ABSTRACT
In physical applications, one often finds that it is functions of several variables that are most frequently encountered; in fact, problems involving vibrating membranes, biological systems and heat conduction, all lead naturally to the consideration of multivariable functions. The definition of differentiability of a complex function is completely analogous to the definition of its real counterpart. In the sequel the reader should keep in mind that all limits are taken with respect to the usual metric for the complex plane. Max-min-type problems include certain constraints on the function. For example, a physicist may wish to determine the maximum velocity of a particle in 3-space where the movement of the particle is restricted to a predetermined surface. The chapter aims to finding a solution to the classical brachistochrone problem. It concludes with another brief departure from the realm of abstraction in order to illustrate one specific context in which partial derivatives arise naturally in a "real" setting.
