Vector differentiation, the ∇ operator,

1. Chapter3
Vectordifferentiation,theÑoperator,
grad,divandcurl.
Syllabustopicscovered:
1.Vectorelds
2.Grad,divandcurloperatorsinCartesiancoordinates.Grad,div,andcurlofproductsetc.
Herewecoverdifferentiationofvectors.NotethatthisdiffersfromthegradientintroducedinChapter1,
whichweobtainedbydifferentiatingascalareld.
3.1Vectorfunctionsofoneormorevariables
(SeeThomas13.1)
Inmanyphysicalcontextsoneisinterestedinvectorsthatvarywithpositionortime.Forexample,the
positionofapointcanbedescribedbyavectorr.Thus,ifweconsideramovingparticle,itspositioncanbe
describedasafunctionoftimetbythevectorr(t),anditsrateofchangewithrespecttotwillbethevelocity
(whichhasmagnitudeanddirection,i.e.isavector:itsmagnitudeisthespeed).Thepositionvectoristhen
afunctionofonevariable.
Anothercontextiswherewehaveavectordenedateachpoint,sayF(r)=F(x;y;z)andacurvewith
aparameteru,say,soitspointsare(x(u);y(u);z(u)).Thenwecandeneavectorfunctionofu,F(u)=
F((x(u);y(u);z(u)).Wecandealwiththisandthemovingparticlecaseasfollows.
Avectorfunctionofascalaru,F(u),canbedenedbyspecifyingitscomponentsasfunctionsofu:
F(u)=(f1(u);f2(u);f3(u)):
ThederivativedF=duofFwithrespecttouisthen:
dF
du
=

df1
du
;
df2
du
;
df3
du

:
Thissimplygoesbacktothefundamentaldenitionofaderivative:
dF
du
=limdu!0
F(u+
du)F(u)
du
:
28

2. Clearlyonecancomputehigherderivatives,suchasd
2
F=du
2
,bydifferentiatingthecomponentsofFthe
requirednumberoftimes.
Example3.1.Ifr(t)isthepositionvectorofaparticle,asafunctionoftimet,thendr=dtisthevelocity
voftheparticle.Alsodv=dtd
2
r=dt
2
istheparticle'sacceleration.
Example3.2.Thecontinuousparametertcantakeallrealvalues.Writedownthederivativesdr=dtand
d
2
r=dt
2
forthevectorr=(sint)i+tj.Also,sketchthecurvewhoseparametricequationisr=r(t).
Therstandsecondderivativesare
dr
dt
=(cost)i+j;
d
2
r
dt
2
=(sint)i:
ThesketchisshowninFig.3.1.
-1 1
¼
2¼
¡¼
¡2¼
t=0
t=¼=2
t=¼
Figure3.1:Sketchofthecurvedenedparametricallybyr=(sint)i+tj
Itiseasytoprove,bywritingoutthecomponentsandcollectingterms,thatifFandGarevectorfunctions
ofu,then
d(F:G)
du
=F:
dG
du
+
dF
du
:G:
Proof:
d(F:G)
du
=
d
du
(f1g1+f2g2+f3g3)
=f1
dg1
du
+f2
dg2
du
+f3
dg3
du
+
df1
du
g1+
df2
du
g2+
df3
du
g3
=F:
dG
du
+
dF
du
:G:Q:E:D:
Exercise3.1.Sketchthecurveswhoseparametricequationsare
29

3. (a)r=(3sin pt)i+(2cos pt)j
(b)r=(cos
pt)j
(c)r=ti+t
2
k
(¥t¥),andwritedownthederivativesdr=dtandd
2
r=dt
2
wheretheyaredened. 2
IfFisavectorfunctionofmorethanonevariable,sayF=F(u;v),thenitisstraightforwardtodeneits
partialderivativeswithrespecttouorv,intermsofpartialderivativesofitscomponents.Thus,forexample,
ifF=(f1(u;v);f2(u;v);f3(u;v)),then
¶F
¶u
=

¶f1
¶u
;
¶f2
¶u
;
¶f3
¶u

:
Onecasewherethisarisesisifthevectorisdenedateachpointonasurface(x(u;v);y(u;v);z(u;v)).
3.2VectorFields
(SeeThomas16.2)
Fortherestofthiscourse,weshallbeconcernedmostlywithvectorsandscalarswhichdependonpositionin
three-dimensionalspace,i.e.whicharefunctionsofthreevariablesx,y,z.Sometimestheymaydependalso
onafourthvariable,suchastimet,orwemayonlybeinterestedintheirvaluesonaparticularpathr(s).A
vectordependingonpositionissaidtoconstituteavectoreld.WewriteavectorFthatvarieswithposition
as
F=F(x;y;z)F(r)
AnexampleisshowninFigure3.2.WecanalsowriteFintermsofitscomponents,whichagaindependon
position:
F=(F1(x;y;z);F2(x;y;z);F3(x;y;z)):
Aphysicalexampleofavectoreldisthevelocityinaowinguid(e.g.thewaterintheoceans,moving
becauseofcurrentsandtides;ortheairintheatmosphere,movingbecauseofwinds).Thevelocityatany
pointintheuidisavectorquantity–ithasmagnitudeanddirection.Ifweattachavelocityvectortoeach
pointoftheowinguid,wehaveavectorelddenedintheregionoccupiedbytheuid.
Anotherphysicalexampleisamagneticeld;nowthingsarenotnecessarilymovingwithtime,butthe
magneticeldhasadirectionandastrengthateachpointinspace;soateachpointinspacewehaveavector;
andthisvector(ingeneral)varieswithpositionsoitisavectoreld.
Wecanaddvectoreldsandmultiplythembyaconstantintheobviousway,soifFandGaretwovector
eldsthenF+Gisalsoavectoreld,andif
lisaconstantthen lFisalsoavectoreld.
Givenavectoreld,wecouldofcoursenowdifferentiatethevectoreldwithrespecttoeachofthe
coordinates(x;y;z)inturn,inthemannerdescribedintheprevioussection;thisgivesusatotalof9quantities
foreachofthex;y;zderivativesofF1;F2;F3.HerewewillbeassumingthatFisasmoothly-varyingfunction
ofposition,soallthesederivativesexistatallpointsofinterestr,(exceptpossiblyforoneortwosingular
points).
Note:thesetofall9derivativesofacomponentbyacoordinateformsaquantityofanewkind,called
atensor.Theseareusedinuidmechanics,solidmechanicsandrelativity,forexample.
30

4. 1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
x
y
Figure3.2:Exampleofaow.Inthiscasethespeedanddirectionateachpointisafunctionoftheposition
(x;y)
However,inthiscoursewewillnotdealwithtensors,wewillrestrictourselvestoformingscalarand
vectorquantitiesfromthesederivatives.Todothis,itwillturnoutthatwehavetotakecertaincombinations
whichare“wellbehaved”ifwerotatethex;y;zaxes;thesewillturnouttobeformingthedotandcross
productsofÑwithF,where
Ñ=i
¶
¶x
+j
¶
¶y
+k
¶
¶z
istheoperatorcalled“del”whichwemetpreviouslyinformingthegradientofascalar.NoteagainthatÑ
isnotatruevector(becauseonitsownwecan'tdeneitslengthordirection),butitisavectordifferential
operator.
3.3TheDivergenceofavectoreld
(SeeThomas16.8)
SupposeF(x;y;z)=F1i+F2j+F3kisavectoreld.ThedivergenceofFisdenedtobe
ÑF=
¶F1
¶x
+
¶F2
¶y
+
¶F3
¶z
HeredivFisascalaranddependsonposition,soitisascalareld.Wecanalsogettheaboveresultifwe
writeoutÑandFincomponents,
ÑF=(i
¶
¶x
+j
¶
¶y
+k
¶
¶z
):(F1i+F2j+F3k)
31

5. andusethepropertiesi:i=1,i:j=0,etc.
Notethat,givenascalareldV,wefoundavectoreldÑV.Here,givenavectoreldF,wehave
producedascalareldÑF.
WecanalsowriteÑFasdivF.Thesenotationsarecompletelyinterchangeable.
Itiseasytoshow,bydirectcalculation,thatdivbehavesasexpectedforadditionandmultiplicationbya
constant,i.e.
Ñ(F+G)=(ÑF)+(ÑG);
Ñ(
lF)=l(ÑF)
Thegeometricalmeaningofthedivergenceisasfollows.Looselyspeaking,ifatsomepointinspace
thedivergenceispositive,andoneconsidersasmallclosedsurfacesurroundingthatpoint,thenonbalance
thevectoreldispointingawayfromthepointandoutofthesurface.Ifthedivergenceisnegative,thenon
balancethevectoreldispointingtowardsthepointandintothesurface.(SeeFig.3.3.)Thisideawillbe
mademoreprecisewhenwecometotheDivergenceTheoreminthenextChapter.
AvectoreldFforwhichÑF=0everywhereiscalleddivergence-freeorsolenoidal.Thereason
forthenamesolenoidalishistorical:thatasolenoidisacoiledwirethatproducesamagneticeld,anda
magneticeldBisanexampleofaeldthathasÑB=0everywhere(thisisanobservationalfact,and
arisesbecausemagneticmonopoleshaveneverbeenfoundinmanysearches).
Example3.3.IfF=3xy
2
i+e
z
j+xysinzk,calculateÑF.
ÑF=
¶(3xy
2
)
¶x
+
¶e
z
¶y
+
¶(xysinz)
¶z
=3y
2
+xycosz:
Exercise3.2.IfF=(yx)i+(zy)j+(xz)k,calculateÑF.[Answer:-3] 2
3.4TheCurlofavectoreld
(SeeThomas16.7)
ThecurlofavectoreldFisdenedtobe
ÑF=

¶F3
¶y

¶F2
¶z

i+

¶F1
¶z

¶F3
¶x

j+

¶F2
¶x

¶F1
¶y

k:
NotethatcurlFisanothervectoreld.
WecanwriteÑFascurlF–againthetwonotationsarecompletelyinterchangeable.Itisconvenient
torememberÑFintermsofadeterminantliketheoneforvw:
ÑF=






i j k
¶=¶x¶=¶y¶=¶z
F1 F2 F3






:
Itiseasytoverify,bywritingoutthedeterminantinfull,thatthisisequivalenttotheoriginaldenition.
Itisalsoeasytoshow,bywritingoutthecomponents,thatifF;Gareanytwovectorelds,
Ñ(F+G)=(ÑF)+(ÑG)
32

6. -2 -1 0 1 2
-2
-1
0
1
2
x
y
Figure3.3:Exampleofavectoreldwithpositivedivergence(everywhere):F=xi+yj.
andif
lisanyconstantthen
Ñ(
lF)=l(ÑF)
Noteintheabovethat
lmustbeindependentofposition;seethenextsectionformultiplyingascalarand
vectoreld.
Thegeometricalmeaningofthecurlisasfollows.Looselyspeaking,ifatsomepointinspacethe
componentofthecurlinthendirectionispositive,itmeansthatinthevicinityofthepointandinaplane
normalton,thevectoreldtendstogoroundinananticlockwisedirectionifonelooksalongvectorn.If
thecomponentofthecurlwerenegative,itwouldmeanthatthevectoreldtendstogoroundinaclockwise
direction.(SeeFig.3.4.)ThisideawillbemademoreprecisewhenwecometoStokes'sTheorem.
AvectoreldFforwhichÑF=0everywhereiscalledcurl-freeorirrotational.
Example3.4.Thevelocityinauidisv=yixj+0k.FindÑv.
Ñv=






i j k
¶=¶x¶=¶y¶=¶z
y x 0






=i(00)+j(00)+k(11)=2k:
Exercise3.3.IfF=(x
2
+y
2
+z
2
)i+(x
4
y
2
z
2
)j+xyzk,ndÑF. 2
Exercise3.4.Findthedivergence(ÑF)andcurl(ÑF)ofthefollowingvectorelds:
F=x
2
i+xzj3zk
F=x
2
i2xyj+3xzk
F=Ñ(1=r)wherer=(x
2
+y
2
+z
2
)
1=2
6=0.
33

7. -2 -1 0 1 2
-2
-1
0
1
2
x
y
Figure3.4:Exampleofavectoreldwithpositivecurl(inthezdirection):F=xjyi.
2
3.5Grad,DivandCurlofproducts
(SeeThomas16.7andtheexercisesto16.8)
Wecannowconsidertheapplicationofgrad,divandcurltoproducts.Wesawabovethatgrad,divandcurl
behaveinthe“obvious”wayforadditionandmultiplicationbyaconstant.
However,wecanalsomultiplyscalarandvectoreldstogether:e.g.ifwehavetwoscalarelds
U(r);V(r)wecanmultiplythem(ateachpointr)togetanewscalareldUV(r);likewiseforascalar
eldUandavectoreldF(r)multiplyingthemgivesUF(r).Also,ifwehavetwovectoreldsF;Gwecan
denetheirdotproductF:G(r)andcrossproductFG(r)intheobviousway,bytakingthedotorcross
productsofeacheldatthesamepointr.
Wecannowapplygrad,divandcurltotheseproducts,butonlyforthefollowingallowedcombinations:
toapplygrad,wehavetohaveaproductwhichisitselfascalareld:thatcanbeanordinaryproductoftwo
scalarelds,sayUV,orascalarproduct(dotproduct)oftwovectorelds,F:G.
Divandcurlcanonlybeappliedtovectorelds,sothepossibleproductswecouldhaveareUF,where
Uisascalareld,orthecrossproductFGoftwovectorelds.
Ifweweredealingwithfunctionsofasinglevariable,thederivativewouldjustgivethewell-known
productruleforderivatives,
d(fg)
dx
=f
dg
dx
+
df
dx
g: (3.1)
Someofthevectorcasesarejustlikethat,butsomearemorecomplicated:wenextgivetheresults,and
discussthedetailsafterwards.
34

8. Forgradofproductswehave:
Ñ(UV)=U(ÑV)+V(ÑU) (3.2)
grad(UV)=UgradV+VgradU
Ñ(F:G)=F(ÑG)+G(ÑF)+(F:Ñ)G+(G:Ñ)F (3.3)
Fordivofproductswehave:
Ñ(UF)=U(ÑF)+(ÑU):F (3.4)
Ñ(FG)=G:(ÑF)F:(ÑG) (3.5)
andforcurlofproducts,wehave:
Ñ(UF)=U(ÑF)+(ÑU)F (3.6)
=U(ÑF)F(ÑU)
Ñ(FG)=F(ÑG)+(G:Ñ)FG(ÑF)(F:Ñ)G (3.7)
Weseeabovethatequations3.2,3.4,3.5and3.6lookquitesimilarto3.1,exceptfortheminussignin
3.5andthepossibleminussignin3.6.
NotealsothatEqs.3.2and3.3aresymmetricalinthetwovariables,while3.5and3.7areantisymmetric,
i.e.theymustchangesignifF;Gareswappedduetotheantisymmetryofthecrossproduct.
Theothertwo3.3,3.7aremorecomplicated,andinvolvethenewoperator(G:Ñ):Herethenotation
(G:Ñ)Fistobeinterpretedas(G:ÑF1;G:ÑF2;G:ÑF3),takingF=(F1;F2;F3);whereforascalareldV,if
G=(G1;G2;G3),
(G:Ñ)V=

G1
¶
¶x
+G2
¶
¶y
+G3
¶
¶z

V=G1
¶V
¶x
+G2
¶V
¶y
+G3
¶V
¶z
;
Thuswritingoutthewholething,wehave
(G:Ñ)F=

G1
¶F1
¶x
+G2
¶F1
¶y
+G3
¶F1
¶z
;G1
¶F2
¶x
+G2
¶F2
¶y
+G3
¶F2
¶z
;G1
¶F3
¶x
+G2
¶F3
¶y
+G3
¶F3
¶z

ThisisessentiallythedirectionalderivativeofvectorFinthedirectionofG,i.e.itisjGjtimesthederivative
dF=dsalongthedirectionoftheunitvectorparalleltoG.
(Warning:theformofthisdenitionwillnotpersistincurvilinearcoordinates,butthedirectionalderiva-
tivewillremainthesame).
Note:youarenotexpectedtomemoriseEqs.3.3and3.7,butyoumaybegiventhoseformulaeinan
examquestion.Youshouldknowthedenitionof(G:Ñ)Fabove.
Example3.5.Letabeaconstantvector,andr=jrjasusual.Then,usingEq3.6,
Ñ(ra)=r(Ña)aÑr
=0
ar
r
35

9. Example3.6.Letabeaconstantvector.Then,usingEquation3.7,
Ñ(ar)=a(Ñr)+(r:Ñ)ar(Ña)(a:Ñ)r
=3a+00a
=2a
(Onthetopline,thetwomiddletermsdifferentiatetheconstantasoarebothzero,anditissimpletocheck
fromthedenitionsthatÑr=3and(a:Ñ)r=a.)
Proofs:
Alloftheequations3.2to3.7canbeproveddirectlyfromthedenitionsbyinsertingcomponents,
expandingoutusingtheordinaryproductruleanddoingsomerearrangement;thiscanbefairlylong,butis
notdifcult.
Foracoupleofexamples:rstlyforEq.3.2itissimple,wehave
Ñ(UV)=i
¶
¶x
(UV)+j
¶
¶y
(UV)+k
¶
¶z
(UV)
=i(U
¶V
¶x
+V
¶U
¶x
)+j(U
¶V
¶y
+V
¶U
¶y
)+k(U
¶V
¶z
+V
¶U
¶z
)
=U

i
¶V
¶x
+j
¶V
¶y
+k
¶V
¶z

+V

i
¶U
¶x
+j
¶U
¶y
+k
¶U
¶z

=U(ÑV)+V(ÑU)QED:
Nextwe'llproveEq.3.6:fromthedenitionofcurl,
Ñ(UF)=i

¶
¶y
(UF3)
¶
¶z
(UF2)

+j

¶
¶z
(UF1)
¶
¶x
(UF3)

+k

¶
¶x
(UF2)
¶
¶y
(UF1)

=i

U
¶F3
¶y
+F3
¶U
¶y
U
¶F2
¶z
F2
¶U
¶z

+j

U
¶F1
¶z
+F1
¶U
¶z
U
¶F3
¶x
F3
¶U
¶x

+k

U
¶F2
¶x
+F2
¶U
¶x
U
¶F1
¶y
F1
¶U
¶y

Nowwejustre-orderthe12termssothatthesixwithaU
¶Ficomerst,thenthesixwithanFi ¶Ucome
next;andfromthedenitions,itbecomesclearthattheresultis
Ñ(UF)=U(ÑF)+(ÑU)F QED:
Theotherscanbeprovedinasimilarway,thoughitgetsquitelongforEqs.3.3and3.7.Muchshorter
proofscanbegivenusingindexnotation,butthisisnolongeronthesyllabus.
3.6Vectorsecondderivatives:applyingÑtwice
Wealsohaveasecondsetofidentitiesarisingfromapplyingtwoofgrad,divorcurlinsuccession.Heregrad
UandcurlFproducevectorelds,towhicheitherdivorcurlcanbeapplied;whiledivFproducesascalar
eld,andthenwecanapplygradtothat.Thisgivesatotalofveallowedcases,whichareasfollows:
36

10. div(gradU)=Ñ(ÑU)Ñ
2
U (3.8)
curl(gradU)=Ñ(ÑU)=0 (3.9)
div(curlF)=Ñ(ÑF)=0 (3.10)
curl(curlF)=Ñ(ÑF)=Ñ(ÑF)Ñ
2
F (3.11)
grad(divF)=Ñ(ÑF)=Ñ(ÑF)+Ñ
2
F (3.12)
Weseeherethattwoofthesecases(curlgradU,anddivcurlF)areidenticallyzero;thisistrueforany
elds,aslongastheyaresufcientlywellbehavedthatthepartialderivativescommute,seebelow.These
twozerocasescanbehelpfullymemorisedbythefactthattheywouldalsogivezeroifÑwasreplacedbyan
ordinaryvectora;butbeware,thissortofruleisnotapplicabletoeveryequationcontainingÑ.
TherstequationaboveEq.3.8introducesanewoperatorÑ
2
calledtheLaplacian;thisisveryimportant
inawiderangeofphysicalproblems,andwewillmeetitextensivelyinChapter8.Incomponents,as
expectedfromthedenitionofÑ,itissimply
Ñ
2
U
¶
2
¶x
2
U+
¶
2
¶y
2
U+
¶
2
¶z
2
U (3.13)
Thisoperatorcanbeappliedtoeitherascalareldoravectoreld,producingaeldofthesametype;so
Ñ
2
FmeansapplyÑ
2
toeachcomponentofFseparately,giving
Ñ
2
F=iÑ
2
F1+jÑ
2
F2+kÑ
2
F3
soÑ
2
Fisanothervectoreld.
Notethatthelasttwooftheaboveequations3.11and3.12arejustarearrangementofeachother,giving
arelationshipbetweencurlcurlF,graddivFandÑ
2
F.
Alloftherelationsabovecanbeprovedbydirectsubstitution,e.g.:
Proofof3.9:
curl(ÑU)=






i j k
¶=¶x¶=¶y¶=¶z
¶U=¶x¶U=¶y¶U=¶z






=

¶
2
U
¶y¶z

¶
2
U
¶z¶y
;
¶
2
U
¶z¶x

¶
2
U
¶x¶z
;
¶
2
U
¶x¶y

¶
2
U
¶y¶x

=0:
[Note,weassumethatthefunctionUissufcientlywell-behavedforitspartialsecondderivativestocom-
mute.]
TherelationcurlgradU=0isparticularlyuseful,sinceitisofteninterestingtoask,givensomevector
eldF,canwendascalareldUsuchthatÑU=F?Ifwecan,thissimpliesthingsfrom3components
to1.NowwecanseeimmediatelythatifcurlF6=0,itisimpossibletondsuchascalareldU:becausefor
anyU,thevectorelddenedbyHÑUwillhavecurlH=0,thereforeH6=F.However,wewillshowin
thenextchapterthatifcurlF=0everywhereinagivendomainthenwecanndsuchascalareldU,and
we'llalsoshowhowtoconstructthedesiredUwithasuitableintegral.
37