Mathematical Physics - I

HRI Graduate School

09 August - 10 December 2010

Things to remember:

To contact me anytime, remember that my office number is 323; the phone extension number is 4357. In case you want to contact me and do not find me in office, send me an email to tirth at hri.res.in.
Your tutor for the course would be Ipsita Mandal <ipsita at hri.res.in>.
Any information/updates/news about the course will be put up at the webpage http://www.hri.res.in/~tirth/Teaching/MM1/index.html. Make sure you check the webpage periodically.
The course will consist of approximately 35-40 lectures (of 1.5 hours each) at the rate of 3 lectures per week. There will be a mid-term break during 11-18 October 2010.
A "self-test" (or Assignment 0) will be given so that you get some idea of the pre-requisites. If you find any of the questions difficult, let me know. Remember that you do not have to submit this set and it will not form part of the final score! It is only to give you an idea of the required background.
There will be several take-home assignment sets and/or unannounced class tests (quizzes), one mid-semester examination (tentatively during the week 04-08 October) and one final examination (tentatively during the week 06-10 December) which will be used for evaluating the course.
The evaluation procedure is the following: Your final average score will be computed giving 40% weightage to final exam, 30% to the mid-semester exam and 30% to the assignments plus quizzes. You need to score a minimum of 60% in the average grade to pass the course.

Course Structure:

Preliminaries:

Sets, Functions, Equivalence relation.
Binary operations.
Groups, Rings, Fields, Integral Domains, Vector Spaces.

Vector Spaces:

Definitions and Examples, Subspaces, Linear Combinations and Spans, Linear Independence, Basis and Dimension
Linear Mappings, Operations with Linear Mappings, Linear Operators, Direct Sums, Matrix Representation of a Linear Operator.
Basis Transformation, Invariant Subspaces, Eigenvalues and Eigenvectors.
Linear Functionals, Dual Spaces.
Real Inner Product Spaces, Orthonormal Bases.
Complex Inner Product Spaces, Linear Functionals in Inner Product Spaces, Linear Operators in Inner Product Spaces, Hilbert Space.

Tensor Analysis:

Concepts and Summation Convention, Tensors on Vector Spaces.
Coordinate Transformations, First Order Tensors, Operations with Tensors.
Orthogonal Coordinates.
Derivatives of a Tensor.

Complex Analysis:

Properties of Complex Numbers, Functions, Limits and Continuity, Derivatives, Integrals.
Cauchy's Theorem, Cauchy's Integral Formulae.
Series, Analytic Continuation.
Residues, Evaluation of Integrals.
Mapping, Conformal Mapping.

Ordinary differential equations:

First Order Differential Equations.
Singular Points, Series Solutions, Second Solution of a Second Order Differential Equation.
Variation of Parameters, Greens Function.
Sturm-Liouville Theory.

Reading List:

Books to be followed closely:

Arfken, George B.; Weber, Hans J. (2005), Mathematical methods for physicists (6th ed.), Elsevier Academic Press
Szekeres, Peter (2004), A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry, Cambridge University Press

Classics:

Jeffreys, Harold; Swirles Jeffreys, Bertha (1956), Methods of mathematical physics (3rd rev. ed.), Cambridge University Press
Mathews, Jon; Walker, Robert L. (1970), Mathematical methods of physics (2nd ed.), W. A. Benjamin
Courant, Richard; Hilbert, David (1989), Methods of mathematical physics, Interscience Publishers