Mathematical Physics and Numerical methods

Paper Code: 
PHY-401
Credits: 
3
Contact Hours: 
45.00
Max. Marks: 
100.00
Objective: 

This course will enable the students to 

The objectives of this paper are to acquaint the students with different types of coordinate systems, tensors, four vectors etc. The students will also learn to make Fourier analysis of complex functions and use various numerical methods for solving different types of Physical Science problems.

Course Outcomes (COs):

 

Course

Learning outcome (at course level)

Learning and teaching strategies

Assessment Strategies

Paper Code

Paper Title

PHY 401

Mathematical Physics and Numerical Methods

(Theory)

The students will be able to –

CO47:Find the divergence, gradient or curl of a vector or scalar field expressed in terms of orthogonal curvilinear coordinates, Learn the curvilinear coordinates which have applications in problems with spherical and cylindrical symmetries, to understand and apply concept of tensor in real life problems and Learn the Dirac delta function its properties, which have applications in various branches of Physics, especially quantum mechanics.

CO48: Apply a range of techniques to solve first & second order partial differential equations and Model physical phenomena using partial differential equations such as the heat and wave equations.

CO49: Learn the Fourier analysis of periodic functions and their applications in physical problems such as vibrating strings etc.

CO50: Know about the basic theory of errors, their analysis, estimation with examples of simple experiments in Physics, Understand problems, methods and techniques of calculus of variations, Find numerical solutions of system of linear equations and check the accuracy of the solutions and Learn about various interpolating and extrapolating methods.

CO51: Solve initial and boundary value problems in differential equations using numerical methods and apply various numerical methods in real life problems.

Approach in teaching:

Interactive Lectures, Discussion, Tutorials, Power point presentation,Problem Solving in tutorials

Learning activities for the students:

Self learning assignments, Effective questions, Seminar presentation, Solving numericals

Class test, Semester end examinations, Quiz, Solving problems, Assignments, Presentations
 

12.00
Unit I: 
Orthogonal curvilinear coordinate system :
Orthogonal curvilinear coordinate system, scale factors, Expressions for gradient, divergence and curl and their application to Cartesian, Circular Cylindrical and Spherical polar coordinate systems.
Tensors:
Coordinate transformations, Transformation of covariant, contra variant and mixed tensors. Addition, subtraction, outer product , contraction and inner product of tensors, Quotient law, Symmetric and antisymmetric tensors, Metric tensor.
Dirac delta function and its properties.

8.00
Unit II: 
Four vectors:
Four vector formulation, four velocity vector, energy-momentum four vector, relativistic equation of motion; invariance of rest mass, orthogonality of four force and four velocity, Lorentz force as an example of four force, transformation of four frequency vector, longitudinal and transverse Doppler’s effect, Compton effect.
 

10.00
Unit III: 
Boundary value problems:
Techniques of separation of variables and its application to the following boundary value problems (i) Laplace’s equation in three dimensional Cartesian coordinate system – line charge between two earthed parallel plates, (ii) Helmholtz equation in circular cylindrical coordinates-Cylindrical resonant cavity, (iii) Wave equation in spherical polar coordinates-the vibrations of a circular membrane, (iv) Diffusion equation in two dimensional Cartesian coordinate system-heat conduction in a thin rectangular plate.
 

8.00
Unit IV: 
Fourier Series and Integrals:

Introduction, Fourier series and coefficients, functions with point of discontinuity, arbitrary period, even and odd functions, half range expansion, Parseval’s theorem.

7.00
Unit V: 
Numerical Methods:
Introduction, Finite-Difference Operators, Differential Operator related to the Difference Operator, Truncation error, Numerical interpolation, Roots of equations, Initial-value problems –Ordinary Differential equations: Taylor’s method, Euler’s method and direct method. Trapezoidal and Simpson’s rule for numerical integration.
 

Essential Readings: 
  •  “Mathematical Methods” , Potter and Goldberg, Prentice Hall of India (1998).
  • “Mathematical methods in Physics”, D.Biswas, New Central Book Agency (P) Ltd.
  • “Mathematical Physics”, M.P.Saxena, P.R.Singh, S.S.Rawat, P.K.Sharma, CBH, Jaipur.
  •  “Applied Maths for Engineers and Physicists”,Pipes and Harvill, McGraw Hill.
  • “Advanced Engineering Mathematics”, Ervin Kreyzig 5thEdition, Wiley Eastern Ltd. 
  • “Numerical Methods”, S. Balachandra Rao, C.K. Shantha, University Press, 1992.
  • “Mathematical Physics”,Ellgnine Butkon, Addisson Wiesley.
  • “Mathematical Physics”,Gupta, Vikas Publishing House.

References: 

Academic Year: 
2021-2022
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