The  3rd week we will pass to Chapter 8 of the BG book, which is about multivariable calculus (constraint extrema, integration, general Stokes theorem, and other). In particular, the theoretical part of 3rd week's material corresponds to the material described in the first Section of Chapter 8 in the BG book, with title ``Functions and mappings on R^n''.   This includes the following concepts:

1) Multivariate functions;

2) Topology of Euclidean spaces;

3)  Directional derivatives, partial derivatives,  derivatives of higher order;

4)  The notion of the Hessian;

5) Multidimensional version of Taylor expansions;

6) Local extrema and their characterization;

7) The differential of a function;

8) The inverse function theorem;

9)  The implicit mapping theorem;

10) Constrained extremas and the theory of Lagrange multipliers.

Examples and applications related to these notions are described in the corresponding left column of the BG book, and in particular from Section A with title ``Multivariate functions'' to Section H with title ``Constrained optimization''.