Syllabus - JAM 2016
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 Variables
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,
skew-symmetric, 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.
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