DEFINITION
Section 16.8
TheorientationofasurfaceS inducesthepositive
orientationoftheboundarycurveC asshownin
thediagram.Thismeansthatifyouwalkinthe
positivedirectionaroundC withyourheadpointing
inthedirectionofn,thenthesurfacewillalwaysbe
onyourleft.
Stokes’ Theorem
STOKES’ THEOREM
NOTATION
LetS beanorientedpiecewise‐smoothsurfacethatis
boundedbyasimple,closed,piecewise‐smooth
boundarycurveC withpositiveorientation.LetF bea
vectorfieldwhosecomponentshavecontinuouspartial
derivativesonanopenregionin thatcontainsS.
Then
Thepositivelyorientedboundarycurveofthe
orientedsurfaceS isoftenwrittenas∂S.Sothe
resultofStokes’Theoremcanbeexpressedas
⋅
curl ⋅
⋅
curl ⋅
NOTE:Stokes’Theoremcanberegardedasahigher‐
dimensionalversionofGreen’sTheorem.
COMMENT 1
Since
⋅
⋅
and
curl ⋅
curl ⋅ Stokes’Theoremsaysthatthelineintegralaroundthe
boundarycurveofS ofthetangentialcomponentofF is
equaltothesurfaceintegralofthenormalcomponent
ofthecurlofF.
RELATIONSHIP TO THE FUNDAMENTAL
THEOREM OF CALCULUS
curl ⋅
⋅
ThereisananalogyamongStokes’Theorem,
Green’sTheorem,andtheFundamental
TheoremofCalculus.Thereisanintegral
involvingderivativesontheleftsideofthe
equationabove(recallthatcurlF isasortof
derivative)andtherightsideinvolvesthe
valuesofF onlyontheboundary ofS.
1
GREEN’S THEOREM AS A
SPECIAL CASE
ThespecialcasewherethesurfaceS isflatandlies
inthe ‐planewithupwardorientation,theunit
normalisk,thesurfaceintegralbecomesadouble
integral,andStokes’Theorembecomes
⋅
curl ⋅
curl
⋅ ThisispreciselythevectorformofGreen’sTheorem
giveninSection16.5.Thus,weseethatGreen’s
TheoremisreallyaspecialcaseofStokes’Theorem.
COMMENT 2
NotethatinthesecondpartofExample2,we
computedasurfaceintegralsimplybyknowingthe
valuesofF ontheboundarycurveC.Thismeansthat
ifwehaveanyotherorientedsurfacewiththesame
boundarycurveC,thenwegetexactlythesamevalue
forthesurfaceintegral.
IngeneralifS1 andS2 areorientedsurfaceswiththe
sameorientedboundarycurveC andbothsatisfythe
hypothesesofStokes’Theorem,then
curl ⋅
⋅
curl ⋅
CURL (CONTINUED)
Let
, ,
beapointinthefluidatlet beasmall
diskwithradius andcenter .Then,
curl
curl
forallpointsP on because
thecurlofF iscontinuous.Thus,byStokes’Theorem,we
getthefollowingapproximationtothecirculationaround
theboundarycircle .
⋅
curl ⋅
curl
curl ⋅ ⋅
curl
⋅
EXAMPLES
1. Let∂S bethetriangleformedbythe
intersectionoftheplane2
2
6
andthethreecoordinateplanes.Verify
Stokes’s Theoremif
, ,
.
, ,
2. VerifyStokes’s Theoremfor
2
,where S isthesurfaceofthe
paraboloid
4
and∂S isthe
traceofS inthe ‐plane.
THE MEANING OF THE CURL
VECTOR
SupposethatC isanorientedclosedcurveandv
representsthevelocityfieldinfluidflow.Considerthe
lineintegral
⋅
⋅ andrecallthatv ·T isthecomponentofv inthe
directionoftheunittangentvectorT.Thismeansthe
closerthedirectionofv istothedirectionofT,the
largerthevalueofv ·T .Thus,
⋅
isameasure
ofthetendencyofthefluidtomovearoundC andis
calledthecirculation ofv aroundC.
CURL (CONCLUDED)
Theapproximationbecomesbetteras → 0 andwe
have
curl
⋅
lim
→
1
⋅
Thisgivestherelationshipbetweenthecurlandthe
circulation.Itshowsthatthecurlv ·n isameasureof
therotatingeffectofthefluidabouttheaxisn.The
curlingeffectisgreatestabouttheaxisparallelto
curl .
2