# 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

- Basics
- Goniometric functions
- Hyperbolic functions
- Calculus
- Limits
- Complex numbers and quaternions
- Complex numbers
- Quaternions

- Geometry
- Triangles
- Curves

- Vectors
- Series
- Expansion
- Convergence and divergence of series
- Convergence and divergence of functions

- Products and quotients
- Logarithms
- Polynomials
- Primes

- Probability and statistics
- Combinations
- Probability theory
- Statistics
- General
- Distributions

- Regression analyses

- Calculus
- Integrals
- Arithmetic rules
- Arc lengs, surfaces and volumes
- eparation of quotients
- Special functions
- Elliptic functies
- The Gamma function
- The Beta function
- The Delta function

- Goniometric integrals

- Functions with more variables
- Derivatives
- Taylor series
- Extrema
- The nabla-operator
- Integral theorems
- Multiple integrals
- Coordinate transformations

- Orthogonality of functions
- Fourier series

- 4. Differential equations
- Linear differential equations
- First order linear DE
- Second order linear DE
- The Wronskian
- Power series substitution

- Some special cases
- Frobenius' method
- Euler
- Legendre's DE
- The associated Legendre equation
- Solutions for Bessel's equation
- Properties of Bessel functions
- Laguerre's equation
- The associated Laguerre equation
- Hermite
- Chebyshev
- Weber

- Non-linear differential equations
- Sturm-Liouville equations
- Linear partial differential equations
- General
- Special cases
- The wave equation
- The diffusion equation
- The equation of Helmholtz

- Potential theory and Green's theorem

- Linear algebra
- Vector spaces
- Basis
- Matrix calculus
- Basic operations
- Matrix equations

- Linear transformations
- Plane and line
- Coordinate transformations
- Eigen values
- Transformation types
- Isometric transformations
- Orthogonal transformations
- Unitary transformations
- Symmetric transformations
- Hermitian transformations
- Normal transformations
- Complete systems of commuting Hermitian transformations

- Homogeneous coordinates
- Inner product spaces
- The Laplace transformation
- The convolution
- Systems of linear differential equations
- Quadratic forms
- Quadratic forms in $I\hspace{-1mm}R^2$
- Quadratic surfaces in $I\hspace{-1mm}R^3$

- Complex function theory
- Functions of complex variables
- Complex integration
- Cauchy's integral formula
- Residue

- Analytical functions definied by series
- Laurent series
- Jordan's theorem

- Tensor calculus
- Vectors and covectors
- Tensor algebra
- Inner product
- Tensor product
- Symmetric and antisymmetric tensors
- Outer product
- The Hodge star operator
- Differential operations
- The directional derivative
- The Lie-derivative
- Christoffel symbols
- The covariant derivative

- Differential operators
- The Gradient
- The divergence
- The curl
- The Laplacian

- Differential geometry
- Space curves
- Surfaces in $I\hspace{-1mm}R$
- The first fundamental tensor
- The second fundamental tensor
- Geodetic curvature

- Riemannian geometry

- Numerical mathematics
- Errors
- Floating point representations
- Systems of equations
- Triangular matrices
- Gauss elimination
- Pivot strategy

- Roots of functions
- Successive substitution
- Local convergence
- Aitken extrapolation
- Newton iteration
- The secant method

- Polynomal interpolation
- Definite integrals
- Derivatives
- Differential equations
- The fast Fourier transform