Syllabus

Module 1: ( 6 hours)
Curvilinear coordinates: transformation of coordinates, orthogonal curvilinear coordinates. Unit vectors, arc length, volume elements, gradient, divergence, curl and Laplacian in curvilinear systems. Special orthogonal coordinate systems cylindrical, spherical coordinate systems.

Module 2: (4 hours)
Generalized functions: step function, Dirac delta function-definition, representation of Dirac delta function in terms of conventional functions, relation between Dirac delta and step function, Fourier representation of the delta function, properties of delta function.

Module 3: (12 hours)
Ordinary differential equations (ODE): linear second order differential equations with variable coefficients, power series method and Frobenius method of solving differential equations. Special functions: Bessel’s function, Legendre function, Hermite function, Laguerre function and their properties: orthogonality, recurrence relations, generating functions.

Module 4: (10 hours)
Partial differential equations (PDEs): first order PDE and its solutions, different classes of second order PDEs and boundary conditions, separation of variables and Laplace equation in cartesian and spherical polar coordinate systems, solution of homogeneous Helmholtz equation with constant coefficient in cartesian, circular cylindrical and spherical polar coordinate systems, inhomogeneous self-adjoint linear ordinary differential operator and Green function, eigenfucntion expansion of Green function. Solving Poisson’s equation by Green function method.

Module 5: (4 hours)
Special matrices: real, symmetric and hermitian matrices, number of independent parameters, independent matrices and linear combinations, orthogonal matrix, independent elements of orthogonal matrix, unitary matrix, independent elements of a unitary matrix, 2x2 special unitary matrix, orthogonal and unitary transformations, transformation of vectors and matrices, direct sum and direct product of matrices.

To learn curvilinear coordinate systems and their corresponding algebra.

Learning various special functions which are useful in advanced courses in Physics.

To learn the power series method and special functions. Applications of special functions in standard problems in Physics.

Solving various physically motivated PDEs via separation of variables in different coordinate systems.

5. To learn the class of matrices and their properties, which are ubiquitous in Physics.

At the end of the course, students will be able to:
CO1: Do necessary calculations by changing into appropriate curvilinear coordinates.

CO2: Apply properties of special functions in simplifying problems.

CO3: Solve complicated differential equations by applying the knowledge of special functions.

CO4: Apply the knowledge of special functions as solutions of differential equations in advanced Physics courses.

CO5: Identify different kinds of matrices and their properties to solve eigenvalue problems.

G. B. Arfken, H. J. Weber, F. E. Harris, Mathematical Methods for Physicists, Elsevier , 7th Edition (2012).

A. W. Joshi., Matrices and Tensors in Physics, New Age International Private Limited (2017).

V. Balakrishnan, Mathematical Physics, Ane Books (2017).

M. T. Vaughn, ntroduction to Mathematical Physics, Wiley India (2007).