Syllabus (PHYSICS)

Course Type: MAJ-11

Semester: 7

Course Code: BPHSMAJ11C

Course Title: Mathematical Methods III

(L-P-Tu): 4-2-0

Credit: 6

Practical/Theory: Combined

Course Objective: This course aims to equip students with advanced mathematical tools essential for theoretical and applied physics. It introduces the concept of Green’s functions for solving inhomogeneous differential equations, and provides a comprehensive understanding of integral transforms—Fourier and Laplace—with applications to wave and diffusion equations. Students will gain a strong foundation in linear algebra, including vector spaces, operators, and matrix representations, along with tensor analysis to handle complex coordinate transformations and covariant formulations. The course further develops group theory concepts and their applications in physics, emphasizing symmetry principles and Lie groups. Practical components focus on numerical methods, matrix operations, curve fitting, and iterative techniques to solve physical and engineering problems computationally.

Learning Outcome: This course aims to equip students with advanced mathematical tools essential for theoretical and applied physics. It introduces the concept of Green’s functions for solving inhomogeneous differential equations, and provides a comprehensive understanding of integral transforms—Fourier and Laplace—with applications to wave and diffusion equations. Students will gain a strong foundation in linear algebra, including vector spaces, operators, and matrix representations, along with tensor analysis to handle complex coordinate transformations and covariant formulations. The course further develops group theory concepts and their applications in physics, emphasizing symmetry principles and Lie groups. Practical components focus on numerical methods, matrix operations, curve fitting, and iterative techniques to solve physical and engineering problems computationally.

CC 11: Mathematical Methods III (6 Credits)

Course Objective:

This course aims to equip students with advanced mathematical tools essential for theoretical and applied physics. It introduces the concept of Green’s functions for solving inhomogeneous differential equations, and provides a comprehensive understanding of integral transforms—Fourier and Laplace—with applications to wave and diffusion equations. Students will gain a strong foundation in linear algebra, including vector spaces, operators, and matrix representations, along with tensor analysis to handle complex coordinate transformations and covariant formulations. The course further develops group theory concepts and their applications in physics, emphasizing symmetry principles and Lie groups. Practical components focus on numerical methods, matrix operations, curve fitting, and iterative techniques to solve physical and engineering problems computationally.

Theory (4 Credits)

Inhomogeneous Differential Equations: Green’s function and its applications. (2 Lectures)

Integrals Transforms

Fourier Transforms: Fourier Integral theorem. Fourier Transform. Examples. Fourier transform of trigonometric, Gaussian, finite wave train and other functions. Representation of Dirac delta function as a Fourier Integral. Fourier transform of derivatives, Inverse Fourier transform, Convolution theorem. Properties of Fourier transforms (translation, change of scale, complex conjugation, etc.). Three dimensional Fourier transforms with examples. Application of Fourier Transforms to differential equations: One dimensional Wave and Diffusion/Heat Flow Equations.

Laplace transforms: Laplace transform and its inverse transform, Transform of derivative and integral of a function; Solution of differential equations using Laplace transforms.

(15 Lectures)

Linear Algebra

Linear vector spaces. Basis and dimension of a vector space. Inner product. Metric spaces. Cauchy-Scwartz inequality. Linear independence and orthogonality of vectors, Gram-Schmidt orthogonalisation procedure. Linear operators. Inverse of an operator. Dual spaces and adjoint operators. Special linear operators. Projection operator. Matrix representation of linear operators. (10 Lectures)

Tensor Analysis

Coordinate transformations, scalars, Covariant and Contravariant tensors. Addition, Subtraction, Outer product, Inner product and Contraction. Symmetric and antisymmetric tensors. Quotient law. Metric tensor. Conjugate tensor. Length and angle between vectors. Associated tensors. Raising and lowering of indices. The Christoffel symbols and their transformation laws. Covariant derivative of tensors. (13 Lectures)

Group Theory

Concept of a group, Definition and examples, Order of a group. Multiplication table and rearrangement theorem, Isomorphism and homomorphism, Distinct groups of a given order, subgroups. Permutation groups. Distinct groups of a given order. Cyclic and non-cyclic groups.

Group representations. Definition of representation. Faithful and unfaithful representations. Equivalent representations. Invariant subspaces, Reducibility of a representation. Irreducible representation. Lie groups and Axial rotation group SO(2). Rotation group SO(3). Special unitary groups SU(2), SU(3) and their applications in physics. (20 Lectures)

Practical (2 Credits)

Numerical integration – Trapezoidal formula, Simpson’s 1/3rd & 3/8th formulae.

Matrix Operation- Matrix summation, subtraction and multiplication, Matrix inversion and solution of simultaneous equation transpose, Jacobi Method of Matrix Diagonalization, eigenvalue and eigenvector determination.

Least square technique: Problems of linear least squares fit, applications.

Linear curve fitting and calculation of linear correlation coefficient, Lagrange interpolation based on given input data

Solution of transcendental or Polynomial equations – Bisection and Newton-Raphson method

Reading References

Theory

1. K.F. Riley, M.P. Hobson, and S.J. Bence: Mathematical Methods for Physics and Engineering (Cambridge)
2. V Balakrishnan, Mathematical Physics (Ane Books Pvt Ltd)
3. T. Dass and S.K. Sharma: Mathematical Methods in Classical and Quantum Physics (University Press)
4. G.B. Arfken, H.J. Weber and F.E. Harris: Mathematical Methods for Physicists (Elsevier)
5. S.L. Ross: Differential Equations (Wiley)
6. A.W. Joshi: Matrices and Tensors in Physics (New Age)
7. A.W. Joshi: Elements of Group Theory for Physicists (New Age)
8. Satya Prakash: Mathematical Physics (Sultan Chand & Sons)

Practical

1. Introduction to Numerical Analysis, S.S. Sastry, 5^th Edn., PHI Learning Pvt. Ltd.
2. Introduction to computation and programming using Python, J. Guttag, Prentice Hall India.
3. Effective Computation in Physics- Field guide to research with Python, A. Scopatz and K.D. Huff, O’ Rielly
4. A first course in Numerical Methods, U.M. Ascher & C. Greif, PHI Learning.
5. Elementary Numerical Analysis, K.E. Atkinson, 3rd Edn., Wiley India Edition.
6. Numerical Methods for Scientists & Engineers, R.W. Hamming, Courier Dover Pub.
7. Scientific computing in python, Abhijit Kar Gupta, Techno World.