MPH111

UNIT 1

Linear vector spaces and operators: Vector spaces and subspaces, Linear dependence and independence, Orthogonality, Basis and Dimensions, linear operators, Matrix representation, Types of matrices, Similarity transformations, Characteristic polynomial of a matrix, Eigen values and eigenvectors, Self-adjoint and Unitary transformation.

UNIT 2

Vector analysis and curvilinear co-ordinates- Gradient, Divergence, their geometrical interpolation and Curl operations, rotational motion, vector potential function, Vector Integration, Gauss’ and Stokes’ theorems, Curvilinear co-ordinates, tangent and normal vectors, contravariant and covariant components, line element and the metric tensor, Gradient, Curl, divergence and Laplacian in spherical polar and cylindrical polar co-ordinates.

UNIT 3

Calculus of variations and Non-linear methods: Concept of variation, Euler’s equation, Missing dependent variables, Applications of the Euler equation, Several independent variables, Hamilton’s principle and Lagrange’s equations, Lagrangian multipliers.

UNIT 4

Ordinary differential equations and Special Functions-I: Linear ordinary differential equations, Separation of Poisson and Helmholtz equations in spherical polar and cylindrical polar coordinates, power Series methods, Series solutions – Frobenius’method, Series solutions.

UNIT 5

Ordinary differential equations and Special Functions-II: Series solutions of the differential equations of Bessel, Legendre, Laguerre and Hermite polynomials, Generating functions, Some recurrence relations, orthogonality properties of these functions, Brief discussion of spherical Bessel functions and spherical harmonics.

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