IN THIS CHAPTER we study the integral calculus of real-valued functions of several variables.
- SECTION 7.1 defines multiple integrals, first over rectangular parallelepipeds in \(\R^n\) and then over more general sets. The discussion deals with the multiple integral of a function whose discontinuities form a set of Jordan content zero, over a set whose boundary has Jordan content zero.
- SECTION 7.2 deals with evaluation of multiple integrals by means of iterated integrals.
- SECTION 7.3 begins with the definition of Jordan measurability, followed by a derivation of the rule for change of content under a linear transformation, an intuitive formulation of the rule for change of variables in multiple integrals, and finally a careful statement and proof of the rule. This is a complicated proof.