Derivations of transport, heat/reaction-diffusion, wave, Poissons equations; well-posedness; characteristics methods for first order PDE's; D'Alembert formula and conservation of energy for wave equations; propagation of waves; Fourier transforms; heat kernel, smoothing effect; maximum principles; Fourier series and Sturm-Liouville eigen-expansions; method of separation of variables, frequencies of wave equations, stable and unstable modes, long time behavior of heat equations; delta-function, fundamental solution of Laplace equation, Newton potential; Greens function and Poisson formula; Dirichlet Principle.