Department of Mathematics,
Faculty of Physics, MSU

Tensor analysis

2 hours of lectures per week

The course is devoted to the presentation of the basics of differential geometry and is expository. It is designed for students 2 - 5 courses

The course content
  1. The tensor algebra. General information about the tensor. Antisymmetric tensor outer product. Approximately 2 lectures.
  2. Factorization of linear space by the subspace. Roughly 1 lecture.
  3. General topology. Topology defined by the system of neighborhoods, the base topology, continuous functions. Homeomorphisms of topological spaces, homeomorphisms coordinate spaces (a large part of the theory is presented without proof). Diffeomorphisms of coordinate spaces. Approximately 4 lectures.
  4. Smooth manifolds. Coordinate map coordinate atlases topology on a smooth manifold. The tangent space, tensor fields, the problem of differentiation of tensor fields. Approximately 4 lectures.
  5. The exterior differential of a differential form. Definition, elementary properties. Gradient, curl, divergence. Approximately 3 lectures.
  6. The concept of the de Rham complex. Definition, elementary properties. Poincare lemma on closed and exact forms on a manifold diffeomorphic to a ball. Approximately 2 lectures.