M.Sc. Mathematics

Semester II

  • COURSE CODE
    COURSE NAME
    CREDITS
  • JMMH-201

    Complex Analysis:

    UNIT-I

    Functions of complex variables; Limit and continuity, Differentiability; Power Series as an analytic function, Exponential and Trigonometric functions, Complex Logarithms, Zeros of analytic functions.

    UNIT-II

    Complex integration, curves in the complex plane , basic properties of complex integrals winding number of a curve; Cauchy–Goursat Theorem, Cauchy’s Integral formula, Morera’s Theorem, Laurent’s series Maximum modulus principle, Schwarz lemma, Liouville’s theorem.

    UNIT-III

    Isolated singularities, removable singularity, poles, Singularity at infinity calculus of residue at finite point, residue at the point at in finite residue theorem, Number of zeros, Poles, Rouche’s Theorem.

    UNIT-IV

    Bilinear transformations, their properties and classifications, Definitions and examples of conformal mappings; spaces of analytic functions, Hurwitz’s theorem, Montel’s theorem, Riemann mapping theorem; Mobius transformations.

    UNIT-V

    Hyper-geometric Series, Generalized Hyper-geometric functions, Gamma function and its properties, Riemann Zeta function, Riemann’s functional equation.

    Books:

    • 1. J.B. Conway, Narosa Complex Analysis, Publishing House.
    • 2. Ruel V. Churchill, Complex Variables and Applications, Tata McGraw-Hill.
    • 3. Foundation of Complex Analysis, S. Ponnusamy , Narosa Publishing House.
    • 4. H.A. Priestly, Introduction to Complex Analysis, Clarendon Press, Oxford.
    • 5. J.B. Conway, Function of one Complex Variable, Springer-Verlag.
    04
  • JMMH-202

    Linear Algebra:

    UNIT-I

    Vector Spaces: Definition, General properties of vector spaces; Vector subspaces; Algebra of subspaces;Linear Spans; Row space of Matrix; Linear dependence and independence of vectors; Finite-dimensional vector spaces; Dimension of vector space and sub-spaces; Quotient spaces; Direct sum of spaces;Coordinates; Disjoint subspaces. .

    UNIT-II

    Vectors in Rn ,Curves in Rn, Vectors in R3,Vector in C3; Matrices: Addition and scalar multiplication,Transpose of matrix, Square matrices; Systems of linear equations; Diagonalisation; Eigen values and Eigen vectors; Minimal polynomial; Cayley-Hamilton Theorem; Hermitian & Skew-Hermitian and unitary matrices; Powers of Matrices; Polynomials in Matrices; Invertible Matrices; Special types of Square Matrices; Complex and Block Matrices.

    UNIT-III

    Linear Transforms; Linear operator; Range and null space of a linear Transformation; Rank and nullity;Product of linear Transformation; Singular Transformation; Representation of linear Transformation by matrix; Dual spaces; Dual Bases; Projections.

    UNIT-IV

    Inner Product Spaces: Definition, Euclidean and unitary spaces; Norm and length of vector; Cauchy Schwarz’s inequality and Applications; Orthogonality, Orthogonal Sets and Basis, Gram-Schmidt orthogonalization process; self-adjoint operators, Complex Inner Product Spaces; Unitary and Normal operators; Projection theorem; Spectral theorem.

    UNIT-V

    Bilinear Forms: Definition, Bilinear form as vectors; Matrix of a bilinear form; Symmetric & skew Symmetric bilinear forms.

    • 1. Vivek Sahai, Vikas Bist; Linear Algebra, Narosa Publishing House.
    • 2. Sharma & Vashistha, Linear Algebra, Krishna Prakashan Media Ltd.
    • 3. Schaum’s series Linear Algebra, Tata McGraw- Hill.
    • 4. Kenneth Hoffman & Ray Kunze, Linear Algebra, Pearson Education.
    04
  • JMMH-203

    Topology:

    UNIT-I

    Metric space, Open sets, closed sets, Convergence, Completeness, Continuity in metric space, Cantor intersection theorem.

    UNIT-II

    Topological space, Elementary concept, Basis for a topology, Open and closed sets, Interior and closure of sets, Neighborhood of a point, Limits points, Boundary of a set, Subspace topology , Weak topology, Product topology, Quotient topology.

    UNIT-III

    Continuous maps, Continuity theorems for Open and closed sets, Homeomorphism, Connected spaces, Continuity and connectedness, Components, Totally disconnected space, Locally connected space, Compact space, Limit point compact, Sequentially compact space, Local compactness, Continuity and compactness, Tychonoff theorem.

    UNIT-IV

    First and second countable space, T1 spaces, Hausdorff spaces, Regular spaces, Normal spaces, Completely normal space, Completely regular space, Tietz- Extention theorem, Metrizability, Uryshon Lemma, Uryshon metrization theorem.

    UNIT-V

    Fundamental group function, Homotopy of maps between topological spaces, Homotopy equivalence, Contractible and simple connected spaces, Fundamental groups of S1, and S1x S1 etc., Calculation of fundamental groups of Sn , n>1 using Van Kampen’s theorem , Fundamental groups of a topological group.

    Books:

    • 1. James R. Munkres, Topology, Pearson Education Pvt. Ltd.
    • 2. J. R Munkres, Topology A First Course, Prentice- Hall.
    • 3. J.L. Kelly, General Topology, Van Nostrand, Reinhold Co.
    • 4. G.F. Simmons, Introduction to Topology and Mordern Analysis.
    • 5. K. D. Joshi, Introduction to General Topology, Wiley Eastern Limited.
    • 6. L. A. Steen and J. A. Seebach Jr, Counterexamples in Topology, H R & Winston.
    04
  • JMMH-204

    Mechanics:

    UNIT-I

    Moments of inertia, Kineticenergy, Angular momentum.

    UNIT-II

    Mechanics of a particle and system ofparticles, kinematics of a rigid body, Euler’s angles.

    UNIT-III

    Euler’s dynamical equations, two dimensional motion of a rigid body, Compound pendulum, Contraints.

    UNIT-IV

    D’Alembert’s principle, Lagrange’s equations of motion, Techniques of calculus of variations.

    UNIT-V

    Hamilton’s principles, Hamilton’s equations of motion, Contact transformation, Lagrange’s and Poison brackets, Integralin variances, Hamilton-Jacobi Poisson equations.

    Books:

    • 1. Principle of Mechanics: Singe and Griffith.
    • 2. Lectures in Analytical Mechanics: F. Gantamacher.
    • 3. Ele. Treatise on the dynamics of particle and rigid bodies: S.L. Loney.
    • 4. A Text Book of Dynamics: F. Choelton.
    • 5. An introduction to the Calculus of Variation: C. Fox.
    • 6. Calculus of Variations: R. Weinstock.
    04
  • JMMH-205

    Viva-Voce
    04
  • Total Credits
     
    20