Course

UNIT 1:VECTOR 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 Stokes, Gausss and Greens theorem.UNIT 2:CURVILINEAR COORDINATES: Cartesian, spherical and cylindrical coordinates. General curvilinear coordinates, Coordinate curves, Scale factors, Unit vectors in curvilinear systems, Arc length, area elements, volume elements. Gradient, divergence, curl, and Laplacian.Special orthogonal coordinate systems: Parabolic and cylindrical coordinates, Paraboloidal coordinates, Elliptic cylindrical coordinates, and applications.UNIT 3:TENSOR ANALYSIS: Definition and basic properties of tensors. Covariant, contravariant, and mixed tensors. The summation convention, Fundamental operations with tensors. Theline element and metric tensor. Tensor algebra, Christoffel symbols and their transformation laws, Covariant differentiation. Tensor form of gradient, divergence, and curl. Geodesic equation, Curvature tensors.UNIT 4: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, Parsevals theorem, Convolution theorem, applications. Greens functionUNIT 5:Gamma, Beta and Error functions definitions, properties and applications. Orthogonal functions, Bessels equation, General solution for non-integer ?; general solution for integer ?; 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 Bessels 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 the methods of mathematical physics and to develop the required mathematical skills to solve advanced problems in theoretical physics.Course Outcomes:After completing the course, the student should be able to:CO1: Understand mathematical methods used in various advanced physics courses and apply the techniques in solving problems involvedCO2: Understand the theory of vector calculus in orthogonal and general curvilinear coordinates and apply it to solve physically relevant problemsCO3: 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 transformsCO4: Understand the properties of the Dirac delta function, various special functions, Fourier series, and integral transforms and application of the same in solving integrals and differential equations

TEXT BOOKS:1.Riley K F, Hobson M P, Bence S J, Mathematical Methods for Physics and Engineering, CUP, 3rd Ed, 20102.Arfken & Weber, Mathematical Methods for Physicists, Elsevier Indian Reprint, 7th Ed., 2012.Reference Books: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.