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Syllabus (2 - Year M.Sc. (Mathematics & Computing))

MATHEMATICS- for 2-year 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. Reimann-integration: necessary and sufficient conditions, Reimann Stieltjes integral as a generalization of Reimann integration, necessary and sufficient conditions for R-S 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 non-negative terms, comparison, Cauchy’s root, condensation, D’ Alembert’s, Raabe’s, De-Morgan 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, Caley-Hamilton 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, Cauchy-Reimann equations, harmonic functions, complex integration, Cauchy’s theorem, Cauchy’s integral formula. Expansion of analytic functions in power series-Taylor’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, t-multiplication and t-division 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 non-linear 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.

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