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Mathematical physics and numerical methods | Physics

Mathematical physics and numerical methods

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Paper Code: 
PHY-411
Credits: 
03
Contact Hours: 
45.00
Max. Marks: 
100.00
Objective: 

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 the Fourier analysis and various numerical methods.

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: 

1.    “Mathematical Methods” , Potter and Goldberg, Prentice Hall of India (1998).

2.     “Mathematical methods in Physics”, D.Biswas, New Central Book Agency (P) Ltd.

3.    “Mathematical Physics”, M.P.Saxena, P.R.Singh, S.S.Rawat, P.K.Sharma, CBH, Jaipur.

 

References: 

1.     “Applied Maths for Engineers and Physicists”,Pipes and Harvill, McGraw Hill.

      2.     “Advanced Engineering Mathematics”, Ervin Kreyzig 5th Edition, Wiley Eastern Ltd.

      3.     “Numerical Methods”, S. Balachandra Rao, C.K. Shantha, University Press, 1992.

4.     “Mathematical Physics”,Ellgnine Butkon, Addisson Wiesley.

5.    “Mathematical Physics”,Gupta, Vikas Publishing House.

 

 

 

 

Academic Year: