1

Chapter 16 – Vector Calculus16.9 The Divergence Theorem

16.9 The Divergence Theorem

Objectives: Understand The

Divergence Theorem for simple solid regions.

Use Stokes’ Theorem to evaluate integrals

16.9 The Divergence Theorem 2

IntroductionIn Section 16.5, we rewrote

Green’s Theorem in a vector version as:

where C is the positively oriented boundary curve of the plane region D.

div ( , )C

D

ds x y dA F n F

16.9 The Divergence Theorem 3

Equation 1If we were seeking to extend this

theorem to vector fields on 3, we might make the guess that

where S is the boundary surface of the solid region E.

div ( , , )S E

dS x y z dV F n F

16.9 The Divergence Theorem 4

Introduction It turns out that Equation 1 is true, under

appropriate hypotheses, and is called the Divergence Theorem.

Notice its similarity to Green’s Theorem and Stokes’ Theorem in that:◦ It relates the integral of a derivative of a

function (div F in this case) over a region to the integral of the original function F over the boundary of the region.

16.9 The Divergence Theorem 5

Divergence TheoremLet:

◦ E be a simple solid region and let S be the boundary surface of E, given with positive (outward) orientation.

◦ F be a vector field whose component functions have continuous partial derivatives on an open region that contains E.

Then, div

S E

d dV F S F

16.9 The Divergence Theorem 6

Divergence TheoremThus, the Divergence Theorem states that:

◦Under the given conditions, the flux of F across the boundary surface of E is equal to the triple integral of the divergence of F over E.

16.9 The Divergence Theorem 7

HistoryThe Divergence Theorem is sometimes

called Gauss’s Theorem after the great German mathematician Karl Friedrich Gauss (1777–1855).

◦ He discovered this theorem during his investigation of electrostatics.

16.9 The Divergence Theorem 8

History In Eastern Europe, it is known

as Ostrogradsky’s Theorem after the Russian mathematician Mikhail Ostrogradsky (1801–1862).

◦ He published this result in 1826.

16.9 The Divergence Theorem 9

Example 1Use the Divergence Theorem to calculate the

surface integral ; that is, calculate the flux of F across S.

SdF S

2( , , ) sin cos ,

S is the surface of the box bounded by the planes

0, 1, 0, 1, 0, 2

x xx y z e y e y yz

x x y y z z

F i j k

16.9 The Divergence Theorem 10

Example 2Use the Divergence Theorem to calculate the

surface integral ; that is, calculate the flux of F across S.

SdF S

2 3 3 4( , , ) 2 ,

S is the surface of the box with vertices 1, 2, 3 .

x y z x z xyz xz

F i j k

16.9 The Divergence Theorem 11

Example 3Use the Divergence Theorem to calculate the

surface integral ; that is, calculate the flux of F across S.

SdF S

3 2 2 2

2 2 2

( , , ) ,

S is the surface of the solid bounded by the hyperboloid

1 and the planes 2, 2.

x y z x y x y x yz

x y z z z

F i j k

16.9 The Divergence Theorem 12

Example 4Use the Divergence Theorem to calculate the

surface integral ; that is, calculate the flux of F across S.

SdF S

2 2( , , ) 2 ,

S is the surface of the tetrahedron bounded by the planes

0, 0, 0, 2 2

x y z x y xy xyz

x y z x y z

F i j k

16.9 The Divergence Theorem 13

Example 5 – pg. 1157 #11Use the Divergence Theorem to calculate the

surface integral ; that is, calculate the flux of F across S.

SdF S

2 2

2 2

( , , ) cos sin ,

S is the surface of the tetrahedron bounded by the paraboloid

and the plane 4.

zx y z z xy xe y x z

z x y z

F i j k

16.9 The Divergence Theorem 14

More Examples

The video examples below are from section 16.9 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. ◦Example 1◦Example 2

16.9 The Divergence Theorem 15

DemonstrationsFeel free to explore these

demonstrations below.◦The Divergence Theorem◦Vector Field with Sources and Sinks

16.9 The Divergence Theorem 16

Review of ChapterThe main results of this chapter

are all higher-dimensional versions of the Fundamental Theorem of Calculus (FTC).

◦ To help you remember them, we collect them here (without hypotheses) so that you can see more easily their essential similarity.

16.9 The Divergence Theorem 17

Review of Chapter

In each case, notice that:

◦On the left side, we have an integral of a “derivative” over a region.

◦The right side involves the values of the original function only on the boundary of the region.

16.9 The Divergence Theorem 18

Fundamental Theorem of Calculus

'b

aF x dx F b F a

16.9 The Divergence Theorem 19

Fundamental Theorem for Line Integrals

Cf d f b f a r r r

16.9 The Divergence Theorem 20

Green’s Theorem

CD

Q PdA Pdx Qdy

x y

16.9 The Divergence Theorem 21

Stokes’ Theorem

curlC

S

d d F S F r

16.9 The Divergence Theorem 22

Divergence Theorem

divE S

dV d F F S