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TYBSc Semester V

PHY-351 | Mathematical Methods in Physics – II

T.Y.B.Sc. | Sem V | Physics

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T.Y.B.Sc. | Sem V | PHY-351
Mathematical Methods in Physics – II

It’s a great pleasure for us to present this book Mathematical Methods in Physics-II (PHY 351, Semester V) for T.Y.B.Sc. Physics students of the Savitribai Phule Pune University, Pune. This book is written as per the revised syllabus of Savitribai Phule Pune University, Pune to be implemented from June 2021.
We have tried to explain the concepts & information in lucid language so that students could understand it easily. Many senior faculty members have inspired us to perform this duty. We feel it’s not a new creations but presentation in simple form for the better understanding of students.

1 Curvilinear Co-ordinates 
1.0 Introduction
1.1 Review of Cartesian, spherical and cylindrical co-ordinate
1.1.1 Cartesian co-ordinate
1.1.2 Spherical polar co-ordinate
1.1.3 Cylindrical co-ordinate
1.2 Transformation equations
1.2.1 Cartesian to spherical co-ordinate
1.2.2 Cartesian to cylindrical co-ordinate
1.3 General Curvilinear co-ordinate system: Co-ordinate surface, co-ordinate lines
1.3.1 Length elements in curvilinear co-ordinate system.
1.3.2 Surface and Volume elements in curvilinear co-ordinate system.
1.3.3 Volume elements in Spherical co-ordinate system
1.3.4 Volume elements in Cylindrical co-ordinate system
1.4 Orthogonal curvilinear co-ordinate system
1.4.1 Cartesian, Spherical polar and Cylindrical co-ordinate system
1.4.2 Gradient, Divergence, Curl and Laplacian in Orthogonal curvilinear co-ordinate system
1.5 Special case: Gradient, divergence and curl in Cartesian, spherical polar and cylindrical co-ordinate system.
1.5.1 Gradient, Divergence and Curl in spherical polar co-ordinates
1.5.2 Gradient, Divergence and Curl in Cartesian co-ordinates
1.5.3 Gradient, Divergence and Curl in cylindrical co-ordinates
Solved Examples

2 The Special Theory of Relativity 
2.0 Introduction and applications,
2.1 Newtonian relativity,
2.2 Galilean transformation equation,
2.3 Michelson-Morley experiment,
2.4 Postulates of special theory of relativity,
2.5 Lorentz transformations,
2.6 Kinematic effects of Lorentz transformation,
2.6.1 Length contraction
2.6.2 Proper time
Solved Problems

3 Partial Differential Equations 
3.0 General definitions
3.1 Partial Differential Equation (PDE)
3.1.1 Basic terminologies
3.1.2 General Differential Equation in Physics
3.2 Methods for solving second order PDEs
3.3 Separation of variables
3.3.1 Two Dimensional Laplace’s Equation in Cartesian Co-ordinate
3.3.2 Laplace’s Equation in Spherical Polar Co-ordinates
3.3.3 Separation of Variable in Laplace’s Equation in Cylindrical Co-ordinates
3.3.4 One dimensional Wave equation
3.3.5 Wave equation in Spherical Polar Co-ordinates
3.4 Singular points
3.5 Fuchs’s theorem
3.6 Frobenius Method
Solved Problems

4 Special Functions 
4.0 Introduction
4.1 Legendre equation, Legendre polynomial (Pn(x)) and generating function
4.2 Properties of Legendre Polynomial
4.2.1 Recurrence relations
4.2.2 Rodrigue’s Formula
4.2.3 Christoffel’s Summation Formula
4.2.4 Orthogonality of Legendre Polynomial
4.3 Hermite’s equation, Hermite polynomial (Hn (x)) and generating function
4.4 Properties of Hermite Polynomial
4.4.1 Recurrence relations
4.4.2 Orthogonality of Hermite Polynomial
4.5 Bessel equation, Bessel polynomial (Jn (x)) and generating function
4.6 Properties of Bessel Polynomial
4.6.1 Recurrence relations
4.7 Applications of special functions
Solved Problems


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