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