notes56

Lecture5 VectorOperators: Grad,DivandCurl Inthefirstlecture ofthesecond partofthiscourse wemovemore toconsider properties offields. Weintroduce three field operators which revealinteresting collecti vefield properties, viz.thegradient ofascalar field,thedivergence ofavector field, andthecurl ofavector field. There aretwopoints togetoverabout each:Themechanics oftaking thegrad, divorcurl, forwhich youwillneed tobrush up your multi variate calculus.Theunderlying physical meaning —thatis,whytheyareworthbothering about. InLecture 6wewilllook atcombining these vector operators. 5.1 The gradient ofascalar field Recall thediscussion oftemperature distrib ution throughout aroom intheovervie w, where wewondered howascalar would varyaswemovedoffinanarbitrary direction. Here wefindouthow. If
   isascalar field, ieascalar function ofposition    in3dimen- sions, then itsgradient atanypoint isdefined inCartesian co-ordinates by
 
 ! 
 "# 
  $&% Itisusual todefine thevector operator which iscalled “del” or“nabla”'     "   $  % 59 60 LECTURE 5.VECTOROPERATORS:GRAD,DIVANDCURL Then

'
% Note immediately that '
isavector field! Without thinking toocarefully about it,wecanseethatthegradient ofascalar field tends topoint inthedirection ofgreatest change ofthefield. Later wewill bemore precise.Workedexamples ofgradient evaluation 1.
  '
   !   "#   $    % 2.
         '
      "#   $            "  $  % 3.
  ,where isconstant. '
     "   $           "  $ % 4.
   ,where    
isafunction of alone so  exists. As
    also,       !  &   % '
      "   $       "   $" But     ,so #  $andsimilarly for  . '
     "  $  &% (' % 5.2.THESIGNIFICANCE OFGRAD 61 PSfragreplacementsgrad


 

 
 
Figure5.1:Thedirectional derivative 5.2 The significance ofgrad Ifourcurrent position is insome scalar field
(Fig. 5.1), andwemoveaninfinitesi- maldistance  ,weknowthatthechange in
is
 
  
  
  % Butweknowthat       "   $  and '
   
   " 
   $ 
  , sothatthechange in
isalsogivenbythescalar product
 '
   % Nowdivide both sides by 
  '
   % Butremember that   ,so  isaunitvector inthedirection of . This result canbeparaphrased as: 
hastheproperty thattherateofchange of
wrtdistance ina particular direction (  )istheprojection of 
onto thatdirection (or thecomponent of  
inthatdirection). Thequantity 
iscalled adirectional derivative,butnote thatingeneral ithasa different valueforeach direction, andsohasnomeaning until youspecify thedirection. Wecould alsosaythat 62 LECTURE 5.VECTOROPERATORS:GRAD,DIVANDCURLAtanypoint P, 
points inthedirection ofgreatest change of
at P,andhasmagnitude equal totherateofchange of
wrtdistance inthat direction. −4−2024 −4−202400.020.040.060.080.1 Another nice property emer gesifwethink ofasurfaceofconstant
–thatisthelocus   for
   
   % Ifwemoveatinyamount within that iso-
surface, there isnochange in
,so
  .Soforany   inthesurface'
    % But   isatangent tothesurface,sothisresult showsthat 
iseverywhere NORMAL toasurfaceofconstant
. Surface of constant UgradU Surface of constant U These are called Level Surfaces 5.3.THEDIVERGENCE OFAVECTORFIELD 63 5.3 The divergence ofavector field Thedivergence computes ascalar quantity from avector field bydifferentiation. If     isavector function ofposition in3dimensions, thatis 
 
 "
 $, then itsdivergence atanypoint isdefined inCartesian co-ordinates by  
 
 
 Wecanwrite thisinasimplified notation using ascalar product with the 'vector differential operator:      "   $    ' Notice thatthedivergence ofavector field isascalar field.Examples ofdivergence evaluation div 1)   2)      "  $ 3)     4) ,for constant     Weworkthrough example 3). The component of  $  is  %(   #   ,andweneed tofind  ofit.  %         %                 %         % Theterms in and aresimilar ,sothat                        5.4 The significance of  Consider atypical vector field, water flow,anddenote itby  .This vector has magnitude equal tothemass ofwatercrossing aunitarea perpendicular tothedirection of perunittime. Nowtakeaninfinitesimal volume elementandfigure outthebalance oftheflowofinandoutof. 64 LECTURE 5.VECTOROPERATORS:GRAD,DIVANDCURL Tobespecific, consider thevolume element      inCartesian co-ordinates, andthink firstabout thefaceofarea  perpendicular tothe axis andfacing out- wards inthenegative direction. (That is,theonewith surfacearea
!    ".) dS = -dxdz j y xz dz dx dyj dS = +dxdz Figure5.2:Elemental volumeforcalculating divergence. Thecomponent ofthevector normal tothisfaceis   "
 ,andispointing inwards, andsotheitscontrib ution totheOUTW ARD fluxfrom thissurfaceis    
     where
  means that
isafunction of .(Bytheway,fluxhere denotes mass per unittime.) Asimilar contrib ution, butofopposite sign, willarise from theopposite face, butwe must remember thatwehavemovedalong byanamount  ,sothatthisOUTW ARD amount is
       
 
      Thetotal outw ardamount from these twofaces is
    
 Summing theother faces givesatotal outw ardfluxof
 
 
   '  Soweseethat 5.5.THELAPLACIAN:
   OFASCALAR FIELD 65 Thedivergence ofavector field represents thefluxgeneration perunitvolume ateach point ofthefield. (Divergence because itisaneffluxnotaninflux.) Interestingly wealso sawthatthetotal effluxfrom theinfinitesimal volume wasequal tothefluxintegrated overthesurfaceofthevolume. (NB: Theabovedoes notconstitute arigorous proof oftheassertion because wehave notprovedthatthequantity calculated isindependent oftheco-ordinate system used, butitwillsufficeforourpurposes.) 5.5 The Laplacian:   ofascalar field Recall that 
ofanyscalar field
isavector field. Recall alsothatwecancompute thedivergence ofanyvector field. Sowecancertainly compute   
,evenif wedon’tknowwhat itmeans yet. Here iswhere the 'operator starts tobereally handy .'  '
     "   $        "   $  
      "   $       "   $   
    
 
 
 
  This lastexpression occurs frequently inengineering science (you will meet itnext insolving Laplace’ sEquation inpartial differential equations). Forthisreason, the operator  iscalled the“Laplacian” 
    
Laplace’ sequation itself is 
 66 LECTURE 5.VECTOROPERATORS:GRAD,DIVANDCURLExamples of  
evaluation
 
1)   (   # 2)           3)   Let’sproveexample (3)(which isparticularly significant –canyouguess why?).                          %                     % %              $  Adding upsimilar terms for and        (   (    5.6 The curl ofavector field Sofarwehaveseen theoperator 'applied toascalar field '
;anddotted with a vector field '. Wearenowoverwhelmed byanirrestible temptation tocross itwith avector field '
This givesthecurl ofavector field'
   Wecanfollowthepseudo-determinant recipe forvector products, sothat'
    " $  


            
  
   
  
  " 
  
  $ 5.7.THESIGNFICANCE OFCURL 67Examples ofcurl evaluation '
1)     "  $ 2)    $ (      " 5.7 The signficance ofcurl Perhaps thefirstexample givesaclue. Thefield      "issketched inFigure 5.3(a). (Itisthefield youwould calculate asthevelocity field ofanobject rotating with    .)This field hasacurlof  $,which isinther-hscrewsense outofthe page. Youcanalso seethatafield likethismust giveafinite value tothelineintegral around thecomplete loop
    % y x ax(y)a(x)yax(y+dy) ay(x+dx) dxdyy yx x+dxy+dy (a) (b) Figure5.3:(a)Aroughsketchofthevector®eld   .(b)Anelementinwhichtocalculate curl. Infactcurlisclosely related tothelineintegralaround aloop. Thecirculation ofavectorround anyclosed curve isdefined tobe
"   andthecurl ofthevector fieldrepresents thevorticity ,orcirculation per unit area,ofthefield. 68 LECTURE 5.VECTOROPERATORS:GRAD,DIVANDCURL Ourproof uses thesmall rectangular element  by  showninFigure 5.3(b). Con- sider thecirculation round theperimeter ofarectangular element. Thefields inthe direction atthebottom andtopare
    
     
  
    where
   denotes
isafunction of ,andthefields inthe direction attheleft andright are
    
    
  
  Starting atthebottom andworking round intheanticlockwise sense, thefour contrib u- tions tothecirculation  aretherefore asfollows,where theminus signs takeaccount ofthepath being opposed tothefield:  
      
     
       
    
     
#
   
    

  
    
     
  
     '
 
where     $. NB: Again,thisisnotacompletely rigorous proof aswehavenotshownthattheresult isindependent oftheco-ordinate system used. 5.8 Some definitions involving div,curl andgradAvector field with zero divergence issaidtobesolenoidal .Avector field with zero curlissaidtobeirrotational .Ascalar field with zero gradient issaidtobe,er,constant . Revised Sep2005 Lecture6 VectorOperator Identities Inthislecture welook atmore complicated identities involving vector operators. The main thing toappreciate itthattheoperators behaveboth asvectors andasdifferential operators, sothattheusual rules oftaking thederivativeof,say,aproduct must be observ ed. There could beacottage industry inventing vector identities. HLTcontains alotof them. Sowhynotleaveitatthat? First, since ,  and describe keyaspects ofvectors fields, theyarise often inpractice, andsotheidentities cansaveyou alotoftime andhacking ofpartial derivatives,aswewillseewhen weconsider Maxwell’ sequation asanexample later. Secondly ,theyhelp toidentify other practically important vector operators. So,al- though thismaterial isabitdry,therelevance oftheidentities should become clear later inother Engineering courses. 6.1 Identity 1:curl grad 
'
&'
    " $     
  
  
     
   
    "   $    as      . Note thattheoutput isanullvector . 69 70 LECTURE 6.VECTOROPERATORIDENTITIES 6.2 Identity 2:divcurl
' '
            


  
   
   
  
  
   
  6.3 Identity 3:divandcurl of Suppose that
 isascalar field andthat  isavector field andweareinterested intheproduct
.This isavector field, sowecancompute itsdivergence andcurl. Forexample thedensity   ofafluid isascalar field, andtheinstantaneous velocity ofthefluid   isavector field, andweareprobably interested inmass flowrates for which wewillbeinterested in     . Thedivergence (ascalar) oftheproduct
isgivenby:' 

 '  '

 
 Inasimilar way,wecantakethecurlofthevector field
,andtheresult should bea vector field:'


'
 '

% 6.4 Identity 4:divof   Life quickly gets trickier when vector orscalar products areinvolved:Forexample, it isnotthatobvious that 
 
        Toshowthis, usethedeterminant:      


     
  
   
  
   
  
   % % %         % % %      curl  6.5.IDENTITY 5:
   71 6.5 Identity 5:         
     " $     
  
 
  
 
  
   sothe component is 
  
    
  
  which canbewritten asthesum offour terms:
     
 
      
  
 
   Adding
     tothefirstofthese, andsubtracting itfrom thelast, anddoing the same with   
   totheother twoterms, wefindthat(you should ofcourse check this):'

   '    '    '    '   where   ' canberegarded asnew,andveryuseful, scalar differential operator . 6.6 Definition oftheoperator    This isascalaroperator,butitcanobviously canbeapplied toascalar field, resulting inascalar field, ortoavector field resulting inavector field:  ' 

 
 
  % 6.7 Identity 6:     foryoutoderive Thefollowing important identity isstated, andleftasanexercise:         where   
    
  "  
  $ 72 LECTURE 6.VECTOROPERATORIDENTITIESExample ofIdentity 6:electr omagnetic waves Q:James Clerk Maxwell established asetoffour vector equations which arefunda- mental toworking outhoweletromagnetic wavespropag ate.Theentire telecom- munications industry isbuiltonthese.
          Inaddition, wecanassume thefollowing, which should allbefamiliar toyou:  , ! ,   , where allthescalars areconstants. Nowshowthatinamaterial with zero freechargedensity ,! ,andwith zero conducti vity, 
,theelectric field must beasolution ofthewaveequation          % A:First, abitofrespect. Imagine youarethefirsttodothis—thisisatingle moment. 
           %
        
                           Butweknow(orrather youworkedoutinIdentity 6)that     ,andusing (c)      
       sointerchanging theorder ofpartial differentation, andusing (a)  
 :                     This equation isactually three equations, oneforeach component:        andsoonfor and . 6.8.GRAD,DIV,CURLAND
INCURVILINEAR CO-ORDIN ATESYSTEMS 73 6.8 Grad, div,curl and incurvilinear co-ordinate systems Itispossible toobtain general expressions forgrad, divandcurl inanyorthogonal curvilinear co-ordinate system bymaking useofthefactors which were introduced inLecture 4. Werecall thattheunitvector inthedirection ofincreasing ,with and being kept constant, is     whereistheposition vector ,and     isthemetric coefficient. Similar expressions apply fortheother co-ordinate directions. Then           % 6.9 Grad incurvilinear coordinates Noting that

 and

   ,andusing theproperties ofthegradient of ascalar field obtained previously'
    
 
  
  
  Itfollowsthat'
            
  
  
  Theonly waythiscanbesatisfied forindependent,, iswhen'
  
     
    
   6.10 Divergence incurvilinear coordinates Expressions canbeobtained forthedivergence ofavector field inorthogonal curvilin- earco-ordinates bymaking useofthefluxproperty . Weconsider anelement ofvolume .Ifthecurvilinear coordinates areorthogonal then thelittle volume isacuboid (tofirstorder insmall quantities) and      % 74 LECTURE 6.VECTOROPERATORIDENTITIES w uy The scale params areh (v) dw h (v+dv) dw h (v+dv) duh (v) duuw w u h dvv functions of u,v,w Figure6.1:Elemental volumeforcalculating divergenceinorthogonal curvilinear coordinates However,itisnotquite acuboid: thearea oftwoopposite faces willdifferasthescale parameters arefunctions of,and ingeneral. Sotheneteffluxfrom thetwofaces inthe  direction showninFigure 6.1is

 
  
   
      
      
       which iseasily shownbymultiplying thefirstlineoutanddropping second order terms (i.e.  !). Bydefinition divistheneteffluxperunitvolume, sosumming uptheother faces:   
      
   
              
      
   
       So,finally ,       
      
     
     6.11.CURLINCURVILINEAR COORDIN ATES 75 6.11 Curl incurvilinear coordinates Recall from Lecture 5thatwecomputed the component ofcurlasthecirculation per unitarea from  
  
    Byanalogy with ourderivation ofdivergence, youwillrealize thatforanorthogonal curvilinear coordinate system wecanwrite thearea as      .Buttheopposite sides arenolonger quite ofthesame length. The lowerofthepair inFigure 6.2is length  ,buttheupper isoflength    y y aa uu u+duu u uv+dv(v+dv) h (v+dv) du dv (v)h (v) du Figure6.2:Elemental loopforcalculating curlinorthogonal curvilinear coordinates Summing thispairgivesacontrib ution tothecirculation
     
         
    andtogether with theother pair:   
   
   Sothecirculation perunitarea is           
   
   76 LECTURE 6.VECTOROPERATORIDENTITIES andhence curlis          
   
        
    
        
    
    Youshould check thatthiscanbewritten as Curl incurvilinear coords:                     
  
 
  6.12 The Laplacian incurvilinear coordinates Substitution ofthecomponents of  
intotheexpression for immediately (!*?) givesthefollowing expression fortheLaplacian ingeneral orthogonal co-ordinates: 
   
    
      
        
 % 6.13 Grad Div,Curl, incylindrical polars Here     
 .Theposition vector is    
    
 "  $,and      ,etc.    
 
      
  
    
 
    

 
  $    
 

 
     

 
   
  
   
   

 $ 
          6.14.GRADDIV,CURL,
INSPHERICAL POLARS 77 6.14 Grad Div,Curl, inspherical polars Here     

.Theposition vector is     
      
 "    $.      
  
          
  
             
 
   
 
    
      

       
    
        

        
     

      

  
         
    
 
          Examples Q1Find  in(i)Cartesians and(ii)Spherical polars when      "  $ . A1(i)InCartesians      " $              "  $ % (ii)Inspherical polars,    
and     "  $ .So     
 
    

  
  % Hence as      "!$#
% &('*) ,+!$#
- '/.10  2* 3,+!$#54 ') !$#(% 6('/.10  7 89!$#
- 5'*) 9!$#(4 '/. 78 LECTURE 6.VECTOROPERATORIDENTITIES       
   
        
         
    
    
     
Checking: these tworesults should bethesame, buttocheck weneed expressions for   interms of etc. Remember thatwecanworkouttheunit vectors  andsooninterms of  etc using               
       "  $ % Grinding through wefind
   
  
   
   
   
   
 

  " $   
  " $  Don’ tbeshock edtoseearotation matrix :weareafter allrotating oneright- handed orthogonal coord system intoanother . Sotheresult inspherical polars is      
    
 "   $   
 
  
 "   
    "     
 $   "  $ which isexactly theresult inCartesians. Q2Find thedivergence ofthevector field   where isaconstant vector (i) using Cartesian coordinates and(ii)using Spherical Polar coordinates. A2(i)Using Cartesian coords:           % % %  %         % % %    % 6.14.GRADDIV,CURL,
INSPHERICAL POLARS 79 (ii)Using Spherical polars 
 
  
  andourfirsttaskistofind
andsoon.Wecan’tdothisbyinspection, andfinding their values requires more workthan youmight think! Recall
   
  
   
   
  
   


  " $    
  " $  Nowthepoint isthesame point inspace whate verthecoordinate system, so
 
  
  
  
  "
  $ andusing theinner product



 

   



 

  " $ 



 
 
  " $ 



 

  " $ 



 
  



 




 




 
  




   



  Forourparticular problem,
   ,etc,where isaconstant, sonowwecan write down
    
(    
(  
    
(   
   
    
( 
( 80 LECTURE 6.VECTOROPERATORIDENTITIES Nowallweneed todoistobash out    
    
       

Inglorious detail thisis    
(    
(         
(  
          
  
( Abitmore bashing andyou’llfind     
    
      This isEXA CTL Ywhat youworkedoutbeforeofcourse. Takehome messages fromthese examples:Justasphysical vectors areindependent oftheir coordinate systems, soarediffer- ential operators.Don’ tforgetabout thevector geometry youdidinthe1styear.Rotation matrices areuseful!Spherical polars were NOTagood coordinate system inwhich tothink about this problem. Letthesymmetry guide you. Revised Sep2005