Last version: May 4, 2008
A list of recent additions and improvements is available.
Contents .................................................................... I
1. 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 ............................................................... 4
1.8.1 Expansion ...................................................... 4
1.8.2 Convergence and divergence of series ........................... 5
1.8.3 Convergence and divergence of functions ........................ 6
1.9 Products and quotients .............................................. 6
1.10 Logarithms .......................................................... 7
1.11 Polynomials ......................................................... 7
1.12 Primes .............................................................. 7
2. 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 ................................................. 11
3. Calculus ................................................................ 12
3.1 Integrals ........................................................... 12
3.1.1 Arithmetic rules .............................................. 12
3.1.2 Arc lengs, surfaces and volumes .............................. 12
3.1.3 Separation of quotients ...................................... 13
3.1.4 Special functions ............................................ 13
Elliptic functies ............................................ 13
The Gamma function ........................................... 13
The Beta function ............................................ 13
The Delta function ........................................... 14
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 nabla-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
4. 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
The wave equation .............................................. 25
The diffusion equation ......................................... 26
The equation of Helmholtz ...................................... 26
4.5.3 Potential theory and Green's theorem ........................... 27
5. 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
Isometric transformations ........................................... 32
Orthogonal transformations .......................................... 33
Unitary transformations ............................................. 33
Symmetric transformations ........................................... 33
Hermitian transformations ........................................... 34
Normal transformations .............................................. 34
Complete systems of commuting Hermitian transformations ............. 35
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 R^2 ............................................ 38
5.14.2 Quadratic surfaces in R^3 ......................................... 38
6. 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
7. Tensor calculus ......................................................... 44
7.1 Vectors and covectors ............................................... 44
7.2 Tensor algebra ...................................................... 45
7.3 Inner product ...................................................... 45
7.4 Tensor product ...................................................... 46
7.5 Symmetric and antisymmetric tensors ................................. 46
7.6 Outer product ....................................................... 46
7.7 The Hodge star operator ............................................. 47
7.8 Differential operations ............................................ 47
7.8.1 The directional derivative .................................... 47
7.8.2 The Lie-derivative ............................................ 47
7.8.3 Christoffel symbols ........................................... 48
7.8.4 The covariant derivative ...................................... 48
7.9 Differential operators ............................................... 48
The Gradient ......................................................... 48
The divergence ....................................................... 48
The curl ............................................................. 48
The Laplacian ........................................................ 49
7.10 Differential geometry ............................................... 49
7.10.1 Space curves ................................................. 49
7.10.2 Surfaces in R^3 .............................................. 50
7.10.3 The first fundamental tensor ................................. 50
7.10.4 The second fundamental tensor ................................ 50
7.10.5 Geodetic curvature ........................................... 50
7.11 Riemannian geometry ................................................. 51
8. Numerical mathematics ................................................... 52
8.1 Errors .............................................................. 52
8.2 Floating point representations ...................................... 52
8.3 Systems of equations ................................................ 53
8.3.1 Triangular matrices ........................................... 53
8.3.2 Gauss elimination ............................................. 53
8.3.3 Pivot strategy ................................................ 54
8.4 Roots of functions ................................................... 54
8.4.1 Successive substitution ........................................ 54
8.4.2 Local convergence .............................................. 54
8.4.3 Aitken extrapolation ........................................... 55
8.4.4 Newton iteration ............................................... 55
8.4.5 The secant method .............................................. 56
8.5 Polynomal interpolation .............................................. 56
8.6 Definite integrals ................................................... 57
8.7 Derivatives .......................................................... 57
8.8 Differential equations ............................................... 58
8.9 The fast Fourier transform ........................................... 59