Differentiation:Basic definitions and theorems, Partial derivatives,
Derivatives (as linear maps), Inverse and Implicit function theorems.
Integration:Basic definitions and theorems, Integrable functions, Partitions of
unity, Change of variables.
Manifolds:Basic definitions, forms on manifolds, Stokes theorem on manifolds,
Volume element, classical theorems (Green’s and divergence).
Suggested books and references:
Rudin, W., Principles of Mathematical Analysis
Apostol, T. M., Mathematical Analysis
,Narosa Publishing House, New Delhi, 1992.
Spivak, M., Calculus on Manifolds
,The Benjamin Publishing Company, New York, 1965.
Munkres, J., R., Analysis on Manifolds
,Addison-Wesley Publishing Company, 1991.