The single First-Order equation. The Cauchy problem. Systems of First-Order equations. Characteristics. The Cauchy-Kowalevski Existence and Uniqueness Theorem. Elliptic equations. TheLaplaceequation.TheLagrange-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. The Heat equation. Prerequisite: Graduate Standing.