Advanced Calculus fi..

Partial DifferentialEquationsA partial differential equation is an equation expressing a relationship between anunknown function of several variables and its derivatives with respect to these variables.For example,are partial differential equations.By a solution of a partial differential equation is meant a particular functionthat satisfies the equation identically in a domain of the independent variables. Forexample,u = eTt sinx,u = x2 - Y2are solutions of (10.1) and (10.2), respectively.Although partial differential equations were not studied as such in the earlierchapters, they occurred in a number of important connections. For example, thesolutions of (10.2) are precisely the harmonic functions of x and y; such functionswere studied in Chapter 2. See also Sections 5.16 and 8.11. The Cauchy-Riemannequations u, = v,, u, = -v, form a system of partial differential equations; they