S.N

Unit

Content of the topics

Learning objects

Teaching Guidelines

Methodology

Time (Hrs)

1

one

 Definition of a group with example and simple properties of groups
 Subgroups and subgroup criteria
 Generation of groups, cyclic groups
 Cosets, Left and right cosets
 Index of a subgroup, Coset decomposition
 Lagrange’s theorem and its consequences
 Normal subgroups, Quotient groups

Students will be able to
1. Geometrical meaning of DE.
2. First order higher degree equations solvable for x,y,p.Lagrange’s equations
3. Clairaut’s equations 
Lecture should be effective so that student can grasp the topics easily.

 Lecture.
 Seminar.
 Discussion/Interaction with Students.
 Assignment.
 Discussion on Assignment.
 Evaluation of Assignment.

15

2

Two

 Homoomorphisms
 Automorphisms of cyclic groups Groups
 Homoomorphisms

Students will be able to
 Orthogonal trajectories in cartesian coordinates and polar coordinates
2. Linear DE with constant coefficients. Homogeneous linear ODE. Transformation of the equation by changing the dependent variable/independent variable. Method of variations of parameters 
Lecture should be effective so that student can grasp the topics easily.

 Lecture.
 Seminar.
 Discussion/Interaction with Students.
 Assignment.
 Discussion on Assignment.
 Evaluation of Assignment

15

3

Three

 Introduction to rings, subrings, integral domains and fields,
 Characteristics of a ring. Ring homomorphisms, ideals (principle, prime and Maximal) and Quotient rings, Field of quotients of an integral domain.
 Euclidean rings, Polynomial rings, Polynomials over the rational field, , Polynomial rings over Unique factorisation domain
 R unique factorization domain implies so is R[X1,X2,…Xn] commutative rings,

To make students familiar with Ordinary simultaneous DE, Solution of simultaneous DE involving operators x (d/dx) or t (d/dt) etc. General method of solving Pdx+Qdy+Rdz=0 by taking one variable constant. 
Lecture should be effective sothat student can grasp the topics easily.

 Lecture.
 Seminar.
 Discussion/Interaction with Students.
 Assignment.
 Discussion on Assignment.
 Evaluation of Assignment

15
