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Mathematical-Methods-2

MATHEMATICAL METHODS -II (PH41008)

Spring 2020-21

About the Course

This is an advanced course on Mathematical methods for Physicsts meant for the Master students in Physics at IIT Kharagpur. The course has two broad segments group theory and partial differential equations. The suggested syllabus of this course can be found here. The contact details of the instructors can be found here.

Suggested References

  • ``Group theory and its applications to physics problems" by M. Hamermesh.
  • ``Group theory and quantum mechanics" by M. Tinkham
  • ``Group theory in a nutshell for physicists" by A. Zee
  • ``Group theory'' By P. Ramond
  • ``Elements of Partial Differential Equations" by I Sheddon.
  • ``Methods of Mathematical Physics: Partial Differential Equations, Volume 2" by Courant and Hilbert.
  • ``Partial differential equations in physics" by Sommerfeld.
  • ``Mathematical Methods for Physics" by Mathews and Walker.
  • ``Mathematical Methods for Physics and Engineering" by Riley, Hobson and Bence.
  • ``Mathematical Methods for Physicists" by Arfken.

Lecturenotes:

  • Lecture-notes-1: Introductory definitions, Finite group. Cayley's theorem. Lagrange's theorem. Cosets. Conjugate classes. The symmetric group S_n, Young Tableau, Invariant subgroup, Factor group, Homomorphism, Direct product. Here is a small write up on equivalence classes.

  • Lecture-notes-2: Concluding remarks on finite groups. Introduction to continuous groups.

  • Lecture-notes-3: Introduction to Representation theory. Finite group representation theory.

  • Lecture-notes-4: Summary of orthogonality theorems. Application to perturbations on modes of a vibrating circular drup.

  • Lecture-notes-5: Surfaces and curves. Introduction to First order PDEs. Method of Characteristics for quasi-linear first order PDEs.

  • Lecture-notes-6: Second order PDEs.

  • Lecture-notes-7: Linear PDEs with constant coefficients; spherical harmonics.

  • Lecture-notes-8: Representations of SO(3) and its relation to spherical harmonics.

Assignments

  • Problem set-1: Basics of Finite groups. Cayley's theorem, Lagrange's theorem. (Solutions by G2, G8)

  • Problem set-2: Cosets, Conjugate classes, Invariant subgroups, Factor groups, Direct products. (Solutions by G1, G5)

  • Problem set-3: Group Presentation, Dihedral groups, SU(2) and SO(3), group representations. (Solutions by G3, G4)

  • Problem set-4: Group representations, Character tables. (Solutions by G6, G7)

  • Problem set-5: Partial differential equaitons. (Solutions by G9, G10 )

Supplementary material

To appear.

Evaluation for the course

The evaluation for this course will have the following components:

  • There will be 3 class exams of duration one hour each. Each exam will be of 20 marks each. These exams will be given a total of 60% weightage. See the dates of the class exams below.

  • You have to write an essay (see below for details). The essay must be typed or may be written in excellent handwriting. This essay will be given a 25% weightage.

  • Your (Group's) performance in assignments will be given a 15% weightage. If you are able to point out MAJOR mistakes in a solution set provided by a different group or in the class-notes provided by me, you can get extra credits in this segment.

Particulars about the essay:

In the final week of this course, you will have to submit `an essay' on any chosen topic on ``The application of Group theory or PDEs''. This application must be based on the new concepts which you have learnt during the course. The choice would be completely up to you. It can be something very mundane like a summary of what you have learned in the course. It can also be an extensive exploration of any special topic related to Group theory or PDE, which attracts your interest. You don't have to be limited by the syllabus for choosing this topic, but it must be related to material taught in this course. This is completely open-ended. It is an opportunity to showcase your innovative ideas, thinking, and interest.

Note that by `an essay' I mean a scientific report. It can have (should have rather) equations, plots, or anything which you wish to include. The minimum length of this essay would be 3 pages (3 sides). There would be no upper limit. Preferably, the document should be typed (use latex or word or anything that you are comfortable with). If you are very confident with your handwriting, it can also be a handwritten note.

Kindly start working on the essay early (as soon as about 50-60% of the course is over). Don't keep it for the last day.

You will be heavily penalized if it is found that you have "copied" directly (word-to-word) either from a classmate or from the internet.

Important dates:

Announcements

The deadline for the submission of the essay has been extended by a day.