Introduction ; Partial differentiation ; Solutions of PDE's and their specification ; PDE's and related arbitrary functions ; Particular solutions of PDE's ; Similarity solutions ; Correctly set problems ; Some preliminary aspects of linear first order PDE's ; Linear first order PDE's with two independent variables ; First order nonlinear PDE's ; Some technical problems and related PDE's ; Linear first order PDE's with two independent variables, general theory ; First order PDE's with multiple independent variables ; Original details of the Fourier approach to boundary value problems ; Eigenfunctions and eigenvalues ; Eigenfunctions and eigenvalues, continued ; Non-orthogonal eigenfunctions ; Further example of Fourier style analysis ; Inhomogeneous problems ; Local heat sources ; An inhomogeneous configuration ; Other eigenfunction/eigenvalue problems ; Uniqueness of solutions ; Alternative representations of solutions ; Other differential equations and inferences therefrom ; Second order ODE's ; Boundary value problems and Sturm-Liouville theory ; Green's functions and boundary value problems ; Green's functions and generalizations ; PDE's, Green's functions, and integral equations ; Singular and infinite range problems ; Orthogonality and its ramifications ; Fourier expansions: Generalities ; Fourier expansions: Varied examples ; Fourier integrals and transforms ; Applications of Fourier transforms ; Legendre polynomials and related expansions ; Bessel functions and related expansions ; Hyperbolic equations