Course Archives Theoretical Statistics and Mathematics Unit
Course: Analysis of Several Variables
Level: Postgraduate
Time: Currently not offered
Past Exams


i) Metric Topology of Rn. Topology induced by lp norms (p = 1,2,8) on Rn and their equivalence. Continuous functions on Rn. Separation properties. Compact subsets of Rn. Path-connectivity. Topological properties of subgroups like GLn(R), GLn(C), O(n), Hilbert-Schmidt norm and operator norm on Mn(R). Sequence and series in Mn(R). Exponential of a matrix.
ii) Differentiation and integration of functions on Rn. Partial derivatives of real valued functions on Rn. Differentiability of maps from Rm to Rn and the derivative as a linear map. Jacobian theorem. Chain Rule. Mean value theorem. Higher derivatives and Schwarz theorem, Taylor expansions in several variables. Inverse function theorem and implicit function theorems. Local maxima and minima, Lagrange multiplier method.
iii) Vector fields on Rn. Integration of vector fields and flows. Picards Theorem.
iv) Riemann integration of bounded real-valued functions on rectangles (product of intervals). Existence of the Riemann integral for sufficiently well-behaved functions. Iterated integral and Fubinis theorem. Brief treatment of multiple integrals on more general domains. Change of variable and the Jacobian formula.
v) Differential forms on Rn. Wedge product of forms. Pullback of differential forms. Exterior differentiation of forms. Integration of compactly supported n-forms on Rn. Change of variable formula revisited. Integration of k-forms along singular k-chains in Rn. Stokes theorem on chains. [Special emphasis on curves and surfaces in R2 and R3. Line integrals, Surface integrals. Gradient, Curl and Divergence operations, Greens theorem and Gausss (Divergence) theorem.]

Suggested Texts :
(a) M. Spivak: Calculus on manifolds, Benjamin (1965).
(b) T.Apostol: Mathematical Analysis. S. Lang, Algebra, GTM (211), Springer (Indian reprint 2002).
(c) K. Mukherjea: Differential Calculus in Normed Linear Spaces.

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Past Exams
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