Scientific Computing IV: Differential Forms, Tensors, and Vector Calculus
Spring/Summer 2019, Otto-von-Guericke Universität Magdeburg
Lecturer: Christian Lessig
Lectures: Mondays, 11:00-13:00, G29-335
Tutorials: Thursdays, 13:00-15:00, G29-426
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.
- 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.
Vector spaces and bases
Dual vector spaces, dual basis and Riesz representation theorem
Coordinate transformations on vectors and co-vectors; linear maps and their duals
Dual maps, pullback and push-forward; tensors and their coordinate representation
Flat and sharp; tensor product; Symmetric and anti-symmetric tensors and symplectic form
Pullback and push-forward of tensors
Tangent space (example curve (Mathematica)
Vector fields, integral curves, flow maps
Covector and tensor fields
Exterior derivative and wedge product
Properties of the exterior derivative, Lie derivative
Integration of differential forms
Hodge-Helmholtz decomposition, Euler equation for ideal fluid
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
- 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.