Contents
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.................................................. 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.................................................. 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 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 grad-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 TheWronskian......................................... 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'sDE......................................... 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............................. 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
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 IR2..................................... 38
5.14.2 Quadratic surfaces in IR3.................................... 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 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 IR3......................................... 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.......................................... 50
8 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