The single first-order equation; the Cauchy problem; systems of first-order equations; characteristics; the Cauchy-Kowalevski existence and uniqueness theorem; elliptic equations; the Laplace equation; the Lagrange-Green identity. the Maximum principle; harmonic and subharmonic functions; Green’s function and the Poisson formula; hyperbolic equations in higher dimensions; the wave equation; the method of spherical means; Hadamard’s method of descent; hyperbolic equations with constant coefficients; solution by the n-dimensional Fourier transform; parabolic equations; and the heat equation. Prerequisite: Graduate Standing.