MATHEMATICS
for 2year M. Sc. (Mathematics & Computing) course:
(a) DIFFERENTIAL CALCULUS: Successive differentiation. Leibnitz’s theorem,
Taylor’s and Mclaurin’s series of one and two variables, partial and total derivatives,
maxima and minima of function of one, two and three variables, curvature and asymptotes.
(b) INTEGRAL CALCULUS: Definite integral, differentiation under integral
sign. Improper Integrals, Beta, Gamma and error functions, double and triple integrals
and their applications. Reimannintegration: necessary and sufficient conditions,
Reimann Stieltjes integral as a generalization of Reimann integration, necessary
and sufficient conditions for RS integrability.
(c) VECTOR CALCULUS: Differentiation of scalar and vector point functions,
Expansion formulae involving gradient, divergence and curl, line, surface and volume
integration of vector function, Green’s, Gauss and Stoke’s theorems and their applications.
Orthogonal curvilinear coordinates.
(d) ALGEBRA: Convergence of series, Cauchy’s general principle of convergence,
convergence of series of nonnegative terms, comparison, Cauchy’s root, condensation,
D’ Alembert’s, Raabe’s, DeMorgan and Bertrand, and logarithmic tests of convergence.
Alternating series, conditional and absolute convergence, Power series. Solution
of cubic and biquadratic equations.
(e) ABSTRACT ALGEBRA: Group: properties, abelian group, cyclic group, permutation
group, order of an element of group, subgroups of a group and their properties.
Normal subgroup, quotient group. Elementary ideas of a ring, integer domain and
field and their properties.
(f) BOOLEAN ALGEBRA: Properties and relation in Boolean algebra, Application
of Boolean algebra in electrical networks, solvability of Boolean equations and
logical puzzles.
(g) MATRIX ALGEBRA: Rank and Inverse of a matrix, normal form of matrix,
consistency conditions, solution of system of linear equations, linear and orthogonal
transformations, eigen values and eigen vectors, CaleyHamilton theorem, reduction
to diagonal form and reduction of quadratic form to canonical form. Orthogonal,
unitary and Hermitian matrices and their eigen values. Vector space and properties.
(h) COMPLEX VARIABLES: Analytic functions, CauchyReimann equations, harmonic
functions, complex integration, Cauchy’s theorem, Cauchy’s integral formula. Expansion
of analytic functions in power seriesTaylor’s and Laurent’s series, residues, evaluation
of integral using residue theorem.
(i) DIFFERENTIAL EQUATIONS: Formation of differential equations, solution
of first order and higher order differential equations with constant and variable
coefficients. Simultaneous linear differential equations. Partial differential equations
of first order. Application of differential equations.
(j) DYNAMICS: Motion in two dimensions: Velocity and acceleration parallel
to coordinate axes, radial and transverse velocities and acceleration, tangential
and normal velocities and acceleration, D’ Alembert’s principle.
(k) LAPLACE TRANSFORMS: Laplace transform of some elementary functions, properties,
Laplace transforms of derivatives, Laplace transform of integrals, tmultiplication
and tdivision theorems, inverse Laplace transform, convolution theorem, applications.
(l) NUMERICAL METHODS: Finite difference, Interpolation in regular or irregular
intervals, numerical differentiation and integration, numerical solution of first
order ordinary differential equation, solution of nonlinear equations, solution
of simultaneous linear equations by Gaussian methods and method of factorization.
(m) STATISTICS: Probability of events, mutually exclusive and independent
events, Baye’s theorem, probability mass and density functions, binomial, Poisson
and normal distributions.
