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Prof. Wolfgang Ertel
Wolfgang Ertel
Professor Dr. rer. nat.

Advanced Mathematics for Engineers

For up to date information on dates and organisation please refer to the LSF-Page

Preparation for the course

Decent knowledge of basic engineering mathematics is a necessary requirement for this master course. Many universities do not provide this and YOU may be one of the students with insufficient mathematics skills. Thus, we want you to refresh your bachelor mathematics, or if these skills are too weak, to learn the required math from scratch. We recommend for all our master students:
Please study the chapters 1, 2 and 3 in the script you find below and solve all the exercises to these chapters. And please watch the video lectures (no. 1 - 13, links below) on Linear Algebra by Gilbert Strang and my lectures no. 01 (Computer Algebra) - 08 (Extrema, Statistics and Probability).

Demos and Course Material

Octave Example Programs

Previous Examinations

Literature:

In addition to the books mentioned in the script, please consult:
  • Gilbert Strang - Introduction to Linear Algebra (978-0980232714)
  • Linear Algebra Review and Reference (Zico Kolter)
  • Matrix Identities (Sam Roweis)
  • Schaum's Outline Lectures (e.g. for Calculus, Advanced Calculus, Linear Algebra, Differential Equations, Probability and Statistics, Numeriacl Analysis, ... are very good for repeating your bachelors maths by working on exercises and thus filling your gaps.

Video lectures:

In addition to the particular videos for this lecture listed below, you can consult the following lectures:
  • Khan Academy offers very good videos on high school mathematics and also some undergraduate topics.
  • videolectures.org offers all kinds of free academic videos.
  • Youtube, Google, ...

Video lectures on Linear Algebra (from Gilbert Strang):

Lec # Topics
1 The geometry of linear equations
2 Transposes, Permutations, Spaces R^n
3 Column Space and Nullspace
4 Solving Ax = 0: Pivot Variables, Special Solutions
5 Independence, Basis, and Dimension
6 The Four Fundamental Subspaces
7 Orthogonal Vectors and Subspaces
8 Properties of Determinants
9 Determinant Formulas and Cofactors
10 Cramer's rule, inverse matrix, and volume
11 Eigenvalues and Eigenvectors
12 Symmetric Matrices and Positive Definiteness
13 Linear Transformations and Their Matrices

Video lectures of winter semester 2011/12:

# Lecture Number, Part Date Content
01Lecture No 01 Part131.10.2011Computer Algebra
02Lecture No 02 Part103.11.2011Sequences, Introduction to Mathematica
03Lecture No 03 Part107.11.2011Introduction to Octave, Series
04Lecture No 04 Part110.11.2011Power series, Continuity, Discontinuity
05Lecture No 05 Part117.11.2011Continuity, Discontinuity, Taylor Series
06Lecture No 06 Part121.11.2011Differential calculus in many variables
07Lecture No 07 Part124.11.2011The total Differential, Extrema without / with constraints
08Lecture No 08 Part101.12.2011Extrema, Statistics and Probability

Video lectures of winter semester 2011/12 on YouTube:

Lec # Topics Date
08 Extrema, Statistics and Probability 01.12.2011
09 Discrete Distributions, Continuous Distributions 05.12.2011
12 Roots of Nonlinear Equations 19.12.2011
13 Roots of Nonlinear Equations 22.12.2011
14 Polynomial Interpolation, Spline interpolation 09.01.2012
15 Spline interpolation 16.01.2012
16 Method of Least Squares and Pseudoinverse 19.01.2012
17 Method of Least Squares and Pseudoinverse 23.01.2012
18 Method of Least Squares and Pseudoinverse 26.01.2012

Video lectures of summer semester 2012 on YouTube:

Watch the videos together with the slides here

Lec # Topics Date
02 Full Video
Kolmogorov Complexity
Compression of Random Number Sequences
Pseudo Random Number Generators
The Symmetry Test
14.03.2012
03 Full Video
The Symmetry Test
BBS Generator (Blum Blum Shub)
19.03.2012
04 Full Video
True Random Numbers
The Neumann Filter
21.03.2012
05 Full Video
Linear Feedback Shift Registers
Derivation of a suitable Speedup Formula
27.03.2012
06 Full Video
Principal Component Analysis (PCA)
02.04.2012
07 Full Video
Principal Component Analysis (PCA)
Estimators
The Gaussian Distribution
11.04.2012
08 Full Video
Maximum Likelihood
Linear Regression
17.04.2012
09 Full Video
An Image compression Example
23.04.2012
10 Full Video
Maximum Likelihood
Bayesian Linear Regression
Linear Regression Summary
Radial Basis Functions (RBFs)
23.04.2012
11 Full Video
Demonstration of Galton Machine
Octave Examples
02.05.2012
12 Full Video
Clustering
k-Means and the EM-Algorithm
Single Value Decomposition
09.05.2012
13 Full Video
Single Value Decomposition
14.05.2012
14 Full Video
Numerical Integration
21.05.2012
15 Full Video
Numerical Integration
Numerical Differentiation
23.05.2012
16 Full Video
Numerical Solution of Ordinary Differential Equations
06.06.2012
17 Full Video
Numerical Solution of Ordinary Differential Equations
Linear Differential Equations with Constant Coefficients
11.06.2012
18 Full Video
Linear Differential Equations with Constant Coefficients
13.06.2012

Video lectures of winter semester 2010/11:

These videos are redundant, but still listed for completeness.
# Lecture Number, Part Date Content
01Lecture No 01 Part104.10.2010Computer algebra, introduction to Mathematica
02Lecture No 02 Part108.10.2010Mathematica introduction
03Lecture No 03 Part111.10.2010Condition analysis, introduction
04Lecture No 04 Part111.10.2010Gauss-Seidel Iteration for linear systems, matrix norm, condition analysis
05Lecture No 05 Part118.10.2010Condition analysis (cont.), proof of triangle inequality for the p-norm
06Lecture No 06 Part118.10.2010Roots of nonlinear equations, interval bisection
07Lecture No 07 Part122.10.2010Fixed point iteration
08Lecture No 08 Part125.10.2010Proof of the Banach fixed point theorem
09Lecture No 09 Part129.10.2010Introduction to Matlab and convergence speed of fixed-point iteration.
10Lecture No 10 Part108.11.2010Newton's method, polynomial interpolation.
11Lecture No 11 Part108.11.2010Tschebychev interpolation, cubic splines.
12Lecture No 12 Part112.11.2010Spline Interpolation, Method of Least Squares
13Lecture No 13 Part115.11.2010Method of Least Squares and Pseudoinverse
15Lecture No 15 Part122.11.2010Summary / Statistics and probability.
16Lecture No 16 Part126.11.2010Solution of the midterm exam
17Lecture No 17 Part129.11.2010Random Numbers
18Lecture No 18 Part129.11.2010The Symmetry Test / BBS Generator
19Lecture No 19 Part106.12.2010True Random Numbers / The Neumann Filter / Calculation of Means
20Lecture No 20 Part106.12.2010Calculation of Means / Likelihood Function
21Lecture No 21 Part113.12.2010Bayesian Linear Regression
22Lecture No 22 Part120.12.2010Numerical Integration
23Lecture No 23 Part120.12.2010Monte-Carlo-Simulation
24Lecture No 24 Part110.01.2011Repeated Richardson Extrapolation, The Romberg Method, Nummerical Differentation
25Lecture No 25 Part114.01.2011No description available