Course

Unit 1

VECTOR CALCULUS
Learning Objectives
1. A deeper understanding of concept of vectors and its theorems in calculus and Cartesian tensors and its application in physically relevant systems.
2. Application of Cartesian tensors in problems involving vector algebra and calculus.

Coordinate transformations, Definition of vectors, Index notation, Cartesian Tensors, Kronecker delta, Levi-Civita tensor and its application in Vector algebra and calculus. The vector differential operators, Integrals of vectors, Integral forms of gradient, divergence and curl, Line, surface and volume integrals – Stoke’s, Gauss’s and Green’s theorem.

Unit 2

CURVILINEAR COORDINATES:
Learning Objectives
1. Good understanding of the properties of general curvilinear coordinates and vector calculus operators in them
2. Describing a system using suitable coordinates and solving the relevant problem

Cartesian, spherical and cylindrical coordinates. General curvilinear coordinates, Coordinate curves, Scale factors, Unit vectors in curvilinear systems, Arc length, area elements, volume elements. Gradient, diver-gence, curl and Laplacian. Special orthogonal coordinate systems: Parabolic and cylindrical coordinates, Pa-raboloidal coordinates, Elliptic cylindrical coordinates and applications.

Unit 3

TENSOR ANALYSIS
Learning Objectives
1. Familiarity with the concept and principles of tensor algebra and calculus
2. Application of tensor calculus in studying physical systems

Definition and basic properties of tensors. Covariant, contravariant and mixed tensors. The summation con-vention, Fundamental operations with tensors. The line element and metric tensor. Tensor algebra, Christof-fel symbols and their transformation laws, Covariant differentiation. Tensor form of gradient, divergence and curl. Geodesic equation, Curvature tensors.

Unit 4

DIRAC DELTA & GENERALISED FUNCTIONS, FOURIER SERIES & INTEGRAL TRANSFORMS
Learning Objectives
Study of generalised function- Dirac delta and its various properties and solving physically relevant problems modelled with Dirac delta function in different dimensions and coordinates
Study and application of Fourier Series and integral transforms

Introduction to Generalised functions, delta sequences. One dimensional Dirac delta function, properties and representations, higher dimensional Dirac delta function. Dirac Delta function in curvilinear coordinates. Heaviside unit step function. Applications and properties of Fourier series and its Complex form, Fourier representation of Dirac Delta. Integral transforms and properties, Parseval’s theorem, Convolution theorem, applications. Green’s function

Unit 5

SPECIAL FUNCTIONS
Learning Objectives
1. Familiarity with various special functions and its properties
2. Application of special functions in solving integrals and as solutions of relevant differential equations describing systems with various symmetries.

Gamma, Beta and Error functions – definitions, properties and applications. Orthogonal functions, Bessel’s equation, General solution for non-integer ν; general solution for integer ν; Bessel function of first kind and second, properties of Bessel functions, Integral representations. Recurrence Relation, Orthogonality, Ro-drigues Formula. Modified Bessel functions, Henkel functions. Equations transformed into Bessel’s equa-tion. Other special functions: Legendre, Hermite, Laugerre functions- Recurrence relations and generating functions-. Applications.

Course Objectives: The purpose of the course is to introduce students to the methods of mathematical physics and to develop required mathematical skills to solve advanced problems in theoretical physics.

Course Outcomes
After completing the course, the student should be able to:
CO 1. Understand mathematical methods used in various advanced physics courses and apply the techniques in solving problems involved

CO 2. Understand the theory of vector calculus in orthogonal and general curvilinear coordinates and apply it to solve physically relevant problems

CO 3. Perform basic operations with tensors in algebra and calculus; formulate and express physical laws in terms of tensors, and simplify it by the use of coordinate transforms

CO4. Understand the properties of Dirac delta function, various special functions, Fourier series and integral transforms and application of the same in solving integrals and differential equations

Skills:
Problem solving skills using various mathematical methods. Mathematical outlook to physical problems.

Text Books

1. Riley K F, Hobson M P, Bence S J, Mathematical Methods for Physics and Engineering, CUP, 3rd Ed, 2010
2. Arfken & Weber, Mathematical Methods for Physicists, Elsevier Indian Reprint, 7th Ed., 2012.

Reference

1. M Boas, Mathematical Methods in Physical Sciences, Wiley Indian Reprint 3rd Ed, 2006.
2. Mathews J and Walker R L, Mathematical Methods of Physics, Pearson India, 2nd Ed, 2004.
3. C. W. Wong, Introduction to Mathematical Physics: Methods & Concepts, Oxford, 2013

CO-PO Mapping

*CA – Can be Quizzes, Assignments, Projects, and Reports.

Justification for CO-PO Mapping