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Mathematical Methods In Physics 2

Prof. Auditya Sharma · IISER Bhopal

Start with lecture 1 At NPTEL

About

This would be the second of a two-part series on Mathematical Methods in Physics. The aim here is to provide a solid mathematical foundation for the budding Physicist eager to climb the ladder of self-learning. This course is designed to cater to an undergraduate student who excitedly embarks on a study of Physics, but is obstructed by the mathematics which appears so forbidding. The approach we will follow is one of showing many examples, and weaving the theory around examples. When possible, we will show how Mathematica can be used to offer extra insight.

Intended audience: Any Interested Learners

Prerequisites: We try to make this self-contained. However,some familiarity with the basic mathematical tools used in say, the first courses on Mechanics, Electromagnetism, would help.INDUSTRIES SUPPORT : Quantitative Finance and Scientific consulting companies

Lectures

Week 1

  1. Introduction to complex numbers YouTube
  2. The triangle inequality YouTube
  3. The de Moivre formula YouTube
  4. Roots of unity YouTube
  5. Functions of a complex variable and the notion of continuity YouTube
  6. Derivative of a complex function YouTube
  7. Differentiation rules for a complex function YouTube
  8. Cauchy-Riemann Equations YouTube
  9. Sufficient conditions for differentiability YouTube
  10. Cauchy-Riemann conditions in polar coordinates YouTube
  11. More persepective on differentiability YouTube
  12. The value of the derivative YouTube

Week 2

  1. Analytic functions YouTube
  2. Harmonic functions YouTube
  3. The exponential function YouTube
  4. Complex logarithm YouTube
  5. Complex exponents YouTube
  6. Trigonometric functions of complex variables YouTube
  7. Hyperbolic functions of complex variables YouTube
  8. Inverse Trigonometric and Hyperbolic functions YouTube
  9. Branch of a multivalued function YouTube

Week 3

  1. Contour Integrals YouTube
  2. Green's Theorem YouTube
  3. Path dependence of the contour intergal YouTube
  4. Antiderivatives YouTube
  5. The Cauchy theorem YouTube
  6. Crossing contours and multiply connected domains YouTube
  7. Cauchy Integral formula YouTube
  8. Derivatives of an analytic function YouTube
  9. Liouville's theorem and the Fundamental theorem of algebra YouTube

Week 4

  1. Taylor Series YouTube
  2. Laurent Series YouTube
  3. Convergence YouTube
  4. Differentiation and integration of power series YouTube
  5. Isolated Singularities YouTube
  6. Residues YouTube
  7. Residue Theorem YouTube
  8. Evaluation of integrals-I YouTube
  9. Evaluation of integrals-II YouTube
  10. Analytic Continuation YouTube

Week 5

  1. Introduction of orthogonal polynomials YouTube
  2. How to construct orthogonal polynomials YouTube
  3. The weight function YouTube
  4. Recursion relations YouTube
  5. Differential equation satisfied by the orthogonal polynomials YouTube
  6. Hermite polynomials YouTube
  7. Properties of Hemite polynomials YouTube
  8. Legendre polynomials YouTube
  9. Legendre polynomials: recurrence relation YouTube

Week 6

  1. Differential equation corresponding to Legendre polynomials YouTube
  2. The generating function corresponding to Legendre polynomials YouTube
  3. Laguerre Polynomials YouTube
  4. Laguerre Polynomials: recurrence relation YouTube
  5. Laguerre polynomials: differential equation YouTube
  6. Laguerre polynomials: generating function YouTube
  7. Bessel functions: series defination YouTube
  8. Bessel functions: recurrence relations YouTube
  9. Bessel functions: differential equation YouTube
  10. Bessel functions of integral order: generating function YouTube
  11. Bessel function: orthogonality YouTube

Week 7

  1. Classification of Second Order PDEs YouTube
  2. Canonical Forms for Hyperbolic PDEs YouTube
  3. Canonical Forms for Parabolic PDEs YouTube
  4. Canonical Forms for Elliptic PDEs YouTube
  5. Tha Laplace Equation YouTube
  6. The Laplace Equation: Separation of Variables YouTube
  7. The Laplace Equation: Dirichlet and Neumann boundary conditions YouTube
  8. The Laplace Equation in Cartesian coordinates YouTube
  9. The Laplace Equation for a 3-D rectangular box YouTube

Week 8

  1. The Laplace Equation in spherical coordinates YouTube
  2. The Laplace Equation in Spherical Coordinates: Solution YouTube
  3. The Laplace Equation in Spherical Coordinates: illustrative examples YouTube
  4. The Poisson's Equation: Green's function solution YouTube
  5. The heat equation: a heuristic discussion YouTube
  6. From the random walk to the diffusion equation YouTube
  7. Solution of the Diffusion equation YouTube
  8. The Diffusion equation with Dirichlet and Neumann boundary conditions YouTube
  9. The Heat equation: illustrative examples YouTube
  10. The Wave equation: Method of characteristics YouTube
  11. The Wave equation: Separation of variables YouTube

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
IISER Bhopal
Department
Mathematics
Teacher
Prof. Auditya Sharma
Level
Undergraduate
Language
English
Subjects
Mathematics
Source
NPTEL
Licence
CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
Attribution
NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge. Coordinated by IIT Madras.
Who can watch
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