Advanced Analysis

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Course TypeCourse CodeNo. Of Credits
Foundation ElectiveSUS1MA5404

Semester and Year Offered: Winter semester and 2013-14

Course Coordinator and Team: Dr Pranay Goswami (cc), Professor Geetha Venkataraman

Email of course coordinator: pranay[at]aud[dot]ac[dot]in

Pre-requisites: SUS1MA508


This course has mainly been designed with the aim of introducing students to the advanced mathematics. The course has three parts one on Metric Spaces which is compulsory and students can choose from one of the other two parts on Complex Analysis or Multivariable functions. Lab work may also be included where it will enhance visualisation and understanding.

Course Outcomes:

After completing this course, students should able to

  1. understand the concepts of Metric spaces,
  2. understand the concepts of Complex Analysis or Multivariable Calculus.

Brief description of modules/ Main modules:

The following topics will be covered in the course under the three main modules as described below.

Metric Spaces

Metric spaces and its examples, continuity, homeomorphism, closed sets and open sets, closure, interiors, and boundary points, inheritance, clustering and condensing, product metrics, completeness and boundedness, compactness, connectedness, covering.

Complex Analysis

Field of complex numbers, The complex plane, Topological aspects of complex plane, Function of complex variables, continuity and differentiability, algebra of differentiation, Analytic functions, Cauchy-Riemann equations, Line integrals, closed curve theorem for entire functions, power series and its convergence, differentiability and uniqueness of power series, Cauchy integral formula , Taylor’s series , Liouville’s theorem and fundamental theorem of algebra

Multivariable Functions

Multiple integrals, change of variables, Triple integrals in Cylindrical coordinates, Triple integrals in spherical coordinates, vector fields and gradients, line integrals, Fundamental theorems of line integrals, Green’s Theorem, curl and divergence, parametric surfaces and their areas, Surface integrals (Oriented surface only), Stoke’s theorem, Gauss Divergence theorem.

Assessment Details with weights:



Date/period in which Assessment will take place



Class test

First week of February



Mid Semester Exam

As per AUD Academic Calendar



Home assignment/Tut

Throughout the semester



Presentation/ Viva

End March/Early April



End Semester Exam

As per AUD Academic Calendar



Reading List:

  • Joseph Bak and Donald J. Newman, Complex Analysis (2nd edition), Springer, 2001.
  • Charles C. Pugh, Real Mathematical Analysis, Springer, 2002.
  • S. Kumaresan, Topology of metric space, Alpha Science International Ltd., 2005.
  • James Stewart, Essential Calculus: early transcendentals, Thomson Brook/ Cole, 2007.