In the fifth week we will  focus on   exterior differential calculus and notions as vector fields, differential forms, integration of differential forms, the notion of manifolds and other. In  particular, the theoretical part of 5th week's material corresponds to the material described in the first  section of Chapter 9 of the BG book, with title ``Exterior differential calculus and integration",  from  the paragraph  9.1.1 till the end of  the paragraph 9.1.9.

This includes the following concepts:

1) Vector fields and exterior k-forms on R^n;

2)  Wedge products of exterior forms;

3) Integration of exterior forms on R^n;

4) Manifolds, vector fields  on manifolds and differential forms on manifolds;

5)  The notion of a partition of unity;

6)  Integration of  k-forms on manifolds.