Chapter 16 – Vector Calculus
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
1
Introduction

In Section 16.5, we rewrote Green’s
Theorem in a vector version as:

C
F  n ds   div F( x, y) dA
D
where C is the positively oriented
boundary curve of the plane region
D.
16.9 The Divergence Theorem
2
Equation 1

If we were seeking to extend this
theorem to vector fields on 3, we
might make the guess that
F

n
dS

div
F
(
x
,
y
,
z
)
dV


S
E
where S is the boundary surface
of the solid region E.
16.9 The Divergence Theorem
3
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
4
Divergence Theorem


Let:
◦ 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,
F

d
S

div
F
dV


S
E
16.9 The Divergence Theorem
5
Divergence Theorem
 Thus,
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
6
History

The 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
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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
8
Example 1

Use the Divergence Theorem to calculate the
surface integral S F  dS ; that is, calculate the
flux of F across S.
F( x, y, z )  e x sin yi  e x cos yj  yz 2k ,
S is the surface of the box bounded by the planes
x  0, x  1, y  0, y  1, z  0, z  2
16.9 The Divergence Theorem
9
Example 2

Use the Divergence Theorem to calculate the
surface integral S F  dS ; that is, calculate the
flux of F across S.
F( x, y, z )  x 2 z 3i  2 xyz 3 j  xz 4k ,
S is the surface of the box with vertices  1, 2, 3 .
16.9 The Divergence Theorem
10
Example 3

Use the Divergence Theorem to calculate the
surface integral S F  dS ; that is, calculate the
flux of F across S.
F ( x, y, z )  x 3 yi  x 2 y 2 j  x 2 yzk ,
S is the surface of the solid bounded by the hyperboloid
x 2  y 2  z 2  1 and the planes z  2, z  2.
16.9 The Divergence Theorem
11
Example 4

Use the Divergence Theorem to calculate the
surface integral S F  dS ; that is, calculate the
flux of F across S.
F( x, y, z )  x 2 yi  xy 2 j  2 xyzk ,
S is the surface of the tetrahedron bounded by the planes
x  0, y  0, z  0, x  2 y  z  2
16.9 The Divergence Theorem
12
Example 5 – pg. 1157 #11

Use the Divergence Theorem to calculate the
surface integral S F  dS ; that is, calculate the
flux of F across S.
F( x, y, z )   cos z  xy 2  i  xe z j   sin y  x 2 z  k ,
S is the surface of the tetrahedron bounded by the paraboloid
z  x 2  y 2 and the plane z  4.
16.9 The Divergence Theorem
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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
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Demonstrations

Feel free to explore these
demonstrations below.
◦ The Divergence Theorem
◦ Vector Field with Sources and Sinks
16.9 The Divergence Theorem
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Review of Chapter

The 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
16
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
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Fundamental Theorem of Calculus

b
a
F '  x  dx  F  b   F  a 
16.9 The Divergence Theorem
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Fundamental Theorem for Line Integrals

C
f  dr  f  r  b    f  r  a  
16.9 The Divergence Theorem
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Green’s Theorem
 Q P 

dA

P
dx

Q
dy


D  x y  C
16.9 The Divergence Theorem
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Stokes’ Theorem
curl
F

d
S

F

d
r


C
S
16.9 The Divergence Theorem
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Divergence Theorem
div
F
dV

F

d
S


E
S
16.9 The Divergence Theorem
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