## Target Audience

Fourth-semester students in the B.Tech. (Engineering Physics) programme.

## Assessment

- Two Quizzes — 15% each
- Assignments— 20%
- Final Examination — 50%

## Course Notes

I plan to include notes (or links to notes by others) related to various topics that are discussed in the class lectures.

Normal Subgroups; Dual Vector Spaces;

Prof. Balakrishnan's Resonance Article on the Dirac Delta Function;

Prof. Tan's notes on Fourier Series and Transforms;

Prof. Tan's notes on Hilbert Spaces;

Legendre Polynomials from Gram-Schmidt Orthogonalisation;

Observe the Gibbs phenomenon in the Fourier Series for the square wave;

Notes on Lie groups and Lie algebras.

Summer reading on some interesting partial differential equations.

## Official Course Content

Scalars, vectors and tensors in index notation. Kronecker and Levi-Civita tensors. Del and Laplacian operators. Vector calculus in index notation.

Dirac delta function, representation and properties, Linear vector spaces. Dual space. Bra and ket notation. Basis sets. Orthogonality and completeness. Hilbert space. Linear operators. Self-adjoint and unitary operators.

Families of orthogonal polynomials as basis sets in function space, Legendre, Hermite, Laguerre, Chebyshev and Gegenbauer polynomials, Generating functions. Expansion of functions. Inversion formulas.

Rotation group in 2 and 3 dimensions. Pauli matrices. Generators of rotations.

Fourier series and Fourier transforms, Fourier expansion and inversion formulas, Convolution theorem.

Elements of analytic function theory, Cauchy-Riemann conditions, Cauchy's integral theorem and integral formula, singularities - poles and essential singularities, residue theorem and contour integration.

Occurrence of Laplace, Poisson, Helmholtz, wave and diffusion equations in physical applications, Elementary properties of these equations and their solutions.

## References

- G. Arfken,
*Mathematical Methods for Physicists*(5th Edition) (Academic Press, 2000). - L.A. Pipes and L.R. Harwell,
*Applied Mathematics for Engineers and Physicists*(McGraw-Hill). - B. Friedman,
*Principles and Techniques of Applied Mathematics*(Dover, 1990). - D.W. Lewis,
*Matrix Theory*(Allied Publishers, 1991). - K.F. Riley, M.P. Hobson and S.J. Bence,
*Mathematical Methods for Physics and Engineering*(Cambridge Univ. Press, 1998) - M. P. Boas,
*Mathematical Methods in the Physical Sciences*(2nd Edition) (Wiley, 1983) - M. Artin,
*Algebra*(Eastern Economy Edition) (Prentice Hall, 1991)