Content: The course provides an introduction to tensors and differential forms, which in the last years have become an important tool for the development of good numerical techniques in computational science and engineering. The courses also relates these to classical vector calculus.

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News

• The lectures in the first two weeks of June will take place Thursdays in the tutorial time.
• We will have an extra lecture on 25/4/2019 to compensate for the one that would have taken place Easter monday.
• There will be no lecture and tutorial in the week of 8. April. Instead, we will have a lecture on 4. April, 13:00 - 15:00 (tutorial time).
• See the Vorlesungsverzeichnis for details.

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Lectures

Week 1: Vector spaces and bases Week 2: Dual vector spaces, dual basis and Riesz representation theorem Week 3: Coordinate transformations on vectors and co-vectors; linear maps and their duals Week 4: Dual maps, pullback and push-forward; tensors and their coordinate representation Week 5: Flat and sharp; tensor product; Symmetric and anti-symmetric tensors and symplectic form (code pendulum) Week 6: Pullback and push-forward of tensors Week 7: Tangent space (example curve (Mathematica)) Week 8: Vector fields, integral curves, flow maps Week 8: Covector and tensor fields Week 9: Codifferential Week 10: Exterior derivative and wedge product Week 10/11: Properties of the exterior derivative, Lie derivative Week 12: Integration of differential forms; Hodge-Helmholtz decomposition Week 13: Discretizations of exterior calculus   Discrete exterior calculus   Finite element exterior calculus

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Exercises

Week 1: Exercise 1 Week 2: Exercise 2 Week 3: Exercise 3 Week 4: Exercise 4 Week 5: Exercise 5 Week 6: Exercise 6 Week 7: Exercise 7 Week 8: Exercise 8 Week 9: Exercise 9 Week 10: Exercise 10 Week 11: Exercise 11 Week 12: Exercise 12 Week 13: Exercise 13 Week 14: Exercise 14 Project: Assignment   Skeleton code

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Software

Occasionally we will be working with python and the Numpy library in this course. On Linux you can install it using your package manager. On other operating systems it is convenient to use the Anaconda distribution which contains all necessary packages. An introduction to python and Numpy can be found here.

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Literature

• J. E. Marsden, T. S. Ratiu, and R. Abraham, Manifolds, Tensor Analysis, and Applications, Springer-Verlag, 2004.
• T. Frankel, The Geometry of Physics, third ed., Cambridge University Press, 2011.
• I. Agricola and T. Friedrich, Vektoranalysis: Differentialformen in Analysis, Geometrie und Physik. Vieweg+Teubner Verlag, 2010.