Section 17.6

Divergence Theorem

Instead of measuring standard flow at the surface boundaries, we can sum up the internal "creation" rate.

Introduction

Also known as Gauss's Theorem. It states that the total outward flux of a vector field through a closed surface is equal to the triple integral of the divergence over the region inside.

"Net flow out equals total expansion inside."

The Theorem

Statement

Conditions: S is a closed surface with outward orientation. E is the solid region bounded by S.

Visualizing Expansion

Interactive: Flux out of a Sphere

Worked Examples

Example 1: Flux Out of a Cube

Calculate flux of out of the unit cube .

1. Without Theorem:

Calculate 6 surface integrals. Painful.

2. Calculate Divergence:

3. Volume Integral:

Example 2: Radial Field Flux

Verify for out of sphere radius a.

1. Divergence Side:

2. Surface Integral Side:

From Lesson 17.4, we found this was also .

The theorem holds!

Example 3: Impossible Integral

Flux of out of unit sphere.

1. Divergence:

The nasty terms () all vanish.

2. Result:

Divergence Theorem simplifies the impossible!