Functions on $R^n$ , directional derivatives, total derivative, higher order derivatives and Taylor series.The inverse and implicit function theorem,
Integration on $R^n$ , differential forms on $R^n$ , closed and exact forms. Green’s theorem, Stokes’ theorem and the Divergence theorem.
Suggested books and references:
Rudin, Principles of Mathematical Analysis, McGraw-Hill, 1986.
B. V. Limaye and S. Ghorpade, A course in Calculus and Real Analysis, Springer.
Spivak, M., Calculus on Manifolds, W.A. Benjamin, co., 1965.