Mathematical Formulary

Dear reader,

This document contains 66 pages with mathematical equations intended for physicists and engineers. It is intended to be a short reference for anyone who often needs to look up mathematical equations.

This, and a Dutch version of this file, can be obtained from the author, Johan Wevers johanw@xs4all.nl.

It can also be obtained on the WWW. See http://www.xs4all.nl/~johanw/index.html, where \(\rm\LaTeX\), html, Postscript and PDF versions are available.

If you find any errors or have any comments, please let me know. I am always open for suggestions and possible corrections to the physics formulary.

This work is licenced under the Creative Commons Attribution 3.0 License. To view a copy of this licence, visit http://creativecommons.org/licenses/by/3.0/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California 94105, USA. which contains a lot equations in physics. It is written at advanced undergraduate/postgraduate level. It is intended to be a short reference for anyone who works with physics and often needs to look up equations.

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.

ir. J.C.A. Wevers

Contents

1. Basics
   1. Goniometric functions
   2. Hyperbolic functions
   3. Calculus
   4. Limits
   5. Complex numbers and quaternions
      1. Complex numbers
      2. Quaternions
   6. Geometry
      1. Triangles
      2. Curves
   7. Vectors
   8. Series
      1. Expansion
      2. Convergence and divergence of series
      3. Convergence and divergence of functions
   9. Products and quotients
   10. Logarithms
   11. Polynomials
   12. Primes

2. Probability and statistics
   1. Combinations
   2. Probability theory
   3. Statistics
      1. General
      2. Distributions
   4. Regression analyses
3. Calculus
   1. Integrals
      1. Arithmetic rules
      2. Arc lengs, surfaces and volumes
      3. eparation of quotients
      4. Special functions
         1. Elliptic functies
         2. The Gamma function
         3. The Beta function
         4. The Delta function
      5. Goniometric integrals
   2. Functions with more variables
      1. Derivatives
      2. Taylor series
      3. Extrema
      4. The nabla-operator
      5. Integral theorems
      6. Multiple integrals
      7. Coordinate transformations
   3. Orthogonality of functions
   4. Fourier series
4. 4. Differential equations
   1. Linear differential equations
      1. First order linear DE
      2. Second order linear DE
      3. The Wronskian
      4. Power series substitution
   2. Some special cases
      1. Frobenius' method
      2. Euler
      3. Legendre's DE
      4. The associated Legendre equation
      5. Solutions for Bessel's equation
      6. Properties of Bessel functions
      7. Laguerre's equation
      8. The associated Laguerre equation
      9. Hermite
      10. Chebyshev
      11. Weber
   3. Non-linear differential equations
   4. Sturm-Liouville equations
   5. Linear partial differential equations
      1. General
      2. Special cases
         1. The wave equation
         2. The diffusion equation
         3. The equation of Helmholtz
      3. Potential theory and Green's theorem
5. Linear algebra
   1. Vector spaces
   2. Basis
   3. Matrix calculus
      1. Basic operations
      2. Matrix equations
   4. Linear transformations
   5. Plane and line
   6. Coordinate transformations
   7. Eigen values
   8. Transformation types
      1. Isometric transformations
      2. Orthogonal transformations
      3. Unitary transformations
      4. Symmetric transformations
      5. Hermitian transformations
      6. Normal transformations
      7. Complete systems of commuting Hermitian transformations
   9. Homogeneous coordinates
   10. Inner product spaces
   11. The Laplace transformation
   12. The convolution
   13. Systems of linear differential equations
   14. Quadratic forms
       1. Quadratic forms in $I\hspace{-1mm}R^2$
       2. Quadratic surfaces in $I\hspace{-1mm}R^3$
6. Complex function theory
   1. Functions of complex variables
   2. Complex integration
      1. Cauchy's integral formula
      2. Residue
   3. Analytical functions definied by series
   4. Laurent series
   5. Jordan's theorem
7. Tensor calculus
   1. Vectors and covectors
   2. Tensor algebra
   3. Inner product
   4. Tensor product
   5. Symmetric and antisymmetric tensors
   6. Outer product
   7. The Hodge star operator
   8. Differential operations
      1. The directional derivative
      2. The Lie-derivative
      3. Christoffel symbols
      4. The covariant derivative
   9. Differential operators
      1. The Gradient
      2. The divergence
      3. The curl
      4. The Laplacian
   10. Differential geometry
       1. Space curves
       2. Surfaces in $I\hspace{-1mm}R$
       3. The first fundamental tensor
       4. The second fundamental tensor
       5. Geodetic curvature
   11. Riemannian geometry
8. Numerical mathematics
   1. Errors
   2. Floating point representations
   3. Systems of equations
      1. Triangular matrices
      2. Gauss elimination
      3. Pivot strategy
   4. Roots of functions
      1. Successive substitution
      2. Local convergence
      3. Aitken extrapolation
      4. Newton iteration
      5. The secant method
   5. Polynomal interpolation
   6. Definite integrals
   7. Derivatives
   8. Differential equations
   9. The fast Fourier transform