Syllabus

Module 1: (10 hours)
Essential complex analysis: Cauchy-Riemann conditions and complex analytic functions, derivative of analytic function, Cauchy’s theorem, singularities and zeros, multivalued functions, branch point and cut, Cauchy’s integral formula, Taylor and Laurent series.

Module 2: (10 hours)
Residue theorem, definite and indefinite integrals using contour integration, Dirichlet integral and Cauchy principal value, singular integral using ‘i?’ prescription, integrals of multivalued functions.

Module 3: (10 hours)
Tensor Analysis: vectors, indices, Einstein summation convention, transformation properties of vectors, covariant and contravariant vectors, vectors to tensors, some examples of tensors in physical problems, covariant and contravariant tensors, rank of tensors, algebraic properties of tensors: addition, subtraction and contraction of tensors, inner product, direct/outer product, Cartesian tensors, symmetric and antisymmetric tensors, generalized Kronecker delta and Levi-Civita symbols, fully antisymmetric tensor, inverse transformations, quotient rule, pseudo tensors, dual tensors.

Module 4: (5 hours)
Metric tensors covariant and contravariant basis, covariant derivatives, Christoffel symbols, tensor derivative operator, Jacobian, geodesics, Riemann and Ricci tensor.

Module 5: (13 hours)
Group theory and application to physical problems: definition of a group, Discrete groups: subgroups and cosets, homomorphism and isomorphism, representations of groups, equivalent representations, unitary representations, reducible and irreducible representation, Schur’s Lemma and the orthogonality theorem, classes and character. Generators of continuous group, SO(2) and SO(3) rotation groups, orbital angular momentum and rotation, SU(2) and SO(3) homomorphism, homogeneous Lorentz group, Lorentz covariance of Maxwell’s equation.

To learn the differential properties of complex analytic functions and use them to analyze real problems.

To learn the integral properties of complex analytic functions and use them to analyze real problems.

To learn about tensors and their properties.

To learn the applications of tensors in general relativity.

5. Group theory and its application to quantum mechanics.

At the end of the course, students will be able to:
CO1: Understand differential properties of complex functions, power series functions, and to use these properties to analyze real problems.

CO2: Understand integration properties of complex analytic functions, various theorems involving complex integration, and to use these properties to analyze real problems.

CO3: Acquire basic ideas of tensors and their properties.

CO4: Learn tensor algebra and its application in general relativity.

CO5: Get knowledge of group theory and its application in physics.

G. Arfken, H. Weber, and F. Harris, Mathematical Methods for Physicists, Academic Press , 7th Edition (2013).

D. Fliech, A students guide to vectors and tensors, Cambridge University Press (2018). , 2nd Edition (2009).

M. Spiegel, S. Lipschutz, J. Schiller, and D. Spellman, Complex analysis: Schaum’s outline series, Tata MGrewHill , 2nd Edition (2010).

H. Georgi, Lie algebra in particle physics, Westview Press