Section 17.9

The Divergence Theorem

SIMPLE SOLID REGIONSA region E is called a simple solid region if it is simultaneously of types 1, 2, and 3. (For example, regions bounded by ellipsoids or rectangular boxes are simple solid regions.) Note that the boundary of E is a closed surface. We use the convention that the positive orientation is outward, that is, the unit normal vector n is direction outward from E.

THE DIVERGENCE THEOREMLet E be a simple solid region and let S be the boundary surface of E, given with positive (outward) orientation. Let F be a vector field whose component functions have continuous partial derivatives on a open region that contains E. Then

ES

dVd FSF div

NOTE: The theorem is sometimes referred to as Gauss’s Theorem or Gauss’s Divergence Theorem.

EXAMPLES1. Let E be the solid region bounded by the coordinate

planes and the plane 2x + 2y + z = 6, and let F = xi + y2j + zk. Find

where S is the surface of E.

2. Let E be the solid region between the paraboloid

z = 4 − x2 − y2

and the xy-plane. Verify the Divergence Theorem for

F(x, y, z) = 2zi + xj + y2k.

S

dSF

EXAMPLES (CONTINUED)

3. Let E be the solid bounded by the cylinder x2 + y2 = 4, the plane x + z = 6, and the xy-plane, and let n be the outer unit normal to the boundary S of E. If F(x, y, z) = (x2 + sin z)i + (xy + cos z)j + eyk, find the flux of F across E.

AN EXTENSION

The Divergence Theorem also holds for a solid with holes, like a Swiss cheese, provided we always require n to point away from the interior of the solid.

Example: Compute

if F(x, y, z) = 2zi + xj + z2k and S is the boundary E is the solid cylindrical shell

1 ≤ x2 + y2 ≤ 4, 0 ≤ z ≤ 2.

S

dSF

FLUID FLOW

Let v(x, y, z) be the velocity field of a fluid with constant density ρ. Then F = ρv is the rate of flow per unit area. If P0(x0, y0, z0) is a point in the fluid flow and Ba is a ball (sphere) with center P0 and very small radius a, then div F(P) ≈ div F(P0) for all point P in Ba since div F is continuous.

FLUID FLOW (CONTINUED)

We approximate the flux over the boundary sphere Sa as follows:

)()(div

)(div

div

0

0

a

B

BS

BVP

dVP

dVd

a

aa

F

F

FSF

FLUID FLOW (CONCLUDED)

The approximation becomes better as a → 0 and suggests that

aSaa

dBV

P SFF)(

1lim)(div

00

This equation says that div F(P0) is the net rate of outward flux per unit volume at P0. This is the reason for the name divergence. If div F > 0, the net flow is outward near P and P is called a source. If div F < 0, the net flow is inward near P and P is called a sink.