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
B. V. Limaye and S. Ghorpade, A course in Calculus and Real Analysis
Spivak, M., Calculus on Manifolds
,W.A. Benjamin, co., 1965.