list of problems
My solutions to problems from Landau–Lifshitz.
LX LTM exam on Quantum Mechanics
XX Applying LL on Sena's collection of problems
Mathematics-1
The format is that you attend in person on a pre-arranged day and are given three problems to solve over the course of a single day (11:00–18:00), with no books, notes, or reference material allowed. The three problems are always of the same three types: (1) an indefinite integral, (2) an ordinary differential equation, and (3) a computation.
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Problem 1 — Indefinite integral of an elementary function
You’ll be given a single integral to compute by hand. It tests whether the standard integration techniques are fully automatic: substitution, integration by parts, partial fractions for rational functions, trigonometric integrals, and quadratic-trinomial/radical substitutions. I could not find any English translation of Collection of Problems in Higher Mathematics by N. M. Günther and R. O. Kuzmin, so instead I decided to use Demidovich’s Problems in Mathematical Analysis, Chapter IV (Indefinite Integrals).
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Problem 2 — Ordinary differential equation solved in quadratures
You’ll get an ODE to solve by reducing it to integrals. The institute explicitly recommends Filippov, sections 1–14.
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Problem 3 — Averaging over random geometric objects
You’ll be asked to compute some average (often a root-mean-square) involving random points or shapes in two or three dimensions.
“Theoretical physics is a unified and indivisible science with unified methods.”
“It is necessary to learn ALL its main branches, and the sequence of study is dictated by their mutual relationship. . . . You must perform all the calculations by yourself, and must not leave it to the authors of the books you have read.”
“The most important thing to master is the technique of working, and the understanding of the fine points will come by itself.”
