Mathematical physics and numerical methods

Orthogonal curvilinear coordinate system, scale factors, Expressions for gradient, divergence and curl and their application to Cartesian, Circular Cylindrical and Spherical polar coordinate systems.

Tensors:

Coordinate transformations, Transformation of covariant, contra variant and mixed tensors. Addition, subtraction, outer product , contraction and inner product of tensors, Quotient law, Symmetric and antisymmetric tensors, Metric tensor.

Dirac delta function and its properties.

Four vector formulation, four velocity vector, energy-momentum four vector, relativistic equation of motion; invariance of rest mass, orthogonality of four force and four velocity, Lorentz force as an example of four force, transformation of four frequency vector, longitudinal and transverse Doppler’s effect, Compton effect.

Techniques of separation of variables and its application to the following boundary value problems (i) Laplace’s equation in three dimensional Cartesian coordinate system – line charge between two earthed parallel plates, (ii) Helmholtz equation in circular cylindrical coordinates-Cylindrical resonant cavity, (iii) Wave equation in spherical polar coordinates-the vibrations of a circular membrane, (iv) Diffusion equation in two dimensional Cartesian coordinate system-heat conduction in a thin rectangular plate.

Introduction, Fourier series and coefficients, functions with point of discontinuity, arbitrary period, even and odd functions, half range expansion, Parseval’s theorem.

Introduction, Finite-Difference Operators, Differential Operator related to the Difference Operator, Truncation error, Numerical interpolation, Roots of equations, Initial-value problems –Ordinary Differential equations: Taylor’s method, Euler’s method and direct method. Trapezoidal and Simpson’s rule for numerical integration.