chapter 3: Calculus 17 1 ∂ ∂ ∂ div a = (h2 h3 au ) + (h3 h1 av ) + (h1 h2 aw ) h1 h2 h3 ∂u ∂v ∂w 1 ∂(h3 aw ) ∂(h2 av ) 1 ∂(h1 au ) ∂(h3 aw ) curl a = − eu + − ev + h2 h3 ∂v ∂w h3 h1 ∂w ∂u 1 ∂(h2 av ) ∂(h1 au ) − ew h1 h2 ∂u ∂v 2 1 ∂ h2 h3 ∂f ∂ h3 h1 ∂f ∂ h1 h2 ∂f f = + + h1 h2 h3 ∂u h1 ∂u ∂v h2 ∂v ∂w h3 ∂wSome properties of the -operator are: div(φv) = φdivv + gradφ · v curl(φv) = φcurlv + (gradφ) × v curl gradφ = 0 div(u × v) = v · (curlu) − u · (curlv) curl curlv = grad divv − 2 v div curlv = 0 div gradφ = 2 φ 2 v ≡ ( 2 v1 , 2 v2 , 2 v3 )Here, v is an arbitrary vectorfield and φ an arbitrary scalar field.3.2.5 Integral theoremsSome important integral theorems are: Gauss: (v · n)d2 A = (divv )d3 V Stokes for a scalar field: (φ · et )ds = (n × gradφ)d2 A Stokes for a vector field: (v · et )ds = (curlv · n)d2 A this gives: (curlv · n)d2 A = 0 Ostrogradsky: (n × v )d2 A = (curlv )d3 A (φn )d2 A = (gradφ)d3 VHere the orientable surface d2 A is bounded by the Jordan curve s(t).3.2.6 Multiple integralsLet A be a closed curve given by f (x, y) = 0, than the surface A inside the curve in I 2 is given by R A= d2 A = dxdyLet the surface A be defined by the function z = f (x, y). The volume V bounded by A and the xy plane is thangiven by: V = f (x, y)dxdyThe volume inside a closed surface defined by z = f (x, y) is given by: V = d3 V = f (x, y)dxdy = dxdydz
  24. 24. 18 Mathematics Formulary by ir. J.C.A. Wevers3.2.7 Coordinate transformationsThe expressions d2 A and d3 V transform as follows when one changes coordinates to u = (u, v, w) through thetransformation x(u, v, w): ∂x V = f (x, y, z)dxdydz = f (x(u)) dudvdw ∂uIn I 2 holds: R ∂x xu xv = ∂u yu yvLet the surface A be defined by z = F (x, y) = X(u, v). Than the volume bounded by the xy plane and F is givenby: ∂X ∂X f (x)d2 A = f (x(u)) × dudv = f (x, y, F (x, y)) 1 + ∂x F 2 + ∂y F 2 dxdy ∂u ∂v S G G3.3 Orthogonality of functionsThe inner product of two functions f (x) and g(x) on the interval [a, b] is given by: b (f, g) = f (x)g(x)dx aor, when using a weight function p(x), by: b (f, g) = p(x)f (x)g(x)dx a 2The norm f follows from: f = (f, f ). A set functions fi is orthonormal if (fi , fj ) = δij .Each function f (x) can be written as a sum of orthogonal functions: ∞ f (x) = ci gi (x) i=0and c2 ≤ f i 2 . Let the set gi be orthogonal, than it follows: f, gi ci = (gi , gi )3.4 Fourier seriesEach function can be written as a sum of independent base functions. When one chooses the orthogonal basis(cos(nx), sin(nx)) we have a Fourier series.A periodical function f (x) with period 2L can be written as: ∞ nπx nπx f (x) = a0 + an cos + bn sin n=1 L LDue to the orthogonality follows for the coefficients: L L L 1 1 nπt 1 nπt a0 = f (t)dt , an = f (t) cos dt , bn = f (t) sin dt 2L L L L L −L −L −L
  25. 25. Chapter 3: Calculus 19A Fourier series can also be written as a sum of complex exponents: ∞ f (x) = cn einx n=−∞with π 1 cn = f (x)e−inx dx 2π −π ˆThe Fourier transform of a function f (x) gives the transformed function f (ω): ∞ ˆ 1 f (ω) = √ f (x)e−iωx dx 2π −∞The inverse transformation is given by: ∞ 1 1 ˆ f (x+ ) + f (x− ) = √ f (ω)eiωx dω 2 2π −∞where f (x+ ) and f (x− ) are defined by the lower - and upper limit: f (a− ) = lim f (x) , f (a+ ) = lim f (x) x↑a x↓aFor continuous functions is 1 2 [f (x+ ) + f (x− )] = f (x).

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