**MTH ****403: Real Analysis II**** ****(4)**

**Pre-requisites**: MTH 303 Real Analysis I

*Learning Objectives*:

This course deals with the study of functions of several real variables and the geometry associated with such functions.

There are two parts to this course. The first part deals with the study of differentiation and integration of such functions. The second part is devoted to the statement and proof of the higher dimensional version of the fundamental theorem of calculus, viz, Stoke's theorem (and its companions).

This is one of the standard courses in any mathematics curriculum. It also serves as a first introduction to differential geometry and topology.

*Course Contents*:

Vector-valued functions, continuity, linear transformations, differentiation, total derivative, chain rule

Determinants, Jacobian, implicit function theorem, inverse function theorem, rank theorem

Partition of unity, Derivatives of higher order

Riemann integration in **R ^{n}**, differentiation of integrals, change of variables, Fubini’s theorem

Exterior algebra, simplices, chains of simplices, Stokes theorem, vector fields, divergence of a vector field, Divergence theorem, closed and exact forms, Poincare lemma

*Suggested Books*:

- David Widder,
*Advanced Calculus*, second edition, Dover, 1989 - M. Spivak,
*Calculus on manifolds*, fifth edition, Westview Press, 1971 - J. Munkres,
*Elementary Differential topology*, Princeton University Press, 1966

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