Functions of Several Variables and Their Derivatives

Abstract

The concepts of limit, continuity, derivative, and integral, as developed in Volume I, are also basic in two or more independent variables. However, in higher dimensions many new phenomena, which have no counterpart at all in the theory of functions of a single variable, must be dealt with. As a rule, a theorem that can be proved for functions of two variables may be extended easily to functions of more than two variables without any essential change in the proof. In what follows, therefore, we often confine ourselves to functions of two variables, where relations are much more easily visualized geometrically, and discuss functions of three or more variables only when some additional insight is gained thereby; this also permits simpler geometrical interpretations of our results.