Course Archives: Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore

Course Archives Theoretical Statistics and Mathematics Unit

Course: Analysis of Several Variables
Instructor: Jaydeb Sarkar
Room: Second floor auditorium
Level: Undergraduate
Time: Currently offered

Syllabus
Past Exams

Syllabus:

Calculus of several variables: Differentiability of maps from Rm to Rn and the derivative as a linear map. Higher derivatives, Chain Rule, Taylor expansions in several variables, Local maxima and minima, Lagrange multiplier.
Multiple integrals, Existence of the Riemann integral for sufficiently well-behaved functions on rectangles, i.e. product of intervals. Multiple integrals expressed as iterated simple integrals. Brief treatment of multiple integrals on more general domains. Change of variables and the Jacobian formula, illustrated with plenty of examples. Inverse and implicit functions theorems (with proofs). More advanced topics in the calculus of one and several variables curves in R2 and R3. Line integrals, Surfaces in R3, Surface integrals, Divergence, Gradient and Curl operations, Greens, Strokes and Gauss (Divergence) theorems.

Reference Texts:

(a) T. M. Apostol: Mathematical Analysis.
(b) T. M. Apostol: Calculus.
(c) S. Dineen: Multivariate Calculus and Geometry.
(d) R. R. Goldberg: Methods of Real Analysis.
(e) T. Tao: Analysis I & II.
(f) Bartle and Sherbert: Introduction to Real Analysis.
(g) H. Royden: Real Analysis.

Evaluation: