SEQUENCES, SERIES AND DIFFERENTIAL CALCULUS:
Sequences and Series of
real numbers: Sequences
and series of real numbers. Convergent and divergent
sequences, bounded and monotone sequences,
Convergence criteria for sequences of real numbers,
Cauchy sequences, absolute and conditional convergence;
Tests of convergence for series of positive terms -
comparison test, ratio test, root test, Leibnitz test for
convergence of alternating series.
Functions of one variable: limit, continuity,
differentiation, Rolle's Theorem, Mean value theorem.
Taylor's theorem. Maxima and minima.
Functions of two real variable: limit, continuity, partial
derivatives, differentiability, maxima and minima.
Method of Lagrange multipliers, Homogeneous
functions including Euler's theorem.
Integral Calculus: Integration as the inverse process
of differentiation, definite integrals and their properties,
Fundamental theorem of integral calculus. Double and triple
integrals, change of order of integration. Calculating surface
areas and volumes using double integrals and applications.
Calculating volumes using triple integrals and applications.
Differential Equations: Ordinary differential equations
of the first order of the form y'=f(x,y). Bernoulli's equation,
exact differential equations, integrating factor, Orthogonal
trajectories, Homogeneous differential equations-separable
solutions, Linear differential equations of second and higher
order with constant coefficients, method of variation of
parameters. Cauchy- Euler equation.
Vector Calculus: Scalar and vector fields, gradient,
divergence, curl and Laplacian. Scalar line integrals and
vector line integrals, scalar surface integrals and vector
surface integrals, Green's, Stokes and Gauss theorems and
Group Theory: Groups, subgroups, Abelian groups,
non-abelian groups, cyclic groups, permutation groups;
Normal subgroups, Lagrange's Theorem for finite groups,
group homomorphisms and basic concepts of quotient
groups (only group theory).
Linear Algebra: Vector spaces, Linear dependence of
vectors, basis, dimension, linear transformations, matrix
representation with respect to an ordered basis, Range
space and null space, rank-nullity theorem; Rank and
inverse of a matrix, determinant, solutions of systems of
linear equations, consistency conditions. Eigenvalues and
eigenvectors. Cayley-Hamilton theorem. Symmetric, skewsymmetric,
hermitian, skew-hermitian, orthogonal and
Real Analysis: Interior points, limit points, open sets,
closed sets, bounded sets, connected sets, compact sets;
completeness of R, Power series (of real variable) including
Taylor's and Maclaurin's, domain of convergence, term-wise
differentiation and integration of power series.