### MA5370 Multivariable Calculus

#### Course Details

Differentiability in R^{n} , directional derivatives, Chain rule, Inverse function theorem, Implicit function theorem, Lagrange multiplier method.

Riemann integral of real-valued functions on Euclidean spaces, Fubini’s theorem, Partition of unity, change of variables.

Differential forms on R^{n} , simplices and singular chains, Stokes’ theorem for integral of differential forms on chains (general version) on R n , closed and exact forms, Poincaré lemma, Classical Green’s theorem, divergence theorem and Stokes’ formula as applications of general form of Stokes’ theorem.

Arbitrary submanifolds of R^{n} not necessarily open, differentiable functions on submanifolds, tangent spaces, vector fields.

#### Course References:

Text Books:

1. Walter Rudin, Principles of Mathematical Analysis, Third Edition, McGraw Hill International
Editions, Mathematical Studies 1976; Paper-back Indian Edition 2017.

2. S. Kumaresan, A Course in Differential Geometry and Lie groups, Hindustan Book Agency, Trim
22, 2002.

Reference Books:

1. W. Fleming, Functions of Several Variables, Second Edition, Springer-Verlag, 1977.

2. J. Munkres, Analysis on Manifolds, Addison-Wesley, 1991.

3. M. Spivak, Calculus on Manifolds, A Modern Approach to Classical theorems of Advanced
calculus, W.A.Benjamin, Inc.,1965.