The material of the 13th week, essentially corresponds to a deeper look at the  first and second variation of functionals and the Euler-Lagrange equations.  In the BG book this week corresponds to  the third section of Chapter 9, with title ``Remarks on variational calculus'',  from the paragraph 9.3.5  to the end of  the paragraph 9.3.8. It includes the following concepts:

1) The first and the second variation of  functionals;

2) Local maximizers/minimizers of variational problems;

3) The Euler-Lagrange equations;

4) Description of examples and some special cases.