Real Analysis

Advanced calculus and mathematical analysis. Real number system, axioms and properties of supremum and infimum, sequences, subsequences and series. Cauchy sequences. Completeness, convergence, continuity, compactness, connectedness. Discontinuities. Differentiable functions. Mean value theorems, continuity of derivatives. Implicit functions, functional dependence. Taylor 's theorem. Maxima and minima of functions of 2 and 3 variables. Language multipliers and multiple integrals. Theorems of Gauss, Stoke and Green.

Real Analysis II

Riemann Stieljes integral, properties, functions. Uniform convergence. Pointwise and Uniform convergence of sequences and series. Weierstrass M Test. Uniform convergence and continuity, integration and differentiation. Convergence theorem of improper integrals.

Algebra I

Theory of groups and subgroups, and abstract algebra. Generators, relations, cyclic groups. Cosets and Lagrange's theorem. Normalizers and centralizers. Center. Conjugacy. Normal subgroups and simple groups. Factor groups, isomorphism and automorphism. Commutators. Permutation groups, Cayley's theorem, rings, integral domains. Field and its characteristic.

Algebra II

Linear algebra and dimensional vector spaces. Linear dependence/independence of vectors. Vector spaces and subspaces. quotient and direct sum space. Linear transformation, rank and nullity. Representation of linear transformations as matrices. Change of bases. Linear functional, dual spaces, annihilators. Eigen vectors and values; Cayley-Hamilton theorem. Diagonalization of matrices. Inner product spaces. Bilinear, quadratic and hermitian forms.

Complex Analysis

Complex variables and applications. Cauchy-Riemann equations. Power series, convergence. Cauchy's theorem, integral formula and related theorems. Contour integration. Singularities, branch points. Taylor's and Laurent's series. Continuation. Residues. Fundamental theorem of algebra. Application of calculus of residues to infinite products. Conformal transformations.

Topology

Topology and functional analysis. Sets. Cardinal numbers. Axiom of choice. Topological spaces. Continuous and open mapping. Homoeomorphism. Axioms of countability, and separation axioms. Regular and normal spaces. Connectedness, open covers, compact spaces, continuity and compactness in metric spaces. Cantor's intersection and Baire's category theorems.

Differential Geometry

Vector and tensor methods. Theory of space curves. Helices, spherical indicatricies, evolutes involutes. Theory of surfaces. Tangent and normal planes. Coordinate transformations. Surface curves, normal and geodesic curvature. First and second fundamental forms. Christoffel symbols. Gauss theorem. Mean and Gaussian curvatures, Euler's theorem, Dupin's indicatricies. Gauss-Weingarten an Gauss-Codazzi equations.

Differential Equations

Ordinary differential equations, constant/variable coefficients, regular/irregular points, series solutions. Frobenious methods: Bessel, Legendre, Laguere, Hermite, Chebychev, Hypergeometric equations. Orthogonal polynomials,. Sturm-Liouville systems. Adjoint and linear systems.

Mechanics

Classical and theoretical mechanics and dynamics. Particle mechanics. D'Alembert's principle, Lagrange's equations. Velocity dependent potentials and dissipation. Application of Lagrangian formulation. Hamilton's principle. Extension of Hamilton's principle to non-holonomic system. Virial theorem. Euler angles and theorem on rigid body motion. Coriolis force. Moment of inertia. Rigid body problem solution and Euler equations of motion. Torque free motion. Heavy symmetrical top with one point fixed.

Numerical Analysis I

Computation error and error analysis. Iterative methods to solve non-linear equations. Convergence and stability. Newton-Rephson and fixed point methods. Acceleration of convergence by Aitken method, LU decomposition method. Iterative methods: Jacobi, Gauss, Seidel, SOR, SUR. Interpolation, Gauss forward/backward method, Bessel and Stirling method, Lagrange interpolation, Newton divided differences. Spline functions, least squares, differentiation, integration. Newton Cotez formula. simpson's rules. Lagrange and divided differences formula. Gaussian quadrature,orthogonal polynomials. Legendre and Laguere polynomials.

Numerical Analysis II

Solution of ordinary differential equation of first order. N-simultaneous first order equations, R-K methods. Eigen values and vectors. Meritian matrix, diagonal and triadigaonal forms, transformations, Heisenurg form, bisection method to find Eigen values, Iterative methods, power, deflation, LR and QR methods.

Functional Analysis

Elements of functional analysis. Normed linear space and Banach spaces. Quotient normed spaces. Bounded linear operators and functionals. dual spaces. Hahn-Banach theorem. Hilberg spaces. Orthogonal and Orthonormal sets. Direct sums. Projection theorems. Riesz's representation theorem. Adjoint of linear operator. Unitary and normal operators.

Additional Mathematics courses are included in various departments.