Advanced Mathematics for Applied Physics and Biomedical Engineering,   WS 2026/27

Lecture:       time/room tba
Excercises:   time/room tba

Final Grade:  90% Final Examination,  20% R-Test in last week before christmas,  maximum at 100% plus 10% bonus-points
allowed auxilliary means:  1 sheet DIN A4, written or printed on both sides, and a simple electronic calculator;
                                              for the R-Test the whole course material, including solutions, can be used


Course Outline: The following material should be covered:

1. Matrices and Linear Algebra
   1.1   Eigenvalues and Eigenvectors of Matrices
   1.2   Diagonalization of Matrices
   1.3   Short Introduction to the R-Software
   1.4   Matrix-Exponential and the Harmonic Oscillator
   1.5   Singular Value Decomposition of Matrices
   1.6   SVD and Data Compression
   1.7   Linear Regression and R-Implementation
    
   
2. Fourier Transformation
   2.1   Discretization of Differential Operators and Discrete Fourier Transform
   2.2   Fourier Series and Fourier Integrals
   2.3   Fourier Series Expansion as a Linear Regression Problem
    
   
3. Vector Analysis and Multiple Integrals
   3.1   Motivation: Examples from Electrostatics and Magnetostatics
   3.2   Transformation to Arbitrary Coordinates in Multiple Integrals
   3.3   Curl and Divergence and Maxwell Equations in Differential Form
   3.4   Theorems of Gauss and Stokes and Maxwell Equations in Integral Form
   3.5   Scalar Potentials and Vector Potentials
   3.6   The Laplace Operator in Arbitrary Coordinates
   3.7   Dirac Delta-Function
    
   
4. Partial Differential Equations
   4.1   Heat Equation
   4.2   Wave Equation: Oscillating Circular Membrane
   4.3   Schrödinger Equation: The Hydrogen Atom