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SIMPLE SOLID REGIONS
Section 16.9
The Divergence Theorem
AregionE iscalledasimplesolidregion ifit
issimultaneouslyoftypes1,2,and3.(For
example,regionsboundedbyellipsoidsor
rectangularboxesaresimplesolidregions.)
NotethattheboundaryofE isaclosed
surface.Weusetheconventionthatthe
positiveorientationisoutward,thatis,the
unitnormalvectorn isdirectionoutward
fromE.
THE DIVERGENCE THEOREM
LetE beasimplesolidregionandletS bethe
boundarysurfaceofE,givenwithpositive(outward)
orientation.LetF beavectorfieldwhosecomponent
functionshavecontinuouspartialderivativesona
openregionthatcontainsE.Then
⋅
EXAMPLES
1. LetE bethesolidregionboundedbythecoordinate
planesandtheplane2
2
6,andlet
.Find
⋅
whereS isthesurfaceofE.
div 2. LetE bethesolidregionbetweentheparaboloid
NOTE:Thetheoremissometimesreferredtoas
Gauss’sTheorem orGauss’sDivergenceTheorem.
4
andthe
‐plane.VerifytheDivergenceTheoremfor
, ,
AN EXTENSION
EXAMPLES (CONTINUED)
3. LetE bethesolidboundedbythecylinder
4,theplane
6,andthe ‐
plane,andletn betheouterunitnormaltothe
boundaryS ofE.If
, ,
sin
findthefluxofF acrossE.
cos
2
TheDivergenceTheoremalsoholdsforasolid
withholes,likeaSwisscheese,providedwe
alwaysrequiren topointawayfromthe
interiorofthesolid.
Example:Compute∬
⋅
if
, ,
2
andS isthe
boundaryE isthesolidcylindricalshell
1
4,
0
2
1
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FLUID FLOW
FLUID FLOW (CONTINUED)
Let
, , bethevelocityfieldofafluidwith
constantdensityρ.Then
istherateof
flowperunitarea.If
, ,
isapointin
thefluidflowand isaball(sphere)with
centerP0 andverysmallradiusa,then
div
div
forallpointP in since
divF iscontinuous.
Weapproximatethefluxovertheboundary
sphere asfollows:
⋅
div div
div
FLUID FLOW (CONCLUDED)
Theapproximationbecomesbetteras → 0 and
suggeststhat
div
lim
→
1
⋅
Thisequationsaysthatdiv
isthenetrateof
outwardfluxperunitvolumeatP0.Thisisthereason
forthenamedivergence.IfdivF >0,thenetflowis
outwardnearP andP iscalledasource.IfdivF <0,
thenetflowisinwardnearP andP iscalledasink.
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