University of Mumbai, Mumbai

**Teaching Pattern for Semester I, II, III and IV**

- Four lectures per week per course. Each lecture is of 60 minutes duration.
- In addition, there shall be tutorials, seminars as necessary for each of the five courses.

**Course Outcomes:**

- Students will be able to understand the notion of dual space and double dual, Annihilator of a subspace and its application to counting the dimension of a nite dimensional vector space, Basics of determinants, applications to solving system of equations, Nilpotent operators, invariant subspaces and its applications, Bilinear forms and spectral theorem with examples of spectral resolution and Symmetric bilinear form and Sylvester's law.
- Students will be able to understand the applicability of the above concepts in different courses of pure and applied mathematics and hence in other disciplins of science and technology.

**Course Outcomes:**

- This course is the foundation course of mathematics, especially mathematical analysis.
- Student will be able to grasp approximation of a differentiable function localized at a point.
- Inverse function theorem helps to achieve homeomorphism locally at a point whereas implicit function theorem justies the graph of certain functions. Indirectly or directly Unit III talks about value of a function in the neighbourhood of a known element.
- In Unit IV, student will be able to understand the concept of Riemann integration.

**Course Outcomes:**

- In this course the students will learn about series of functions and power series. The concept of radius of convergence will be introduced and calculated.
- This course gives insight of complex integration which is dierent from integration of real valued functions. In particular, Cauchy integral formula will be proved.
- The students will learn that if a function is once (complex) dieffrentiable then it is innitely many times differentiable. This will be a sharp contrast with the theorems of real analysis.
- The various properties of obius transformations that have a wide variety of applications along with major theorems of theoretical interest like Cauchy-Goursat theorem, Morera's theorem, Rouche's theorem and Casorati-Weierstrass theorem will be studied.

**Course Outcomes:**

- Through this course students are expected to understand the basic concepts of existence and uniqueness of solutions of Ordinary Differential Equations (ODEs).
- In case of nonlinear ODEs, students will learn how to construct the sequence of approximate solutions converges to the exact solution if exact solution is not possible.
- Students will be able to understand the qualitative features of solutions.
- Students will be able to identify Sturm Liouville problems and to understand the special functions like Legendre's polynomials and Bessel's function.
- Students will be to understand the applicability of the above concepts in dierent disciplins of Techonolgy.

**Course Outcomes:**

- Students will solve Linear Diophantine equations, cubic equation by Cardano's Method, Quadratic Congruence equation. Students will learn the multiplicativity of function τ, σ and φ.
- Students will be able to understand the proof of Erdos- Szekers theorem on monotone sub-sequences of a sequence with n²+1 terms and the applicability of Forbidden Positions.
- Stundent will learn the Fibonacci sequence, the Linear homogeneous recurrence relations with constant coecient, Ordinary and Exponential generating Functions, exponential generating function for bell numbers, the applications of generating Functions to counting and use of generating functions for solving recurrence relations.
- Stundents will be able to understand Polyas Theory of counting, Orbit stabilizer theorem, Burnside Lemma and its applications, Applications of Polya's Formula.

Copyright © 2015 - All Rights Reserved : Department of Mathematics, University of Mumbai, Mumbai - 400 098.