Mathematics

Gilbert Strang's gold-standard MIT undergraduate course building geometric and algebraic intuition for systems of equations, vector spaces, orthogonality, determinants, and eigenvalues.

Flows on the line and circle, saddle-node and pitchfork bifurcations, two-dimensional phase planes, limit cycles, Hopf bifurcations, Lorenz equations, strange attractors, and fractals.

Harvard's celebrated probability sequence building story proofs, conditioning, Bayes' theorem, indicator random variables, continuous distributions, moment generating functions, and Markov chains.

Dedekind cuts construction of the reals, metric space topology, compactness, Heine-Borel theorem, connected sets, Cauchy sequences, uniform continuity, and function spaces.

Vector geometry, matrices, partial derivatives, Lagrange multipliers, multiple integrals, line and surface integrals, Green's theorem, Stokes' theorem, and the divergence theorem.

Direction fields, first-order and second-order linear ODEs, complex exponentials, resonance, Fourier series, Laplace transforms, Dirac delta impulses, and linear systems.