1 Introduction
1.1 Recapitulation of basic requirements
1.2 Notations
1.3 Books
2 Algebra
2.1 Complex numbers: Definition
2.2 Addition and Multiplication of complex numbers
2.3 Complex e-function
2.4 Other complex functions
2.5 Some formal definitions in linear Algebra
2.6 Vectors in N-dimensional space
2.7 Matrices
2.7.1 Vectors as special matrices
2.7.2 Matrix multiplication commutative?
2.8 Identity matrix as multiplicative unity
2.9 Square matrices and determinants
2.9.1 Examples for the calculation rules for determinants
2.10 Regular matrix, inverse matrix
2.10.1 Examples: Using cofactors for matrix inversion
2.10.2 Example: Using Gauß-Jordan algorithm for matrix inversion
2.11 Classification of N x N Matrices:
2.12 Systems of Linear Equations
2.13 Eigenvalues and Eigenvectors
2.13.1 Example: 2D mirroring
2.13.2 Example: 2x2 and 4x4 matrix
2.13.3 Calculation of Eigenvectors
2.14 Scalar product
2.14.1 Examples: Scalar product
2.15 Vector product
2.15.1 Mathematical description of a plane wave
2.16 Hermite and unitary matrices with complex components
3 Calculus I: Functions of one Variable
3.1 Recapitulation: Derivatives and Integrals
3.2 Calculation rules for derivatives
3.2.1 Prove of product rule and chain rule
3.3 Calculation rules for integrals
3.4 Sequences and Series
3.4.1 Examples: Convergence of infinite series
3.5 Taylor series and their application
3.5.1 Example: Taylor expansion of exponential function
3.5.2 Example: Taylor expansion of sin function
3.5.3 Example: Taylor expansion of cos function
3.5.4 Example: Taylor expansion of the logarithm function
3.6 Taylor series and error estimation
3.6.1 Example: Transformation of a non linear problem into a linear problem
3.6.2 Example: Taylor series of arctan function and pi-calculation
3.7 Linear Optimization
3.8 Fitting to an orthonormal set of functions
3.9 Functions as vectors
3.10 Schmidts orthonormalization procedure
3.11 Fourier series
3.11.1 Example: Periodic step function
3.11.2 Example: Positive part of sin function
3.11.3 Example: Absolute of sin function
3.11.4 Example: Periodic parabolic function
3.12 Fourier series in complex description
3.12.1 Example: Fourier-Series with larger periodicity length
3.13 From Fourier series to Fourier-Transformation
3.14 Fourier-Transformation: Definition
3.14.1 Examples for Fourier transformation
3.15 Fourier Transformation: Properties
3.15.1 Convolution Theorem: Proof and example
3.16 Fourier Transformation: Solving DEQs
3.16.1 Forced oscillations: 2. Example for solving DEQ with Fourier Transformation
3.17 Important non-elementary functions: Gamma function
3.18 Important non-elementary functions: Delta function
3.19 Important non-elementary functions: Gauss- and Error function
3.20 Aspects of probability theory
3.21 Aspects of noise analysis
4 Calculus II: Functions of multiple variables
4.1 Basic requirements
4.2 Functions of more than one variable
4.3 Partial derivatives
4.4 Derivatives in certain directions
4.5 Total Derivatives
4.5.1 Examples: For Jacobi matrix and Jacobi determinant
4.6 Curvilinear coordinates
4.7 Minimization problems
4.8 Criteria for finding extreme values in N-dimensions
4.8.1 Examples for finding extrema
4.9 Simple N-dimensional integrals
4.10 Continuity equation using divergency
4.11 Integrals using grad, div, and curl