Chapter 22: The Fundamental Theorem of Calculus in Higher Dimensions; Additive Measures, Stokes Theorem and the Divergence Theorem

Introduction

We seek, (and find) analogues to the fundamental theorem of calculus for surface and volume integrals. To do this we examine constructs with the same additive properties as our integrals because any theorem that relates an integral to something else must relate it to something else with the same additivity property. With clues from this fact, we find the results we seek, which are Stokes Theorem and the Divergence Theorem, which are the most powerful tools in this subject.

Topics

22.1  General Comments

22.2  Measures on Surfaces and Stokes Theorem

22.3  Measures on Volumes and the Divergence Theorem