MA5370 Multivariable Calculus
Differentiability in Rn , 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 Rn , 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 Rn not necessarily open, differentiable functions on submanifolds, tangent spaces, vector fields.
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
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.