Section 17.9

The Divergence Theorem

SIMPLE SOLID REGIONS

A 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 THEOREM

Let

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 

S

F



d

S

 

E

div

F

dV

NOTE: The theorem is sometimes referred to as

Gauss’s Theorem

or

Gauss’s Divergence Theorem

.

EXAMPLES

1. Let

E

be the solid region bounded by the coordinate planes and the plane 2

x

+ 2

y

+

z

= 6, and let

F

= xi where

S

+

y

2

j

+ zk . Find 

F

 dS

S

is the surface of

E

.

2. Let

E

be the solid region between the paraboloid

z

= 4 −

x

2 −

y

2 and the

xy

-plane. Verify the Divergence Theorem for

F

(

x

,

y

,

z

) = 2 zi + xj +

y

2

k

.

EXAMPLES (CONTINUED)

3. Let

E x

2 +

y

2 be the solid bounded by the cylinder = 4, the plane

x

+

z

= 6, and the

xy

plane, and let

n

be the outer unit normal to the ( boundary

xy S

+ cos

z

)

j

of

E

. If

F

(

x

,

y

,

z

) = (

x

2 +

e y

k

, find the flux of

F

+ sin

z

)

i

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.

if

F

(

x

,

y

,

z

) = 2 zi + xj 

S

+

z

2

k

and

S

is the boundary

E

is the solid cylindrical shell 1 ≤

x

2 +

y

2 ≤ 4, 0 ≤

z

≤ 2.

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

P

0 (

x

0 ,

y

0 ,

z

0 ) is a point in the fluid flow and

B a

is a ball (sphere) with center

P

0 and very small radius

a

, then div

F

(

P

) ≈ div

F

(

P

0 ) for all point

P

in

B a

since div

F

is continuous.

FLUID FLOW (CONTINUED)

We approximate the flux over the boundary sphere

S a

as follows:

S



a

F



d

S

 

B a

div

F

dV

 

B a

div  div

F

(

P

0 )

dV

F

(

P

0 )

V

(

B a

)

FLUID FLOW (CONCLUDED)

The approximation becomes better as

a

div

F

(

P

0 ) 

a

lim  0

V

( 1

B a

) → 0 and suggests that

S



a

F



d

S

This equation says that div

F

(

P

0 ) is the net rate of outward flux per unit volume at

P

0 . 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

.