** Linear algebra:
Review, concepts & applications
Vector space (linear space), eigenvalue problem, diagonalization
Systems of linear differential equations
Nonlinear differential equations & qualitative methods

** Fourier analysis:
Fourier series and integrals
Fourier transformation (vs. Laplace transformation)
Sturm-Liouville theory (eigenfunction expansions)
Orthogonal expansions (special functions: Bessel, Legendre)
Other transformations (Hankel,

** Vector analysis:
Curves, surfaces, tangents, and normals
Vector differential / integral calculus
Line and surface integrals; gradient, divergence, and curl
Green’s thm, Gauss (divergence) thm, Stokes thm, and potential theory

**PDEs:
initial & boundary value problems
bounded vs. unbounded domain
separation of variables and integral transformation method
heat & wave eqn

** Complex variables:
complex functions
Cauchy integral thm, residue thm, and applications
Conformal mapping & special applications