Sr.No. 
Subject of Study 
Part I 

1 
Algebra 
2 
Differential Calculus & Vector Calculus 
3 
Integral Calculus & Trigonometry 
4 
Vector Analysis & Geometry 
Part II 

1 
Advanced Calculus 
2 
Mathematical Method 
3 
Differential Equations 
4 
Mathematics 
Part III 

1 
Analysis 
2 
Abstract Algebra 
3 
Numerical Analysis 
4 
Differential Geometry 
Name of the course 
Topics Covered 
Calculus 
Hyperbolic functions, Leibniz rule and its applications to problems of type eax+bsinx, eax+bcosx, (ax+b)n sinx, (ax+b)n cosx, Reduction formulae, rotation of axes and seconddegree equations, etc. 
Algebra 
Polar representation of complex numbers, nth roots of unity, De Moivre’s theorem for rational indices and its applications, Equivalence relations, Functions, Composition of functions, Systems of linear equations, Introduction to linear transformations, the matrix of a linear transformation, etc. 
Real Analysis 
Review of Algebraic and Order Properties of R,ßœneighborhood of a point in R, Idea of countable sets, uncountable sets and uncountability of R, Sequences, Bounded sequence, Convergent sequence, Limit of a sequence, Infinite series, convergence and divergence of infinite series, Cauchy Criterion, etc. 
Differential Equations 
Differential equations and mathematical models, Introduction to compartmental model, exponential decay model, lake pollution model etc., General solution of a homogeneous equation of second order, a principle of superposition for a homogeneous equation, Equilibrium points, Interpretation of the phase plane, predatorprey model, and its analysis, etc. 
Theory of Real Functions 
Limits of functions (ß³àµ†ßœapproach), a sequential criterion for limits, divergence criteria, Differentiability of a function, Caratheodory’s theorem, Cauchy’s mean value theorem, Riemann integration, Riemann conditions of integrability, Improper integrals, Pointwise and uniform convergence of the sequence of functions, Limit superior and Limit inferior. Power series, a radius of convergence, etc. 
Group Theory 
Definition and examples of groups including permutation groups and quaternion groups (illustration through matrices), Properties of cyclic groups, classification of subgroups of cyclic groups, External direct product of a finite number of groups, Group homomorphisms, properties of homomorphisms, Cayley’s theorem, Characteristic subgroups, Commutator subgroup and its properties, etc. 
PDE and Systems of ODE 
Partial Differential Equations – Basic concepts and definitions, Derivation of the Heat equation, Wave equation and Laplace equation, Systems of linear differential equations, types of linear systems, differential operators, etc. 
Multivariate Calculus 
Functions of several variables, limit and continuity of functions of two variables, Chain rule for one and two independent parameters, directional derivatives, Double integration over rectangular region, Triple Integrals, Triple integral over a parallelepiped and solid regions volume by triple integrals, Line integrals, Applications of line integrals, Green’s theorem, surface integrals, integrals over parametrically defined surfaces, etc. 
Complex Analysis 
Limits, Limits involving the point at infinity, continuity, Analytic functions, examples of analytic functions, exponential function, Logarithmic function, trigonometric function, An extension of Cauchy integral formula, consequences of Cauchy integral formula, Liouville’s theorem, Laurent series and its examples, absolute and uniform convergence of power series, uniqueness of series representations of power series etc. 
Rings and Linear Algebra 
Definition and examples of rings, properties of rings, integral domains and fields, characteristic of a ring. Ideals, ideally generated by a subset of a ring, operations on ideals, prime and maximal ideals. Ring homomorphisms, properties of ring homomorphisms, polynomial rings over commutative rings, division algorithm, Eisenstein criterion. Vector spaces, subspaces, algebra of subspaces, quotient spaces, etc., Linear transformations, null space, range, rank, and nullity of a linear transformation, etc., Dual spaces, dual basis, double dual, the transpose of a linear transformation and its matrix in the dual basis, annihilators etc. 
Mechanics 
Moment of a force about a point and an axis, couple and couple moment, Moment of a couple about a line, resultant of a force system etc., Laws of Coulomb friction, application to simple and complex surface contact friction problems, transmission of power through belts, screw jack, wedge, first moment of an area and the centroid, other centers, etc., Conservative force field, conservation for mechanical energy, workenergy equation, kinetic energy and workkinetic energy expression based on center of mass, etc. 