CALCULUS ON MANIFOLDS
Functions of several variables, Directional derivatives and continuity, total derivative, mean value theorem for differentiable functions, Taylor's formula.
The inverse function and implicit function theorems, extreme of functions of several variables and Lagrange multipliers. Sard's theorem.
Manifolds: Definitions and examples, vector fields and differential forms on manifolds, Stokes theorem.
Recommended books
Spivak, M., Calculus on Manifolds, W.A. Benjamin, co., 1965.
Hirsh, M.W., Differnential Topology, Springer-Verlag, 1997.