Course Details
Subject {L-T-P / C} : PH4001 : Mathematical Methods in Physics { 3-1-0 / 4}
Subject Nature : Theory
Coordinator : Dr. Abhay Pratap Yadav
Syllabus
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, residue theorem,
definite integrals using contour integration, Dirichlet integral and Cauchy principal value, singular
integral using ‘i?’ prescription, integrals of multivalued functions. 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, algebraric proerties 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. Metric tensors covariant
and contravariant basis, covariant derivatives, Christoffel symbols, tensor derivative operator,
Jacobian, geodesics. 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.
Course Objectives
- To learn about complex analysis.
- To learn about Tensor Analysis
- To learn about Group theory and application to physical problems.
Course Outcomes
The student will develop understanding about important concepts in Mathematical Physics.
Essential Reading
- M. R. Spiegel, S. Lipschutz, J. J. Schiller and D. Spellman, Complex Variables, McGraw-Hill Education
- P. Dennery and A. Krzywicki, Mathematics for Physicists, Dover Publications Inc.
Supplementary Reading
- Sadri Hassani, Mathematical Methods for Students of Physics and Related Fields, Springer
- M. T. Vaughn, Introduction to Mathematical Physics, Wiley-VCH