18.300 Principles Continuum Appl Math

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Welcome to 18.300!

The class will meet on TTh from 2:30pm to 4:00pm in 4-145. The office hours will be on TTh at 4:00pm in 2-247.

   Instructor: Laurent Demanet          Q&A on piazza           Psetpartners        

 

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Left: Munich GeoCenter. Right: Dedalus, Keaton Burns

The class will give a broad introduction to mathematical physics. The subject is open to students in all majors who want to learn the core mathematical concepts that underlie (continuum) applied mathematics and theoretical physics. There will only be a small overlap with the content in years prior to 2024.

There will be no recitations, but there will be office hours. We are running a piazza site for Q&A, with the ability to post anonymously or just to the instructor. Use psetpartners if you want to join a study team.

Syllabus

  • The major field equations:
    • Laplace/Poisson
    • Heat equation
    • Wave equation
    • Electromagnetism (Maxwell)
    • Gas dynamics and fluids (Euler, Navier-Stokes)
  • More field equations if time permits:
    • Nonlinear conservation laws (Burgers)
    • Quantum mechanics (Schrödinger, Dirac)
    • Some field equations in machine learning
  • Derivations from phenomenology
  • Derivations from first principles: calculus of variations and classical field theory
    • Functional derivatives
    • Lagrangians, Hamiltonians, Least-action principle
    • Change of basis/coordinates, Tensors, Relativity
    • Symmetry groups
  • Solution methods
    • Separation of variables, Fourier transforms, Fourier series
    • Integral equations, Green's functions, theory of distributions
    • Method of characteristics if time permits

Prerequisites

Required: Calculus II and III (18.02 and 18.03)

Not required but encouraged: Physics I and II (8.01 and 8.02), and some linear algebra (18.06 or 18.C06 or 18.700)

Reference material

The current version of the pdf notes is under Files. Work in progress.

Not required, just FYI: the material will be inspired from various sources, including

  • "Classical electrodynamics" by Jackson
  • "Classical theory of fields" by Landau and Lifschitz
  • "Introduction to partial differential equations" by Folland
  • "Linear and nonlinear waves" by Whitham
  • "Methods of mathematical physics" by Courant and Hilbert
  • "Classical mechanics" by Goldstein, Safko, and Poole

Evaluation

The evaluation will consist of problem sets and a final project. Breakdown: 50% psets, 50% project. Tentative due dates for the problem sets: no earlier than Feb 13, Feb 27, Mar 13, Mar 31, Apr 10, Apr 24.

The problem sets will be in the assignments tab. Collaboration is allowed, but the copy your turn in must be your own work.

The project can consist in writing and presenting a review paper on

  • an equation that was not seen in class (here is a list Links to an external site. you can choose from); or
  • a chapter in one of the recommended books in the list above; or else
  • some topic related to your own interest/research.

Structure for the review paper: at most 5 pages including figures; preferably typeset in LaTeX: \LaTeXLATEX; and with every source carefully listed (i.e., where you got the information). There will be three project meetings: one meeting where you propose a topic; one meeting where you report on your partial progress (both to be scheduled with the instructor); and one final presentation on May 8 or May 13. In these meetings you will be asked to showcase your understanding of the material, in the area of your project broadly defined.