ABSTRACT
Most of the ideas discussed in the previous chapters can be extended to cases where the functions of interest have more than one variable. In this chapter we will briefly review some extensions that will be particularly relevant, and we will develop some fairly deep results concerning integrals of functions of several variables. (In fact, the primary reason this chapter was written was to discuss those “deep” results, and to prevent us from having to prove two to four special cases of each of these results at various widely scattered spots in this text.)1
For convenience, we will limit ourselves to discussing functions of two variables. That will suffice for most of our needs. Also, it covers the hard part of extending one-dimensional results to multi-dimensional results, at least for the results we will be needing. Once you’ve seen the basic ideas expressed here, you should have no trouble extending the definitions and results described in this chapter to corresponding definitions and results for functions whose variables number three or four or five or ….
