Chapter III. Theory of Calculus in Several Real Variables

Abstract

This chapter gives a rigorous treatment of parts of the calculus of several variables.

Sections 1–3 handle the more elementary parts of the differential calculus. Section 1 introduces an operator norm that makes the space of linear functions from $\mathbb{R}^n$ to $\mathbb{R}^m$ or from $\mathbb{C}^n$ to $\mathbb{C}^m$ into a metric space. Section 2 goes through the definitions and elementary facts about differentiation in several variables in terms of linear transformations and matrices. The chain rule and Taylor's Theorem with integral remainder are two of the results of the section. Section 3 supplements Section 2 in order to allow vector-valued and complex-valued extensions of all the results.

Sections 4–5 are digressions. The material in these sections uses the techniques of the present chapter but is not needed until later. Section 4 develops the exponential function on complex square matrices and establishes its properties; it will be applied in Chapter IV. Section 5 establishes the existence of partitions of unity in Euclidean space; this result will be applied at the end of Section 10.

Section 6 returns to the development in Section 2 and proves two important theorems about differential calculus. The Inverse Function Theorem gives sufficient conditions under which a differentiable function from an open set in $\mathbb{R}^n$ into $\mathbb{R}^n$ has a locally defined differentiable inverse, and the Implicit Function Theorem gives sufficient conditions for the local solvability of $m$ nonlinear equations in $n+m$ variables for $m$ of the variables in terms of the other $n$. The Inverse Function Theorem is proved on its own, and the Implicit Function Theorem is derived from it.

Sections 7–10 treat Riemann integration in several variables. Elementary properties analogous to those in the one-variable case are in Section 7, a useful necessary and sufficient condition for Riemann integrability is established in Section 8, Fubini's Theorem for interchanging the order of integration is in Section 9, and a preliminary change-of-variables theorem for multiple integrals is in Section 10.

Sections 11–13 give a careful treatment of integrals of scalar-valued and vector-valued functions on simple arcs and other curves in $\mathbb{R}^n$. The main theorem, proved in Section 13, is Green's Theorem for the plane, which for a suitably nice region of $\mathbb{R}^2$ relates a line integral over the boundary to a double integral over the region. Section 13 concludes with some remarks about higher-dimensional generalizations.