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Mathematical Physics & Numerical Methods
This course will enable the students to –
The objectives of this paper are to acquaint the students with different types of coordinate systems, and to train them to solve problems related to 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):
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Course |
Learning outcome (at course level) |
Learning and teaching strategies |
Assessment Strategies |
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Paper Code |
Paper Title |
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PHY 411 |
Mathematical Physics And Numerical Methods (Theory) |
The students will be able to – CO95: 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. CO96: 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. CO97: Learn the Fourier analysis of periodic functions and their applications in physical problems such as vibrating strings etc. CO98: 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. CO99: 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, Demonstration
Learning activities for the students: Self-learning assignments, Effective questions, Seminar presentation. |
Class test, Semester end examinations, Quiz, Solving problems , Assignments, Presentations |
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.
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.
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.
Introduction, Fourier series and coefficients, functions with point of discontinuity, arbitrary period, even and odd functions, half range expansion, Parseval’s theorem.
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.
- “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 5th Edition, 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.
