This book presents a detailed development of the divergence theorem. The framework is that of Lebesgue integration-no generalized Riemann integrals of Henstock-Kurzweil variety are involved. The first part of the book establishes the divergence theorem by a combinatorial argument involving dyadic cubes. Only elementary properties of the Lebesgue integral and Hausdorff measures are used. The second part introduces the sets of finite perimeter and the last part proves the general divergence theorem for bounded vector fields.

part Part 1|45 pages

Dyadic figures

chapter Chapter 1|17 pages


chapter Chapter 2|13 pages

Divergence theorem for dyadic figures

chapter Chapter 3|11 pages

Removable singularities

part Part 2|102 pages

Sets of finite perimeter

chapter Chapter 4|23 pages


chapter Chapter 5|47 pages

BV functions

chapter Chapter 6|28 pages

Locally BV sets

part Part 3|82 pages

The divergence theorem

chapter Chapter 7|30 pages

Bounded vector fields

chapter Chapter 8|11 pages

Unbounded vector fields

chapter Chapter 9|11 pages

Mean divergence

chapter Chapter 10|14 pages


chapter Chapter 11|12 pages

The divergence equation