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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
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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
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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
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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.
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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
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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.
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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.
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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.
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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
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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
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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
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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
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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
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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
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16.9 The Divergence Theorem 15
DemonstrationsFeel free to explore these
demonstrations below.◦The Divergence Theorem◦Vector Field with Sources and Sinks
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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.
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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.
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16.9 The Divergence Theorem 18
Fundamental Theorem of Calculus
'b
aF x dx F b F a
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16.9 The Divergence Theorem 19
Fundamental Theorem for Line Integrals
Cf d f b f a r r r
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16.9 The Divergence Theorem 20
Green’s Theorem
CD
Q PdA Pdx Qdy
x y
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16.9 The Divergence Theorem 21
Stokes’ Theorem
curlC
S
d d F S F r
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16.9 The Divergence Theorem 22
Divergence Theorem
divE S
dV d F F S