Syllabus (PHYSICS)

Course Type: MAJ-6

Semester: 5

Course Code: BPHSMAJ06C

Course Title: Mathematical Methods

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

Credit: 6

Practical/Theory: Combined

Course Objective: Mathematical Methods

Learning Outcome: Mathematical Methods

CC6: Mathematical Methods – I (6 Credits)

Course Objective:

Theory (4 Credits)

Calculus: Recapitulation - Limits, continuity, average and instantaneous quantities, differentiation. Plotting functions. Intuitive ideas of continuous, differentiable, etc. functions and plotting of curves. Approximation: Taylor and binomial series (statements only). (3 Lectures)

First Order and Second Order Differential equations: First Order Differential Equations and Integrating Factor. Homogeneous Equations with constant coefficients. Wronskian and general solution. Statement of existence and Uniqueness Theorem for Initial Value Problems. Particular Integral. (12 Lectures)

Calculus of functions of more than one variable: Partial derivatives, exact and inexact differentials. Integrating factor with simple illustration. Constrained Maximization using Lagrange Multipliers. (6 Lectures)

Partial Differential Equations: Solutions to partial differential equations, using separation of variables: Laplace's Equation in problems of rectangular, cylindrical and spherical symmetry. Wave equation and its solution for vibrational modes of a stretched string, rectangular and circular membranes. Diffusion Equation. (12 Lectures)

Fourier Series: Periodic functions. Orthogonality of sine and cosine functions, Dirichlet Conditions (Statement only). Expansion of periodic functions in a series of sine and cosine functions and determination of Fourier coefficients. Complex representation of Fourier series. Expansion of functions with arbitrary period. Expansion of non-periodic functions over an interval. Even and odd functions and their Fourier expansions. Application. Summing of Infinite Series. Term-by-Term differentiation and integration of Fourier Series. Parseval Identity.

(11 Lectures)

Some Special Integrals: Beta and Gamma Functions and Relation between them. Expression of Integrals in terms of Gamma Functions. Error Function (Probability Integral). (4 Lectures)

Orthogonal Curvilinear Coordinates: Orthogonal Curvilinear Coordinates. Derivation of Gradient, Divergence, Curl and Laplacian in Cartesian, Spherical and Cylindrical Coordinate Systems. (6 Lectures)

Introduction to probability: Independent random variables: Probability distribution functions; binomial, Gaussian, and Poisson, with examples. Mean and variance. Dependent events: Conditional Probability. Bayes' Theorem and the idea of hypothesis testing. (4 Lectures)

Dirac Delta function and its properties: Definition of Dirac delta function. Representation as limit of a Gaussian function and rectangular function. Properties of Dirac delta function.

(2 Lectures)

Practical (2 Credits)

Introduction and Overview: Computer architecture and organization, memory and Input/output devices (2 Lectures)

Basics of scientific computing: Binary and decimal arithmetic, Floating point numbers, algorithms, Sequence, Selection and Repetition, single and double precision arithmetic, underflow &overflow emphasize the importance of making equations in terms of dimensionless variables, Iterative methods (2 Lectures)

Errors and error Analysis: Truncation and round off errors, Absolute and relative errors, Floating point computations. (2 Lectures)

Introduction to programming in python: Introduction to programming, constants, variables and data types, dynamical typing, operators and expressions, modules, I/O statements, iteration, compound statements, indentation in python, the if-else-if-else block, for and while loops, nested compound statements, lists, tuples, dictionaries. (18 Lectures)

Programs: Sum & average of a list of numbers, largest of a given list of numbers and its location in the list, sorting of numbers in ascending descending order, Binary search, odd and even numbers, factorial, fibonacci series (4 Lectures)

Reading References

Theory

1. Introduction to Mathematical Physics, C Harper, PHI
2. Mathematical Methods for Physicists, G.B. Arfken, H.J. Weber and F.E. Harris, Elsevier
3. Mathematical Methods in Physical Sciences, M.L. Boas, Wiley.
4. Mathematical Physics, V. Balakrishnan, Ane Books Pvt Ltd
5. Differential Equations, S.L. Ross, Willey
6. Fourier Analysis, M.R. Spiegel, Tata McGraw-Hill.
7. Higher Engineering Mathematics, B.S. Grewal, Khanna Publishers, 44^th Edn.
8. Mathematical Physics, H.K. Dass and R. Verma, S Chand & Co

Practical

1. Scientific computing in Python – Abhijit Kar Gupta, Techno World
2. Learning with Python-how to think like a computer scientist, J. Elkner, C. Meyer, and
   1. Downey, Dreamtech Press.

Introduction to computation and programming using Python, J. Guttag, Prentice Hall India.