## Topics of mathematics tutoring, crash courses and courses

### Mathematics Basics

Power rules, logarithm laws, exponential functions

Resolve and change equations and inequalities

Linear equations, linear equation systems

Quadratic equations (pq formula, midnight formula)

polynomial

Differential calculus: calculate derivatives (differentiation), derivation rules

Curve discussion: Range of Definition, Range of Values, Calculate Zero, Check Continuity, Monotony, Calculate Peaks & Turning Points, Inverse Functions

Integral calculus: Compute integrals (integration), use integrals for area calculation

Analytic geometry: points and lines in the plane and in space

Stochastics: combinatorics, calculate probabilities, calculate the normal distribution, confidence intervals, check hypotheses

Higher Mathematics: Analysis

Set theory: sets, set operations, Venn diagrams, number ranges

### Complex numbers

Limits / Convergence of sequences and series (eg geometric series)

Power series: convergence radius, binomial series, coefficient comparison

Taylor series / Taylor polynomials & Fourier series

Polynomials, rational functions, trigonometric functions

Consistency, constant differentiability

Main theorem of differential and integral calculus, mean value theorem

Differential calculus in one and several variables: chain rule, product rule, quotient rule, partial derivatives, directional derivative, gradient

Integral calculus: sum rule, substitution, partial integration, double integral, triple integral, improper integrals

Volume integrals: multidimensional integration, theorem of Gauss (Divergence theorem)

Surface integrals and curve integrals

Vector fields, gradient fields, potentials, sources, sinks, divergence, rotation, Laplace operator

### Higher Mathematics: Linear Algebra

propositional logic

Algebraic structures and equations

Linear systems of equations, Gaussian elimination method

Vectors & matrices, calculation rules (matrix multiplication, scalar product)

determinants

Transposed matrices, symmetric matrices, inverse matrices

Vector spaces, subspaces, linear combination, base, dimension, rank

Linear mappings, lengths, angles, orthogonal matrices, base change, coordinate transformation

Eigenvalues and eigenvectors, diagonalization, Jordan normal form

Square shapes, major axis transformation, positive definite matrices

### Higher Mathematics: Numerics / Numerical Mathematics

Number representation & rounding error

Interpolation: Lagrange interpolation, cubic splines

Numerical Differentiation: Finite differences, Taylor development

Numerical integration: trapezoid rule, Simpson rule

Linear systems of equations: LR decomposition & iterative methods (Jacob’s method, Gauss-Seidel method)

Nonlinear systems of equations, fixed point iteration: Newton’s method

Linear compensation calculation / linear regression

Eigenvalue calculation: Gerschgorin circles

Initial and boundary value problems: Euler method, Runge-Kutta, Finite differences, Finite elements, Finite volumes

### Business Mathematics / Financial Mathematics

Interest calculation, compound interest

pension bill

simulation

Investitionsrechnung

Operations Researach: Linear Optimization, Nonlinear Optimization

Econometrics

Stochastic financial mathematics