M.Sc. (Mathematics)

  Sem. I

Sr. No.

Subjects of Study

A

Advanced Abstract Algebra

Unit I

Direct product of groups (External and Internal). Isomorphism theorems - Diamond isomorphism theorem, Butterfly Lemma, Conjugate classes, Sylows theorem, p-sylow theorem.

II

Commutators, Derived subgroups, Normal series and Solvable groups, Composition series, Refinement theorem and Jordan-Holder theorem for infinite groups.

III

Polynomial rings, Euclidean rings. Modules, Sub-modules, Quotient modules, Direct sums and Module Homomorphisms. Generation of modules, Cyclic modules.

IV

Field theory - Extension fields, Algebraic and Transcendental extensions, Separable and inseparable extensions, Normal extensions. Splitting fields. Galois theory - the elements of Galois theory, Fundamental theorem of Galois theory

V

Galois theory - the elements of Galois theory, Fundamental theorem of Galois theory, Solvalibility by radicals.

  Sem. I

B

Real Analysis

Unit I

Algebra and algebras of sets, Algebras generated by a class of subsets, Borel sets, Lebesgue measure of sets of real numbers, Measurability and Measure of a set, Existence of Non-measurable sets, Measurable functions. Realization of nonnegative measurable function as limit of an increasing sequence of simple functions.

II

Realization of non-negative measurable function as limit of an increasing sequence of simple functions. Structure of measurable functions, Convergence in measure, Egoroff's theorem.

III

Weierstrass's theorem on the approximation of continuous function by polynomials, Lebesgue integral of bounded measurable functions, Lebesgue theorem on the passage to the limit under the integral sign for bounded measurable functions.

IV

Summable functions, Space of square summable functions. Fourier series and coefficients, Parseval's identity, Riesz-Fisher Theorem.Lebesgue integration on R2.

V

Lebesgue integration on R2, Fubini's theorem. Lp-spaces, Holder-Minkowski inequalities. Completeness of Lp-spaces.

  Sem. I

C

Advanced Differential Equations

Unit I

Non-linear ordinary differential equations of particular forms. Riccati's equation -General solution and the solution when one, two or three particular solutions are known. Total Differential equations.

II

Partial differential equation of first order – formulation and classification of partial differential equations, Lagrange’s linear equation, particular forms of non– linear partial differential equations, Charpit’s method.

III

Linear Partial differential equations with constant coefficients. Homogeneous and non- Homogeneous equation. Partial differential equation of second order with variable coefficients- Monge's method,

IV

Classification of linear partial differential equation of second order, Cauchy's problem, Method of separation of variables, Laplace, Wave and diffusion equations, Canonical forms.

V

Linear homogeneous boundary value problems. Eigen values and eigen functions. Strum-Liouville boundary value problems. Orthogonality of eigen functions. Reality of eigen values.

  Sem. I

D

Differential Geometry-I

Unit I

Theory of curves- Space curves, Tangent, Contact of curve and surface, Osculating plane, Principal normal and Binormal, Curvature, Torsion, Serret- Frenet's formulae.

II

Osculating circle and Osculating sphere, Existence and Uniquenss theorems, Bertrand curves, Involute, Evolutes.

III

Ruled surface, Developable surface, Tangent plane to a ruled surface. Necessary and sufficient condition that a surface =f () should represent a developable surface.

IV

Metric of a surface, First, second and third fundamental forms. Fundamental magnitudes of some important surfaces, Orthogonal trajectories. Normal curvature, Meunier's theorem,.

V

Principal directions and Principal curvatures, First curvature, Mean curvature, Gaussion curvature. Umbilics. Radius of curvature of any normal section at an umbilic on z = f(x,y). Radius of curvature of a given section through any point on z = f(x,y). Lines of curvature, Principal radii, Relation between fundamental forms. Asymptotic lines,

  Sem. I

E

Dynamics of a Rigid body

Unit I

D'Alembert's principle. The general equations of motion of a rigid body. Motion of centre of inertia and motion relative to centre of inertia. Motion about a fixed axis: Finite forces (Moment of effective forces about a fixed axis of rotation, angular momentum, kinetic energy of a rotating body about a fixed line. Equation of Motion of the body about the Axis of Rotation, Principle of Conservation of energy. The compound pendulum (Time of a Complete Oscillation, Minimum time of oscillation), Centre of percussion.

II

Motion of a rigid body in two dimensions: Equations of motion in two dimensions, Kinetic energy of a rigid body, Moment of Momentum, Rolling and sliding Friction, Rolling of a sphere on a rough inclined plane , Sliding of a Rod, Sliding and Rolling of a Sphere on an inclined plane, Sliding and Rolling of a sphere on a fixed sphere. Equations of motion of a rigid body under impulsive forces, Impact of a rotating Elastic sphere on a fixed horizontal Rough plane. Change in K.E. due to the action of impulse.

III

Motion in three dimensions with reference to Euler's dynamical and geometrical equations. Motion under no forces, Motion under impulsive forces. Conservation of momentum (linear and angular) and energy for finite as well as impulsive forces.

IV

Lagrange's equations for holonomous dynamical system, Energy equation for conservative field, Small oscillations, Motion under impulsive forces. Motion of a top.

V

Hamilton's equations of motion, Conservation of energy, Hamilton's principle and principle of least action.

Sem. II

A

Linear Algebra

Unit I

Linear transformations, properties of linear transformation, Matrix of a linear transformation, Change of basis, Orthogonal and Unitary transformations. Linear functionals- Dual space and Bidual space. Adjoint of a linear transformation,

II

Matrix: Rank and Nullity, Eigen values, Eigen vectors of linear transformations, Characteristics equation of a matrix, Annihilators- Definition, Properties and dimension.

III

Determinants: Matrix inversion- inverse of a matrix by partitioning, Characteristic polynomial, Minimal polynomial. Cayley-Hamilton theorem and Diagonalisation.

IV

Canonical and Bilinear Forms: Jordan Forms The rational forms, Bilinear forms: Definition and examples. The matrix of a Bilinear form, Orthogonality, Classification of Bilinear forms.

V

Inner Product Spaces: Real inner product, Norm of a vector and normed vector space, Orthogonality, Principal axis theorm, Unitary spaces, Cauchy Schwarz’s inequality.

  Sem. II

B

Topology

Unit I

Metric Spaces-Definition and examples, Open spheres and Closed spheres, Open sets and Closed sets, Neighbourhood, Sequence in metric space. Continuous mapping and Completeness in metric space.

II

Topological Spaces-Definition and examples, Closed sets, Neighbourhood, Open base and sub base. Limit points, Adhere points and derived sets, Closure of a set, Subspaces, Continuity and Homeomorphism.

III

Compact and Locally Compact spaces, Connected spaces- Connected and Locally connected spaces, Continuity and Compactness, Continuity and Connectedness,

IV

Separation axioms: To space, T1 space, T2 space or Hausdroff space, T3 space- Regular space, T4 space- Normal space.

V

Product spaces: weak topologies, Product space of two spaces, Product invariant properties for finite products, General product spaces.

  Sem. II

C

Special Functions

Unit I

Calculus of variation - Functionals, Variation of a functional and its properties, Variational problems with fixed boundaries, Euler's equation, Extremals, Functional dependent on several unknown functions and their first order derivatives, Functionals dependent on higher order derivatives, Functionals dependent on the function of more than one independent variable.

II

Gauss hypergeometric function and its properties, Integral representation, Linear transformation formulas, Contiguous function relations, Differentiation formulae, Linear relation between the solutions of Gauss hypergeometric equation.

III

Linear relation between the solutions of Gauss hypergeometric equation., Kummer's confluent hypergeometric function and its properties, Integral representation, Kummer's first transformation.

IV

Legendre polynomials and functions Pn(x) and Qn(x). Christoffel’s summation formula, Bessel functions Jn(x), Modified Bessel’s Functions, Solutions of Bessel’s equations.

V

Hermite polynomials Hn(x), Laguerre polynomials- Recurrence relations, Generating functions, Orthogonal properties.

   Sem. II

D

Differential Geometry-II &Tensors

Unit I

Geodesics, Differential equation of a geodesic, Single differential equation of a geodesic, Geodesic on a surface of revolution, Geodesic Curvature and Torsion, Gauss-Bonnet Theorem.

II

Gauss's formulae, Gauss's characteristic equation, Weingarten equations, Mainardi-Codazzi equations. Fundamental existence theorem for surfaces, Parallel surfaces, Gaussian and mean curvature for a parallel surface, Bonnet's theorem on parallel surfaces.

III

Tensor Analysis- Kronecker delta. Contravariant and Covariant tensors, Symmetric tensors, Quotient law of tensors, Relative tensor. Riemannian space. Metric tensor, Indicator, Permutation symbols and Permutation tensors.

IV

Christoffel symbols and their properties, Covariant differentiation of tensors. Ricci's theorem, Intrinsic derivative, Geodesics, Differential equation of geodesic, Geodesic coordinates.

V

Reimann-Christoffel tensor and its properties. Covariant curvature tensor, Einstein space. Bianchi's identity. Einstein tensor, Flate space, Isotropic point, Schur's theorem.

  Sem. II

E

Hydrodynamics

Unit I

Kinematics of ideal fluid. Lagrange's and Euler's methods. Equation of continuity in Cartesian, cylindrical and spherical polar coordinates. Boundary surface. Stream-lines, path-lines and stream lines velocity potential irrotational motion.

II

Euler's hydrodynamic equations. Bernoulli's theorem. Helmholtz equations. Cauchy's integral, Motion due to impulsive forces.

III

Motion in two-dimensions, Stream function, Complex potential. Sources, Sinks, Doublets, Images in two dimensions- image of a source with regard to a plane, image of a source with regard to a circle.

IV

Irrotational Motion: Motion of a fluid element (General and Cartesian coordinates), vorticity, Body forces, Surface forces, Stress analysis at a point, Strain analysis, Flow and circulation, Kelvin’s circulation theorem, connectivity, Irrotational motion in multiple connected space. Acyclic and cyclic motion. Kelvin’s minimum energy theorem.

V

Irrotational motion in two dimensions: Introduction, General motion of a cylinder in two dimensions, Motion of a circular cylinder in a uniform stream, liquid streaming past a fixed circular cylinder, two co-axial cylinders, Circulation about a circular cylinder, Blasius’s theorem, Streaming and circulation for a fixed circular cylinder, Equation of a motion of a circular cylinder.