## JAM MATHS SYLLABUS

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MATHEMATICS (MA)

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
their applications.

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 unitary matrices.

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.

Last modified: Monday, 25 November 2013, 9:39 AM