# Mathematical Physics

## Mathematical Physics

(Timing: Wednesday and Thursday, 08:00 AM – 09:00 AM, and Friday, 12:00 PM to 02:00 PM)

### Unit I

Orthogonal Curvilinear co-ordinate system. Scale factors, expression for gradient, divergence and curl and their applications to Cartesian, circular, cylindrical and spherical polar co-ordinate systems.

Co-ordinate transformation and Jacobian. Transformation of Covariant, Contravariant and Mixed Tensors. Addition, Multiplication and Contraction of Tensors. Metric tensor and its use in transformation of tensors. Dirac Delta Function and its properties.

### Unit II

Fourier series: Fourier theorem and computation of Fourier coefficients. Even and odd functions, half range expansion, sums and scale changes, forced oscillations, Expansion Techniques: integration and differentiation. Introduction to Fourier transform and its simple applications.

### Unit III

Solution of differential equations - Series method: Properties of power series, solution of ordinary differential equation: Legendre's Equation, Legendre Polynomials and Functions, Hermite Polynomials.

The method of Frobenius: Solution about regular singular points, The Gamma function, the Bessel-Clifford equation.

Roots differing by an integer: Series method, Solution of Bessel equation for:

- Roots not differing by an integer
- Equal roots
- Roots differing by an integer

Basic identities involving Bessel Functions. Basic properties like orthogonality recurrence relation and generating functions of Bessel, Hermite, Legendre, and associated Legendre's function (simple applications)

### Unit IV

Solution of partial differential by separation of variable technique and its application to following Boundary Value Problems:

- Laplace equation in three dimensional Cartesian co-ordinate system - line charge between two earthed parallel plates.
- Laplace equation in two dimensional Cartesian co-ordinate system - Heat conduction in a thin rectangular plate.
- Wave equation in two dimensional Cartesian co-ordinate system - Heat conduction in a thin rectangular plate.
- Diffusion equation in cylindrical co-ordinate system.

### Unit V

Matrices: Inverse of a matrix, adjoint, Hermition adjoint, Solution of liner equations using matrix.

Normas and inner products, orthogonal sets and matrices, the Gram Schmidt process and the Q-R factorization theorem. Projection matrices. Least square fit of data. Eigen values and Eigen vectors, diagonalization of matrices. Examples involving up to 3×3 matrices and for the case of real symmetric and simple matrices. Solution of linear differential equations for the homogeneous and non-homogeneous cases.

## Reference books

- Mathematical Methods bby Potter and Glodberg (Prentice Hall of India Pvt. Ltd.) (I followed.)
- Applied Mathematics for Engineers and Physicists by Pipes and Harvill (McGraw Hill Book Co.)