r Curl is associated w/rotation X F


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1 13.5 ul nd ivegence ul is ssocited w/ottion X F ivegence is F Tody we define two opetions tht cn e pefomed on vecto fields tht ply sic ole in the pplictions of vecto clculus to fluid flow, electicity, nd mgnetism. Hs nyone studied these in othe couses? If so, hve you hed of cul o divegence of vecto? ul F They nme it cul ecuse the cul vecto (nd it is vecto) is ssocited with ottions. Let s sy F epesents the velocity field in fluid flow. Pticles ne point (,, ) the xis tht points in the diection of cul F ( xyz,, ) how quickly the pticles move ound the xis tht points in the diection of the cul F ( xyz,, ) x yz in the fluid tend to otte out. The length of this cul vecto is mesue of. ( x, y, z ) cul F ( xyz,, ) Swiling fluid Hee s n exmple of vecto field u F with no ottion. Fluid does u not otte ne the u point. Exmple: F = x, y,0, culf = 0= 0,0,0 If cul F = 0 t point, then the fluid is fee fom ottions t tht point nd F is clled iottionl. So thee is just no otting o culing going on. (In othe wods, thee is no whilpool o eddy t tht point) A tiny pddle wheel moves with the fluid ut doesn t otte out its xis. (The cul of gdient vecto field, f, is 0, thee e NO ottions going on) Hee s n exmple of vecto field u u u F with ottion. Exmple: F = y, x,0, culf = 0,0, 2 uling you ight hnd in the diection of the flow lines, you thum points into the pge 2 k o 0,0, 2. If cul F 0, the pddle wheel ottes out its xis nd cetes ottions o culing.
2 Pticles otte the fstest on the plne tht is pependicul to the cul vecto. cul F = X F. It s the diection the pticles otte the fstest. Anywhee else on this field pticles my otte, just not the fstest. To ememe how to find the cul F think of del (of nothing; n empty del) with the vecto components of u F. ul nd oss poduct oth stt with the lette so it will e esy to ememe. =,, F = PQR,, nd F is just ou vecto field. cul F = X F ul is oss;, esy to ememe It s 3 3 deteminnt with twist. Rememe the negtive in font of j. cul F = i j k P Q R ( R) ( Q) ( R) ( P) ( Q) ( P) = i j + k y z x z x y So R Q P R Q P cul F =,, y z z x x y signs switch It s vecto!! HW # 2: =. Find the cul F of the vecto field F ( x, y, z) x yz, xy z, xyz
3 Theoem 3: Pge 942 If f is function of thee viles (tht hs continuous secondode ptil deivtives), then ( f ) cul = 0. The cul of gdient vecto field is zeo ecuse the is no ottionl movement going on ecuse the gdient of f is lwys pependicul to f. So no swiling movement. Rememe gdient of f, f, is vecto field, so you cn compute its cul (ecuse you tke the cul of vecto fields not scls). It s 0 o 0,0,0 ecuse think of the diection of f (pependicul to f ), so no culing o otting is hppening. Let s pove it: cul cul F = X F, so ( f ) = X f Set up the 3 x 3 deteminent: = i j k x y z f f f f f f f = fx, fy, fz o,, f f f f f f i y( z) z( y) j = x( z) z( x) k + x( y) y( x) = i 0 j 0 + k 0 ( ) ( ) ( ) = 0,0,0 so ( f ) cul = 0, 0, 0 And since consevtive vecto field is one whee F = f If F is consevtive, then cul F = 0., Theoem 3 cn e ephsed s: So, if the cul F 0, then (y Theoem 3) F is not consevtive vecto field. This will e get to use! The convese of Theoem 3 is not genelly tue, ut if u F is vecto field defined on ll of 3 d dimension u u whose component functions hve continuous ptil deivtives nd cul F = 0, then F is consevtive vecto field. Ou vectos fields will hold tue to these conditions. If F is consevtive vecto field, then cul F = 0. If the cul F = 0 (nd the domin of u F is in 3dim.), u F is consevtive vecto field. Now we cn use this (find the cul F ) to test whethe o not vecto field in 3dimensions is consevtive o not. If it is, we cn find potentil function f such tht F = f.
4 Exmple 1: etemine whethe o not the vecto field is consevtive. If it is, find function f so tht F = f F xyz,, = 3 z, cos y, 2xz ( ) 2. If cul F = 0, F is consevtive (we cn find f, potentil function) If cul F 0, F is not consevtive (we cn t find potentil fn. f ) At this point, eed wht cul F is (fist pge of notes) Then go on to divegence.
5 ivegence (o flux density) (net flow of fluid out of smll ox centeed t ( x, yz), ) If F ( xyz,, ) the mss of fluid flowing fom the point ( x, yz, ) pe unit volume. is the velocity of fluid, then div F epesents the net te of chnge (with espect to time) of In othe wods, it mesues the tendency of the fluid to divege fom the point. Tke point in the velocity field nd the div F gives the tendency to flow though it. This u exmple fom u ou fist pge of notes hs divegence of 2. F = x, y,0, divf = 2 lled souce. If div F > 0 t tht point, fluid is flowing wy fom the point t fste te thn is flowing in towds the point. It is dining quickly. It s diveging fom the point. Net flow outwd is clled souce. (Exmple: The expnding cloud of mtte in the Big Bng Theoy of the oigin of the univese.)  the vectos tht end ne the point e shote thn those tht stt ne the point  Expnding fluid o gs.  vecto field shows diection nd speed of the fluid. Mesues the tendency of the fluid to divege fom the point.
6 If div F < 0 t tht point, fluid is flowing towds the point t fste te thn it s leving. It s dining slowly like sink.  Incoming ows (vectos) e longe thn outgoing ows. The net flow inwd is clled sink.  ompessing fluid o gs This u is n exmple u fom ou 1 st pge of notes hs divegence of 0. F = y, x,0, divf = 0 Pictue ox sitting in vecto field nd sk youself to estimte the mount of fluid going in nd coming out of the ox. Hee the mount going in nd coming out would equl ech othe so you get zeo. If divf = 0 t tht point, then F is sid to e incompessile. The vectos though the point e siclly the sme. Mgnetic fields e divegent fee.  Incoming nd outgoing ows (vectos) e siclly the sme  Incompessile
7 ivegence If F = PQR,, we cn think of div F = F div F =,, P, QR, ( P) ( Q) ( R) div F = + + ivegence, ot poduct, esy to ememe P Q R div F = + + (You tke the divegence of vecto field. The nswe is scl function, not vecto function.) HW #2: = Find the divegence of the vecto field F ( x, y, z) x yz, xy z, xyz If F is vecto field on 3dimensions, then the cul F le to compute its divegence. Let s do it: div ( cul F ) div ( X F div F = F ) ( X F ) is lso vecto field on 3dim.. So we should e R Q P R Q P,,,, y z z x x y = = R Q P R Q P + + x y z y z x z x y R Q P R Q P + + x y x z y z y x z x z y = 0 ecuse eveything cncels div cul F = 0 So ( ) just 0 not 0, 0, 0 not vecto div Theoem 11 (pge 944) So thee is zeo tendency of the fluid to divege fom the point. ( cul F ) the mount it's otting ound = 0
8 div cul F = 0 So, the ( ) mens tht the divegence of the mount of ottion doesn t ffect the net flow in o out. Whteve goes out, comes ck nd = 0. No net flow in o out. At this point eed div F onsevtive Vecto Fields F( x, y, z) u If ll thee components hve continuous 1 st ptil deivtives thoughout ll of 3dim. nd F( x, y, z) consevtive then we lso know: the line integl, u F d, is independent of pth Fo evey piecewisesmooth closed cuve, u F d = 0 cul u u F = XF = 0 o 0,0,0 F is clled iottionl u F( x, y, z) u is gdient vecto field so F = f fo some potentil function f. u is Vecto Foms of Geen s Theoem The cul nd divegence opetos llow us to ewite Geen s Theoem in vesions tht will e useful lte. Q P  F d = x y da Geen s Theoem. u F = P x, y, Q x, y Note: ( ) ( ) is vecto field on 2dim., u In 3dim. with thid component 0, F = P( x, y, Q( x, y), 0, we hve Pdx+ Qdy. cul F = i j k Q  = 0, 0, x Pxy (, ) Qxy (, ) 0 so now we cn wite Geen s Theoem s P y Those two ptils w/espect to z=0 ecuse thee e only x s nd y s in P&Q.. F d = cul F 0,0,1 da ( ) * wok ( cul ) F d = = F k da o
9 u u u F d = F t t dt = F T ds = = cul F k da? Rememe fom 13.2, Wok = ( ( )) ( ) now ( ) This expesses the line integl of the tngentil ( T u ) integl of the veticl component of the cul F component of F long cuve s the doule ove the egion (enclosed y cuve ). Hee s visuliztion in 2dimensions: uve is in lck is smooth closed loop, vecto field F is in lue nd t cetin points on cuve in geen, nd the unit tngent vecto T is in ed. Pticle is moving counteclockwise. ot poducts F T e shown(tngentil comp) (If they don t ememe the tngentil component, go ck to 13.2 notes nd explin it; pge 10 o so.) Now we e looking t diffeent dot poduct. A diffeent component of F long cuve. Wht is 0,0,1 o k physiclly on? In ny hoizontl plne like the xyplne z=0 o z=1, ll of the noml vectos point stight up, the n s point stight up. n T u n n n n = 0,0,1 n T u T u The veticl component of the cul F ove the egion e the unit n oml vectos to, 0,0,1. Stoke s Theoem in section 13.7 shifts up fom egion to sufce S nd sys u F d = ( cul F ) d S = ( cul F ) n ds Note the diffeence etween Stoke s S S F d = wok = F k da nd Geen s Theoem. ( cul ) In Stoke s now, these will e the n oml vectos to the sufce S insted of the n oml vectos 0,0,1 to flt egion in Geen s.
10 We ll intoduce sufce integls in 13.6 nd then use them with Stoke s Theoem in We expessed the line integl of the tngentil ( T u ) of the veticl component of the cul F component of F long cuve s the doule integl u ove the egion with F T ds. We lso deive simil fomul involving n (t), the noml component of F (not the n oml vectos to egion ) insted of the tngentil component of F. Eveything is lying flt in 2dim. nd the noml vectos e pependicul to the tngent vectos. We e on cuve. n T u n F nds F x y da * = div (, ) n The line integl of the noml component of F long cuve is equl to the doule integl of the divegence of F ove the egion enclosed y. *You ll see this show up in the ivegence Theoem in section 13.8 (ut shifted up fom line integl ove cuve, to sufce integl ove sufce S). Hee s how: If is given y the vecto eqution () t = x( t), y( t), t then the unit tngent vecto then () t = x () t, y () t (sect. 10.2) x () t y () t so T() t =, ( t) () t T () t () t = t u Since T() t nd nt () e pependicul to one nothe they e opposite ecipocls of one nothe, so the y () t x' () t outwd unit noml vecto to is n() t =, ' t ' t () () ( ) n u T( t) slope nt () F = P x t, y t, Q x t, y t = P, Q ( ( ) ( )) ( ( ) ( )) F nds= ( F n)() t () t dt This is shothnd fo this. F dot n s of t ds 2 2 slope () () () () y t x' t fo ese nd n() t =, the dot poduct ' t ' t
11 () Q x () t () t dt fcto out '( t) () t () t P y t = nd educe () () = P y t Q x t dt nottionl chnge dy dx = P Q dt dt dt cncel dt s = P dy Qdx P Q = + da x y This is just the div F By Geen s Theoem, if Pdx Qdy Q P + = x y da c ( ) then Q P Pdy Qdx= + da y x = div F da F n ds F x y da So = div (, ) div F = F =,, x y P Q P Q = + x y So, gin, this sys tht the line integl of the noml component of F long is equl to the doule integl of the divegence of F ove the egion enclosed y. In 13.8 the ivegence Theoem will e shifted up to S n n F n ds = div F dv. See wht I men out shifting up? Oveview of 13.5 cul F  ssocited with ottion, vecto function Pticles ne point ( x, yz, ) tend to otte out the xis tht points in the diection of cul F ( xyz,, ) The length of this cul vecto is mesue of how quickly the pticles move ound the xis. cul F = X F  cosspoduct cul, cosspoduct, culing (ottion): ll c s If cul F = 0, then F is consevtive vecto field nd F is clled iottionl. Then you cn find potentil function f so tht F = f If cul F 0, then F is NOT consevtive. E n n.
12 ivegence F  scl function div F  mesues the tendency of fluid to divege fom point div F = F  dot poduct ivegence, ot Poduct, Tendency to ivege: ll s If div F = 0, then F is clled incompessile. This eltes line integl of F 's tngentil component to the cul of F : u F d = F T ds = cul F k da ( ) This eltes the line integl of F 's noml vecto to the divegence of F. F nds= div F( xy, ) da *Hints fo HW #12 A L: We tke the cul of vecto fields (not scl) nd the esults e vectos themselves. We tke the div of vecto fields (not scl) nd the esults e scl functions. Thinking it though: So line integl ove sufce f(x,y) mesued in sucs ds (e of cutin o side of fence) f ( xy, ) ds = f( ( t) ) () t dt Smooth cuve is given y vecto function t (), t. c egul line integl c f ( xyz,, ) ds = f( ( t) ) () t dt djustment fo c length = t compct vecto nottion ove solid, mesued fo ( ) in sucs, lose visul A line integl ove vecto fields (wok ppliction) F d F ( () t ) = () t dt = F T ds c vecto field wok ppliction c
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