chap3 integral theorems
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Typeset lecture-notes chapter from a university vector calculus course, citing Anton and Bourne as textbook references. It defines domains, convex and semi-convex surfaces, orientability and simple curves. It states the Divergence Theorem with a proof for semi-convex surfaces, then Green's theorem with the area formula, and Stokes' theorem with remarks. The text is cut off partway through the Stokes section. Authorship of the notes is not shown.
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Chapter 3
Integral Theorems
[Anton,pp.1124{1130, pp.1145{1160] &[Bourne, pp.195{224]
Firstofallsome de nitions whichwewillneedinthefollowing:
De nition 3.1. (a)Adomain (region)
isanopenconnected subset ofRn.
(b)Adomain
R3isbounded ,ifthere exists anR>0suchthat
BR,where BRis
theballwithradius Randcentre0.
(c)Asurface SR3isopen,ifforallx1;x262Sthere exists acontinuouscurvefromx1
tox2whichdoesnotcrossS.Asurface SR3isclosed ,ifitisnotopen.
(d)Aclosed surface SR3isconvex,ifeverystraigh tlineintersects (meets) Sattwo
pointsatmost. Examples.
(e)Aclosed surface SR3issemi{con vex,ifwecanchooseacoordinate system 0xyz
sothateverystraigh tlineparalleltothecoordinate axesintersects Sattwopointsat
most. Examples.
Note. Recall also(Remark 1.24) thatasurface Sissmooth,ifitsparametrisation iscontin-
uously di eren tiable. Sispiecewise smooth,ifS=Sn
i=1SiandSismooth.
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3.1TheDivergence Theorem ofGauss
Theorem 3.2(Divergence Theorem). Let
R3beaboundeddomain withpiecewise
smooth,closedboundary (surfac e)S.SupposealsothatF:
!R3isacontinuously di er-
entiable vector eld.Then
ZZZ
rFdV=ZZ
SFdS: (3.1)
Proof.(Only forSsmoothandsemi{con vex).
LetDbetheprojection of
ontothe(x;y){plane.
Consider thelineLthrough thepoint(x;y;0)parallel
tothez{axis. Since Sissemi-con vex,Lintersects
Sattwopoints(x;y;f(x;y))Tand(x;y;g(x;y))T,
where f(x;y)g(x;y)forall(x;y)2D(otherwise
change thecoordinate system).
Hence,PSfrag replacemen ts
xyz
(x;y;0)(x;y;f(x;y))(x;y;g(x;y))
DL
S0S1
(i)Letus rstshowthat
ZZ
SF3kdS=ZZ
Dn
F3(x;y;g(x;y)) F3(x;y;f(x;y))o
dxdy: (3.2)
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(ii)NowweshowthatZZZ
@F3
@zdV=ZZ
SF3kdS: (3.3)
(iii)Similarly ,byprojecting ontothe(x;z)-plane andontothe(y;z)-plane wecanestablish
ZZZ
@F2
@ydV=ZZ
SF2jdS; (3.4)
ZZZ
@F1
@xdV=ZZ
SF1idS; (3.5)
and
Remark 3.3.Thisproofcanbeextended inastraigh tforwardwaytodomains
withpiecewise
smoothandnon-semi-con vexboundary S,if
=Sn
i=1
i,where eachofthe
ihasasmooth,
semi-con vexboundary Si,e.g.torus.
Example 3.4.FindRR
SFdSwhere Sisthesurface oftheunitcubeandF:=(x2;y2;z2)T.
Corollary 3.5.Let
andSbeasinTheorem3.2.Supposef:
!RandF:
!R3are
continuously di erentiable. Then
ZZZ
rfdV=ZZ
SfdS (3.6)
ZZZ
r^FdV= ZZ
SF^dS (3.7)
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Proof.Leta2R3beconstan t.
(i)Apply theDivergence Theorem toG:=fa:
(ii)Apply theDivergence Theorem toG:=a^F:
3.2Green's Theorem inthePlane
Note. Inthissection weworkinR2notinR3!
De nition 3.6. (a)Aclosed curveCR2,issimple ,ifitdoesnotintersect itself, e.g.
PSfrag replacemen ts
simple notsimple.
(b)Aclosed curveCR2isconvex,ifeverystraigh tlineintersects Cat2pointsatmost.
(c)Aclosed curveCR2issemi{con vex,ifwecanchooseacoordinate system 0xyso
thateverystraigh tlineparalleltothecoordinate axesintersects Cat2pointsatmost.
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Theorem 3.7(Green's Theorem inthePlane). Let
R2beaboundeddomain with
simple, piecewisesmoothboundary (curve) CR2describ edintheanticlo ckwise sense. Sup-
posethat:
!R2isacontinuously di erentiable vector eldinR2,i.e.=1i+2j.
Then ZZ
@2
@x @1
@y
dxdy=I
Cdr: (3.8)
Proof.SeeHandout or[Bourne, pp.210{213].
Remark 3.8.Green's Theorem intheplane issometimes alsoreferred toasStokes'Theorem
intheplane (e.g.in[Bourne, pp.210{213]).
Corollary 3.9.Theareaboundedbyasimple, closed,piecewisesmoothcurve CR2isgiven
by
1
2 I
C( yi+xj)dr :
Proof.Apply Green's Theorem intheplane with1(x;y)= yand2(x;y)=x.
3.3Stokes'Theorem
De nition 3.10. (a)Aclosed curveCR3,issimple ,ifitdoesnotintersect itself.
(b)Asurface SR3isorientable,ifaunique normal canbeassigned ateachpointx2S.
Example. AMobius stripforexample isnotorientable:
PSfrag replacemen ts
P
(c)LetSbeanopen,orientablesurface withsimple boundary (curve)C.Let^nbetheunit
normal onS.Imagine aperson walking along thecurveC(inthepositivedirection)
withitsheadpointinginthedirection of^n.
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PSfrag replacemen ts^n
CS
L
R
Then SandCaresaidtobecorresp ondingly orientated ,ifthesurface istotheleft
oftheperson. [Anton,p.1154], [Bourne, p.210].
Theorem 3.11(Stokes'Theorem). LetSR3beanopen,orientable, piecewisesmooth
surfacewithcorrespondingly orientate d,simple, piecewise smoothboundary (curve) CR3.
Supposethatthevector eldFiscontinuously di erentiable (inaneighb ourhoodofS).Then
ZZ
S(r^F)dS=I
CFdr: (3.9)
Proof.SeeHandout or[Bourne, pp.213{216].
Remark 3.12.(a)Stokes'Theorem implies thatthe
uxofr^Fthrough asurface S
dependsonlyontheboundary CofSandistherefore independen tofitsshape.Inother
words,
ZZ
S(r^F)dSisthesame for PSfrag replacemen ts
C CS1S2
andfor
(b)NotethatTheorem 3.7isaspecialcaseofTheorem 3.11. Toseethis,assume thatSin
Theorem 3.11is
at,i.e.SR2f0g.Then
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