Functions of Several Real Variables

Abstract

In this more difficult stage of calculus, we turn from the functions ‘of one variable’, which have so far occupied us, to functions ‘of several variables’. There are two important reasons for the difficulties: first, the material studied is more complicated, and second, the traditional notation is a marvel of brevity. In the hands of experts, this brevity is all very well, but it is definitely confusing to beginners, and may be a reason for the comparative neglect of the subject by many teachers of analysis. We shall attempt here to show how the language of set-theory can resolve this notational difficulty; but we realize that our notation looks complicated, and will be disliked by those who have mastered the traditional notation. For such experts we naturally recommend no change. However, let us say that nothing is achieved by giving apparently simple formulae, at the outset, for complicated matters; and it is better, in introducing a topic, to use symbols in a complicated but self-explanatory formula, rather than to accompany a simple-looking formula with a paragraph of words.