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Section 13.8The Divergence Theorem

Math 21a

April 30, 2008

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.Image: Flickr user Gossamer1013

http://flickr.com/photos/gossamerwings/



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Announcements

Review sessions:Tuesday, May 6 - Hall D 4-5:30pmThursday, May 8 - Hall A 4-5:30pmMonday, May 12 - Hall C 5-6:30pm

Final Exam: 5/23 9:15 am (tentative)

Ofce hours during reading period TBD



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Outline

Last time: Stokes’s Theorem

The Divergence TheoremDivergence is innitesimal uxStatement of the TheoremProof

Worksheet

A trinity of theorems

Next Time: Recap and Movie Day!


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Theorem (Stokes’s Theorem)Let S be an oriented piecewise-smooth surface that is bounded by a simple,closed, piecewise-smooth boundary curve C with positive orientation. Let Fbe a vector eld whose components have continuous partial derivatives on

an open region in R3

that contains S. Then

C

F · d r = S

curl F · d S



Outline


The Divergence TheoremDivergence is innitesimal uxStatement of the TheoremProof

Worksheet





Divergence is innitesimal ux

Given a vector eld F and a point ( x 0 , y 0 , z0), let Sε be the sphere of radius ε centered at ( x 0 , y 0 , z0). Dene

I(ε) =1

4πε 2 Sε

F · d S .

Then a straightforward (albeit long ) calculation shows that

dI

d ε ε= 0

=1

3

div F ( x 0 , y 0 , z0).

So div F is measuring the “net ux” around a point.



Interpretation of the divergence

div F ( x 0 , y 0 , z0) > 0means ( x 0 , y 0 , z 0) is a“source”

div F ( x 0 , y 0 , z0) < 0means ( x 0 , y 0 , z 0) is a“sink”

div F = 0 means F is

incompressible



Statement of the Theorem

Theorem (Gauß’s Divergence Theorem)Let E be a simple solid region and S the boundary surface of E, given withthe positive (outward) orientation. Let F be a vector eld whose

component functions have continuous partial derivatives on an open regionthat contains E. Then

S

F · d S = E

div F dV



Proof of the TheoremSee page 967

We must show

S

(P i + Q j + Rk ) · n dS = E

∂ P ∂ x

+∂ Q∂ y

+∂ R∂ z

dV

It sufces to show

S

P i · n dS = E

∂ P ∂ x

dV S

Q j · n dS = E

∂ Q∂ y

dV

S

Rk · n dS = E

∂ R∂ z dV

We’ll show the last. The others are similar.



Assume E is a region of type 1:

E = { ( x , y , z) | ( x , y ) ∈D, u1( x , y ) ≤ z ≤ u2( x , y ) }

Then S = ∂ E = S1 ∪S2 ∪S3 , where

S1 = { ( x , y , z) | ( x , y ) ∈D, z = ≤ u1( x , y ) }S2 = { ( x , y , z) | ( x , y ) ∈D, z = ≤ u2( x , y ) }S3 = { ( x , y , z) | ( x , y ) ∈∂ D, u1( x , y ) ≤ x ≤ u2( x , y ) }



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On the boundary surface S2 , which is the graph of u2:

n dS =∂ u2

∂ x ,

∂ u2

∂ y , 1 dA

So

S2

Rk · n dS = D

R( x , y , u2( x , y )) dA



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On S1 , the outward normal points down, so

S1

Rk · n dS = −

D

R( x , y , u1( x , y )) dA



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Finally, on S3 , n · k = 0, so

S3

Rk · n dS = 0



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Putting it all together we have

S

= D

(R( x , y , u2( x , y )) − R( x , y , u1( x , y ))) dA

= D u2( x , y )

u1( x , y )

∂ R∂ z dzdA

= E

∂ R∂ z

dV



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Outline


The Divergence TheoremDivergence is innitesimal uxStatement of the Theorem

Proof

Worksheet





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Example (Worksheet Problem 1)Use the Divergence Theorem to calculate the surface integral

S

F · d S , where

F = e x sin y i + e x cos y j + yz2 k

and S is the surface of the box bounded by the planes x = 0, x = 1, y = 0, y = 1, z = 0, and z = 2.



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S

F · d S , where

F = e x sin y i + e x cos y j + yz2 k

and S is the surface of the box bounded by the planes x = 0, x = 1, y = 0, y = 1, z = 0, and z = 2.

SolutionWe have div F = 2 yz so

S

F · d S = E

div F dV

= 1

0 1

0 2

02 yzdzdydx = 2.



E l (W k h P bl 2)



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S

F · d S , where

F = ( cos z + xy 2) i + xe− z j + ( sin y + x 2 z) k

and S is the surface of the solid bounded by the paraboloid

z = x 2

+ y 2

and the plane z = 4.SolutionLet E be the region bounded by S. We have

SF · d S = E

div F dV = E( x

2

+ y 2

) dV

Switch to cylindrical coordinates and we get

( x 2 + y 2) dV =2π

0

2

0

z

r 2r 2 · rdzdrd θ =

32π3



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Outline



Proof

Worksheet





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Theorem (Green’s Theorem)Let C be a positively oriented, piecewise smooth, simple closed curve in theplane and let D be the region bounded by C. If P and Q have continuouspartial derivatives on an open region that contains D, then

C (P dx + Q dy ) = D

∂ Q∂ x −

∂ P ∂ y dA

or

C

F · d r = D

d F dA



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Theorem (Stokes’s Theorem)Let S be an oriented piecewise-smooth surface that is bounded by a simple,closed, piecewise-smooth boundary curve C with positive orientation. Let Fbe a vector eld whose components have continuous partial derivatives on

an open region inR 3

that contains S. Then

C

F · d r = S

curl F · d S



. . . . . .

Theorem (Gauß’s Divergence Theorem)Let E be a simple solid region and S the boundary surface of E, given withthe positive (outward) orientation. Let F be a vector eld whosecomponent functions have continuous partial derivatives on an open region

that contains E. Then

S

F · d S = E

div F dV

M if ld D ti



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Manifold Destiny

A manifold is a smooth subset of R n (roughly speaking,something you can do calculus on)

Examples: Solid regions, surfaces, curvesA manifold can have a boundary, which is of one dimensionlower.



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Each of the operators gradient, curl, divergence, and even the

two-dimensional curl ∂ Q∂ x

− ∂ P ∂ y

are a kind of derivative.

Each of the big three theorems takes the form

d △ = ∂ △ ,

where is a manifold, △ is a vector eld, is whatever integral

and d is whatever derivative is appropriate from the context.

But wait there’s more



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But wait, there’s more

We can have zero-dimensional manifolds, too: they’re points.

The integral over a zero-dimensional manifold is just evaluationat the points.

Other fundamental theorems in calculus can expressed using

this extension of the d △ = ∂

△ formalism



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Theorem (The Fundamental Theorem of Line Integrals)Let C be a smooth curve given by the vector function r (t ) , a ≤ t ≤ b, and let f be a differentiable function of two or three variables whose gradient vector ∇ f is continuous on C. Then

C ∇ f · d r = f (r (b)) − f (r (a))



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Theorem (The Fundamental Theorem of Calculus)Let F ( x ) be a function on [a, b] with continuous derivative. Then

b

aF ′ ( x ) dx = F (b) − F (a).

Outline



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Outline



Proof

Worksheet



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