Sitabhra Sinha
The great book of Nature lies ever open before our eyes and the true
philosophy is written in it ... But we cannot read it unless we have first
learned the language and the characters in which it is written ... It is
written in mathematical language and the characters are triangles, circles
and other geometrical figures.
- Galileo Galilei, 1623 Il Saggiatore (tr. George Polya) p.232
Class Schedule: Tuesday and Thursday (11:30-1:00)
7/8/12: Introduction
9/8/12: Basic Stuff
14/8/12: Linear Vector Space
16/8/12: Linear Independence, Basis and Norm
23/8/12: Metric space, Algebra of Linear operators
28/8/12: Algebra of Linear Operators; Angular Momentum; System of masses connected by elastic springs
30/8/12: Inverse operators; Hermitian and Unitary operators
4/9/12: Trace and Determinant of a matrix; Affine transform
6/9/12: Eigenvalues of Hermitian operator; Orthogonalization theorem
11/9/12: Gram-Schmidt orthogonalization; Normal modes; Qualitative theory of 2 coupled first-order differential equations
13/9/12: Using the Jacobian to evaluate stability of solutions of coupled ODEs; Lotka-Volterra equations
18/9/12: Jordan canonical form; Spectral mapping theorem
20/9/12: Infinite-dimensional vector space; Greens function; Hilbert space
25/9/12: Introduction to Complex Analysis: Analytic functions; Mandelbrot and Julia set; Continuity and Derivative of Complex Functions; Cauchys criterion
27/9/12: Mid-term examination
2/10/12: Holiday
4/10/12:
9/10/12: Infinite sequences and series; Convergence tests
11/10/12: Series of functions; Uniform convergence; Weierstrass M-test and Abel test; MacLaurin series; Complex sequence and series
16/10/12: Cauchy's theorem
18/10/12: Cauchy's Theorem in multiply connected region; Morera theorem; Cauchys Integral representation
23/10/12: Local behavior of an analytic function; Analytic contnuation; Taylor series and Laurent series
25/10/12: Classification of singularities; Weierstrass Theorem; Calculus of Residues
30/10/12: Evaluation of integrals using the Residue Theorem
9/11/12: Multi-valued function, branch cuts and Riemann surface
12/11/12: Numerical Solution of differential equations; Integral Transforms: Fourier Transform
13/11/12: Integral Transforms: Laplace Transform and its application to solve differential equations
15/11/12: Theory of Differential Operators and Greens Function
16/11/12: Theory of Greens Function
17/11/12: Greens Function (continued); WKB approximation
29/11/12: End-term examination
Assignments
Assignment 1 (due August 16, 2012)
Assignment 2 (due September 25, 2012)
Assignment 3 (due November 16, 2012)
Textbooks:
Philippe Dennery and Andre Krzywicki: Mathematics for Physicists (Dover, 1996)
George F. Simmons: Differential Equations, with Applications and
Historical Notes (Tata McGraw-Hill, 1991) (There is now a new 2007 edition)
Daniel T. Finkbeiner: Introduction to Matrices and Linear
Transformations (W H Freeman, 1966)
Harry Hochstadt: The Functions of Mathematical Physics (Dover, 1986)
George B. Arfken, Mathematical Methods for Physicists
(Academic Press, 2005)
Jon Mathews and Robert L. Walker: Mathematical Methods of Physics
(W A Benjamin, 1970)
Essential web resources for mathematical methods in physics:
James Nearing: Mathematical Tools for Physics