260203 VO Introduction to vector and tensor calculus II (2020S)

Goals:
Students will get acquainted with curvilinear coordinates, the corresponding basis vectors and their behaviour under transformations. Starting with the length of curves and the metric tensor they recognize the significance of Riemann space. In connection with the description of the spacial variability of vectors the students acquire a well-founded understanding of the covariant derivative and its applications to simple physical problems. Further the covariant derivative leads to a characterization of curvature of a Riemann space.
Contents:
Description of curves and surfaces, tangential and normal vectors. Curvilinear coordinate systems, definitions of coordinate lines and coordinate surfaces, as well as covariant and contravariant vector bases, transformation behavior. Length of curves, definition of metric tensor, Riemann space, flat space, Euklidian space. Definition of covariant derivative of scalars and vectors, definition of Christoffel symbols, vector differential operators in curvilinear coordinates, applications to cylindrically and spherically symmetric physical problems. Properties of the covariant derivative, higher covariant derivatives, Riemann curvature tensor, Einstein tensor, parallel displacement of vectors.
Method:
Lecture course with predominant use of the blackboard, opportunity for questions and discussion. Several examples are mentioned, where the subject matter of the lecture course can be autonomously applied by the students.