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**