Suggestions for Qualifier Course

Calculus of Variations

• Functionals, Frechet and Gateaux derivatives. Derive Euler-Lagrange equations. Examples+ ODE motivation (1 wk)
• ODEs. Existence, uniqueness + Green's functions (2 wks)
• Variational problems: constrained and unconstrained problems; various are subject to various boundary conditions. Chain rule for functionals and coordinate invariance. Examples. (2 wks)
• Lagrangians, Hamiltonians, and use in mechanics and continuum problems (vibrating strings, drum heads). (1 wk)
• Rayleigh-Ritz principle, Courant-Fischer minimax theorem; variation methods for finding eigenvalues and eigenfunctions. (1 wk)

Operators on Hilbert space

• Unbounded operators, self-adjoint operators (1 wk)
• Spectral theorem, spectral transform; derive Fourier transform, plus other transforms (2 wks)

Fourier transforms

• Definition, inverse transform, convolution theorem, Parseval's Theorem.
• Sampling theorem and uncertainty principle

Schwartz space and tempered distributions

• Schwartz space, tempered distributions,
• Fourier transforms of tempered distributions

PDEs

• Wave equation
• heat equation
• potential equation
• Helmholtz equation