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.
Spivak, M., Calculus on Manifolds, W.A. Benjamin, co., 1965.