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 S. No. Topic Learning Objective Teaching guidelines Methodology Time (Hrs) 1 VECTOR SPACES, TENSORS AND MATRICES ·      Vector spaces: Introduction, definition of linear vector space ·      linear independence, basis and dimension, scalar product, orthonormal basis, ·      Gram-Schmidt Orthogonalization process, Linear operators, ·      Covariant and Contravariant tensors, ·      symmetric and skew-symmetric tensor, ·      product of tensors, ·      Metric tensors, Matrices, ·      Orthogonal, Unitary and Hermition Matrices, ·      Eigen values & Eigen vectors, ·      Matrix diagonalization. To study multilinear algebra, functions of several variables those are linear in each variable separately. It also includes a study of matrices, vectors, tensors, and linear transformations. The present unit will provide the better way to learn linear algebra and multilinear algebra. The present unit will provide the better way to learn linear algebra and multilinear algebra. To cover  vector spaces, linear independence, basis and dimension, scalar product, orthonormal basis, Gram-Schmidt Orthogonalization process, Linear operators, Covariant and Contravariant tensors, symmetric and skew-symmetric tensor, product of tensors, Metric tensors, Matrices, Orthogonal, Unitary and Hermition, Matrices, Eigen values & Eigen vectors, Matrix diagonalization. ·      Lecture. ·      Seminar. ·      Discussion/Interaction with Students. ·      Assignment. ·      Discussion on Assignment.         Evaluation of Assignment 10 2 DIFFERENTIAL EQUATIONS AND SPECIAL FUNCTIONS ·      First order equation, second order equation with variable coefficients. ·      ordinary point, singular point, series solution around an ordinary point and regular singular point, ·      solution of Legendre equation, ·      solution of Bassel’s equation, ·      solution of Hermite & Lageurre equations. ·      Definitions  and properties of special functions: ·      Bessel functions, ·      Legendre polynomials, ·      Hermite polynomials and ·      Lageur repolynomials along with Recurrence relations. Introduce the student to Integral Transforms and their application to the solution of Ordinary Differential Equations Expose the student to some special functions fundamental specifically Bessel, Legendre and Hermite. To cover  first order equation, second order equation with variable coefficients, ordinary point, singular point, series solution around an ordinary point and regular singular point,  solution of Legendre equation,  solution of Bassel’s equation, solution of Hermite & Lageurre equations.  Definitions and properties of special functions:  Bessel functions, Legendre polynomials, Hermite polynomials and Lageur repolynomials along with Recurrence relations. ·      Lecture. ·      Seminar. ·      Discussion/Interaction with Students. ·      Assignment. ·      Discussion on Assignment.         Evaluation of Assignment 10 3 COMPLEX VARIABLES ·      Function of complex variable, ·      limit, continuity and differentiability of function of complex variables, ·      Analytic function, Cauchy-Riemann conditions, ·      Cauchy s integral theorem, ·      Cauchy s Integral formula, ·      Taylor s and Laurent s series, ·      singular points, residues, evaluation of residues, ·      Cauchy’s residue theorem, ·      Jordan s lemma, ·      evaluation of real definite integrals To introduction to the basic theory of complex analytic functions and some applications, in order to get acquainted with a number of methods and techniques applicable to other parts of mathematics, engineering or economics. The aim of the course is to teach the principal techniques and methods of analytic function theory. To cover  Function of complex variable, limit, continuity and differentiability of function of complex variables, Analytic function, Cauchy-Riemann conditions, Cauchy s integral theorem, Cauchy s Integral formula, Taylor s and Laurent s series, singular points, residues, evaluation of residues, Cauchy’s residue theorem, Jordan s lemma, evaluation of real definite integrals ·      Lecture. ·      Seminar. ·      Discussion/Interaction with Students. ·      Assignment. ·      Discussion on Assignment.         Evaluation of Assignment 10 4 INTEGRAL TRANSFORMS ·      Fourier series, Dirichlet’s conditions, ·      Fourier series of arbitrary period, ·      Half-wave expansions, development of the Fourier integral, ·      Fourier integral theorem, ·      Fourier transforms, Properties of Fourier transform, Convolution theorem, ·      Fourier transform of Dirac Delta function, ·      Laplace transform, first and second shifting theorem, ·      Laplace transforms of derivatives and integral of a function, convolution theorem, ·      Inverse Laplace transform by partial fraction and by using convolution theorem, ·      application of Laplace transform in solving differential equations. ·   To provide the standard methods for solving differential equations as well as methods based on the use of matrices or Laplace transforms. ·   To study Fourier series and solve boundary values problems. ·   To develop numerical methods for solving differential equations. To cover  Fourier series, Dirichlet’s conditions, Fourier series of arbitrary period, Half-wave expansions, development of the Fourier integral,  Fourier integral theorem, Fourier transforms, Properties of Fourier transform, Convolution theorem, Fourier transform of Dirac Delta function, Laplace transform, first and second shifting theorem, Laplace transforms of derivatives and integral of a function, convolution theorem, Inverse Laplace transform by partial fraction and by using convolution theorem. ·      Lecture. ·      Seminar. ·      Discussion/Interaction with Students. ·      Assignment. ·      Discussion on Assignment.         Evaluation of Assignment 10