Chapter 1 Vector Analysis 1.1 Vector Algebra: Vector Operations (I)

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1 Chpte Vecto nlsis. Vecto le:.. Vecto Opetions I Vectos: Quntities he oth mnitude nd diection, denoted oldfce,, nd so on. cls: Quntities he mnitude ut no diection denoted odin tpe. In dims, ectos e denoted ows: the lenth of the ow is popotionl to the mnitude of the ecto, nd the owhed indictes its diection. Minus - is ecto with the sme mnitude s ut of opposite diection. Vectos he mnitude nd diection ut not loction... Vecto Opetions II i ddition of two ectos: Plce the til of t the hed of. Commuttie: ssocitie: CC Vecto Opetions III.. Vecto Opetions IV ii Multipliction scl: Multiplies the mnitude ut lees the diection unchned. Distiutie: iii Dot poduct of two ecto scl poduct: he dot poduct of two ectos is defined cosθ, whee θ is the nle the fom when plced til-to-til. Commuttie: Distiutie: C C θ i Coss poduct of two ecto ecto poduct: he coss poduct of two ectos is defined sinθ n, whee is unit ecto pointin pependicul to the plne of nd. ht is used to desinte the unit ecto nd its diection is detemined the iht-hnd ule. Distiutie: C C not commuttie: - n θ 4

2 .. Vecto le: Component fom I, ŷ nd ẑ Let e unit ectos pllel to the,, nd es, espectiel. n it ecto cn e epessed in tems of these sis ectos. ŷ ẑ he numes,, nd e clled components... Vecto le: Component fom II Refomulte the fou ecto opetions s ule fo mnipultin components: i o dd ectos, dd like components. ŷ ẑ ŷ ẑ ŷ ẑ ii o multipl scl, multipl ech component. ŷ ẑ ŷ ẑ Vecto le: Component fom III iii o clculte the dot poduct, multipl like components, nd dd. ŷ ẑ ŷ ẑ i o clculte the coss poduct, fom the deteminnt whose fist ow is, ŷ nd ẑ, whose second ow is in component fom, nd whose thid ow is. ŷ ẑ ŷ ẑ 7.. iple Poducts I ince the coss poduct of two ectos is itself ecto, it cn e dotted o cossed with thid ecto to fom tiple poduct. i cl tiple poduct: C. Geometicll, C is the olume of pllelepiped eneted these thee ectos s shown elow. C C C In component fom C C C C 8

3 .. iple Poducts II ii Vecto tiple poduct: C. he ecto tiple poduct cn e simplified the so-clled C-C ule. Notice tht C C C C C C C C C Polem.6 Unde wht conditions does C C? ns: Eithe is pllel to C,..4 Position, Displcement, nd eption Vectos I Position ecto: he ecto to tht point fom the oiin. Its mnitude the distnce fom the oiin Its diection unit ecto pointin dill outwd o is pependicul to nd C 9 he infinitesiml displcement ecto, fom,, to d, d, d, is d l d d d..4 Position, Displcement, nd eption Vectos II In electodnmics one fequentl encountes polems inolin two points: souce point,, whee n electic field is locted field point,, t which ou e clcultin the electic field. Diffeentil Clculus.. Odin Deities uppose we he function of one ile, f. Wht does the deitie, df/d, do fo us? shot-hnd nottion fo the seption ecto fom the souce point to the field point is, mnitude unit ecto in the diection fom to is ns: It tells us how pidl the function f ies when we chne the ument tin mount, d. df df d d In wods, if we chne n mount d, then, f chnes n mount df. he deitie df/d is the slope of the ph of f esus.

4 .. Gdient I uppose we he function of thee iles. Wht does the deitie men in this cse? theoem on ptil deities sttes tht H H H dh d d d H H H d d d H dl mountin hill H,, he dient of H is ecto quntit, with thee components. H H H H.. Gdient II Geometicl intepettion: Like n ecto, the dient hs mnitude nd diection. dot poduct in stct fom is: dh H dl H dl cosθ wheeθ is the nle etween H nd dl. he dient H points in the diection of mimum incese of the function H. nloous to the deitie of one ile, nishin deitie sinls mimum, minimum, o n inflection. 4 Emple. & Polem. Emple. Find the dient of ns : Polem. Let how tht? ' ' ' [ ' ' ' ' ' ' ' ' '? ' ' ' [ ' ' ']/ ] 5.. he Opeto I he dient hs the foml ppence of ecto,, multiplin, scl H. H H del is ecto opeto tht cts upon H, not ecto tht multiplies H. mimics the ehio of n odin ecto in itull ee w, if we tnslte multipl ct upon. It is melous piece of nottionl simplifiction. 6

5 7.. he Opeto II n odin ecto cn e multipl in thee ws:. Multipl scl :. Multipl nothe ecto dot poduct:. Multipl nothe ecto coss poduct:. On scl function H: H Gdient. On ecto function dot poduct: dieence. On ecto function coss poduct: cul Coespondinl, thee e thee ws the opeto cn ct: 8..4 he Dieence Dieence of ecto is: is mesue of how much the ecto sped out fom the point in question. eo positie positie 9 Emple.4.,, ns : c Emple.4 uppose the functions in oe thee fiues e Clculte thei dieences..,, c..5 he Cul Cul of ecto is: is mesue of how much the ecto cul ound the point in question.

6 Emple.5..6 Poduct Rules I he sum ule: Emple.5 uppose the functions in oe two fiues e, Clculte thei culs. d df d f d d d f f ns : he ule fo multiplin constnt: d d df kf k kf kf d k k k k he poduct ule: d d..6 Poduct Rules II scl : f ecto : f df d f f d d f f f f f f f f f scl : ecto : Chps. 8 nd he quotient ule: d d f f df d..6 Poduct Rules III f f f d d scl : f ecto : 4

7 5..7 econd Deities I pplin twice, we cn constuct fie species of second deities. : dient Dieence of,, fist deities hee : dient of Cul dieence : Gdient of cul : Dieence of 4 : cul of Cul 5 Chps. 8 nd e impotnt lws eo lws eo educe to othes 6..7 econd Deities II he Lplcin of ecto is simil: the Lplcin of he poof hines on the equlit of coss deities:,, 7..7 econd Deities III 4 lws eo Cn we use the followin ecto identit? 5 C C C We will encounte this deitie when delin with the ecto potentil mnetism. 8. Intel Clculus.. Line, ufce, nd Volume I In electodnmics, the line o pth intels, sufce intels o flu, nd olume intels e the most impotnt intels. l P, d Line intels: line intel is n epession of the fom Whee is ecto function, dl is the infinitesiml displcement ecto, nd the intel is to e cied out lon pescied pth P fom point to point. Put cicle on the intel, in the pth in question foms closed loop. l d

8 .. Line, ufce, nd Volume II he lue of line intel depends citicll on the pticul pth tken fom to, ut thee is n impotnt specil clss of ecto functions fo which the line intel is independent of the pth, nd is detemined entiel the end points, e.. W P F dl foce tht hs this popet is clled consetie. Emple.6 Clculte the line intel of the function, fom the point,, to the point,,, lon the pths nd in Fi... Wht is the loop intel tht oes fom to lon nd etuns to lon? he stte hee is to et eethin in tems of one ile. 9.. Line, ufce, nd Volume III ufce intels: line intel is n epession of the fom d, whee is ecto function, nd d is the infinitesiml ptch of e, with diection pependicul to the sufce. he lue of sufce intel depends on the pticul sufce chosen, ut thee is specil clss of ecto functions fo which it is independent of the sufce, nd is detemined entiel the ound. Emple.7 Clculte the sufce intel of the function oe fie sides of the cuicl o. Let upwd nd outwd e the positie diection, s indicted the ow. ol :kin, 5, the sides one t time : d dd, d 4 d dd 4dd d d d dd, d d d 4 6 d dd dd

9 .. Line, ufce, nd Volume IV c Volume intels: line intel is n epession of the fom, whee is scl function, nd is n infinitesiml olume element. In Ctesin coodintes, ddd Fo emple, if is densit of sustnce, then the olume intel would ie the totl mss. Emple.8 Clculte the olume intel of the function oe the pism in Fi..4. ol : Let's do fist to ; then fom to- ; finll fom to. ddd d { d d} he olume intels of ecto functions: 9 9 d he Fundmentl heoem of Clculus Fundmentl theoem of clculus: df d df f f d Geometicl Intepettion: two ws to detemine the totl chne in the function:. o step--step ddin up ll the tin incements s ou o. sutct the lues t the ends... he Fundmentl heoem fo Gdients scl function of thee iles,, chnes smll mount. d dl he totl chne in in oin fom to lon the pth selected is: dl Fundmentl theoem fo dient. he intel of deitie oe n intel is ien the lue of the function t the end points ound. 5 Geometicl Intepettion: Mesue the hih of skscpe.. Mesue the hih of ech floo nd dd them ll up.. Plce n ltimete t the top nd the ottom, sutct the edins t the ends. 6

10 .. he Fundmentl heoem fo Gdients II dl the iht side of this eqution mkes no efeence to the pth---onl to the end points. hus dients he specil popet tht thei line intels e pth independent. Cooll : dl is independent of pth tken fom to. Cooll : dl, since the einnin nd end points e identicl, nd hence -. consetie foce m e ssocited with scl potentil ene function, whees non-consetie foce cnnot. 7 Potentil Ene nd Consetie Foces Potentil ene defined in tems of wok done the ssocited consetie foce. U U F ds *Consetie foces tend to minimie the potentil ene within n sstem: It llowed to, n pple flls to the ound nd spin etuns to its ntul lenth. Non-consetie foce does not impl it is dissiptie, fo emple, mnetic foce, nd lso does not men it will decese the potentil ene, such s hnd foce. c 8 Distinction etween Consetie nd Non-consetie Foces he distinction etween consetie nd nonconsetie foces is est stted s follows: consetie foce m e ssocited with scl potentil ene function, whees non-consetie foce cnnot. U U Fc U F ds c 9 Consetie Foce nd Potentil Ene Function How cn we find consetie foce if the ssocited potentil ene function is ien? consetie foce cn e deied fom scl potentil ene function. Fc U he netie sin indictes tht the foce points in the diection of decesin potentil ene. Git U pin U sp m; k ; du F m d du sp F k d 4

11 ..4 he Fundmentl heoem fo Dieences he fundmentl theoem fo dieences sttes tht: he intetion of deitie in this cse the dieence oe eion in this cse olume is equl to the lue of the function t the ound in this cse the sufce tht ounds the olume d his theoem hs t lest thee specil nmes: Guss s theoem, Geen s theoem, o the dieence theoem. Geometicl Intepettion: Mesue the totl mount of fluid pssin out thouh the sufce, pe unit time.. Count up ll the fucets, ecodin how much ech put out.. Go ound the ound, mesuin the flow t ech point, upplement Guss s dieence theoem nsfomtion etween olume intels nd sufce intels n d Rioous poof cn e found in: Ewin Kesi, dnced Enineein Mthemtics John Wile nd ons, New nd dd it ll up. 4 n d Yok, 99, 7th ed. Chp. 9, pp Rouh poof: nd n cosα cos β cosγ whee α, β, nd γ e the nles etween n nd -, - nd - is, espectiel. d τ ddd dd dd dd cosα cos β cos γ d Emple. Check the dieence theoem usin the function nd the unit cue situted t the oiin. ol : In this cse ddd d d dd d o elute the sufce intel we must conside septel the si sides of the cue. he totl flu is..5 he Fundmentl heoem fo Culs I he fundmentl theoem fo culs---tokes theoem--- sttes tht: d dl he intetion of deitie hee, the cul oe eion hee, ptch of sufce is equl to the lue of the function t the ound in this cse the peimete of the ptch. Geometicl Intepettion: Mesue the twist of the ectos ; eion of hih cul is whilpool. P 4 44

12 ..5 he Fundmentl heoem fo Culs II miuit in tokes theoem: Concenin the ound line intel, which w e we supposed to o ound clockwise o counteclockwise? he iht-hnd ule. Cooll : d depends onl on the ound lines, not on the pticul sufce used. Cooll : d fo n closed sufce, since the ound line shinks down to point. upplement tokes theoem nsfomtion etween sufce intels nd line intels d Rioous poof cn e found in: Ewin Kesi, dnced Enineein Mthemtics John Wile nd ons, New Yok, 99, 7th ed. Chp. 9, pp P dl hese coollies e nloous to those fo the dient theoem Comments: dute leel efeence onl Geen s theoems: Let f f f f n f n Geen's fist fomul: f f f d n f Geen's second fomul: f f f d n n Geen s theoem in the plne s specil cse of tokes theoem Let e ecto function in the -plne. n d d d P 47 Emple. uppose 4 Check tokes theoem fo the sque sufce shown elow. ol : 4 ; d 4 dd he line intel of d dd 4 the fou sements 48

13 d df d f f d d d Intetin oth sides nd inokin the fundmentl theoem d Left f d f d d df Riht f d d d d df df f d d d d f..6 Intetion Pts f f f Intete it oe olume nd inikin the dieence theoem. Left Riht f f f f f f d f d f f d f not ioous poe Homewok # Polems:.5,.7,.,.6,., Cuiline Coodintes.4. pheicl Pol Coodintes I he spheicl pol coodintes, θ, φ of point P e defined elow; : the distnce fom the oiin the mnitude of the position ecto. θ: the nle down fom the -is clled pol nle. φ: he nle ound fom the -is cll the imuthl nle. sinθ cosφ sinθ sinφ cosθ Mu R pieel, Vecto nlsis.4. pheicl Pol Coodintes II,, θ φ he diection of the coodintes: the unit ecto he constitute n othoonl mutull pependicul sis set just like,,. o n ecto cn e epessed in tems of them: θ θ φ In tems of Ctesin unit ecto McFRW-Hill, New Yok, 989, 6th ed. Chp φ

14 .4. pheicl Pol Coodintes III,, θ φ Wnin: e ssocited with pticul point P, nd the chne diection s P moes ound. Fo emple, lws points dill outwd, ut dill outwd cn e the diection, the diection, o n othe diection, dependin on whee ou e. Notice: ince the unit ectos e function of position, we must hndle the diffeentil nd intel with ce.. Diffeentite ecto tht is epessed in spheicl coodintes.. Do not tke the unit ectos outside n intel pheicl Pol Coodintes IV he enel infinitesiml displcement : dl d dθθ sinθdφφ he infinitesiml sufce element of sphee. d dl θ dl dl φ sinθdθdφ he infinitesiml olume element dl θ dl φ sinθddθdφ d fo the sufce pheicl Pol Coodintes V he ecto deities in spheicl coodintes:.4. Clindicl Coodintes I he clindicl coodintes s, φ, of point P e defined elow: scos φ, ssin φ, s: the distnce fom the is. φ: the sme menin s in spheicl coodintes. : the sme s Ctesin. he unit ectos e he infinitesiml displcement d l dss sdφφ d : 56 57

15 .4. Clindicl Coodintes II he ecto deities in clindicl coodintes:.5 he Dic Delt Function.5. he Dieence of / / Conside ecto function he dieence of this ecto function is: he sufce intel of this function is: d π π sinθdθ π π dφ 4π sinθ dθdφ he dieence theoem is flse? 58 No he Dic delt function he One-Dimensionl Dic Delt Function he -D Dic delt function cn e pictued s n infinitel hih, infinitesimll now spike, with e just. if δ with δ d if - echnicll, δ is not function t ll, since its lue is not finite t. uch function is clled the enelied function, o distiution..5. he One-Dimensionl Dic Delt Function II If f is some odin function let s s tht it is continuous, then the poduct fδ is eo eewhee ecept t. It follows tht fδfδ. In pticul, f δ d f δ d f We cn shift the spike fom to some othe point. if δ with δ d if - enelied intetion eqution: 6 f δ d f δ d f 6

16 .5. he One-Dimensionl Dic Delt Function III lthouh δ is not leitimte function, intels oe δ e pefectl cceptle. It is est to think of the delt function s somethin tht is lws intended fo use unde n intel sin. In pticul, two epessions inolin delt function e consideed equl if: f D d f D fo ll "odin" function of Emple.4 Elute the intel d f. δ d δ 4 d 6 6 Emple.5 how tht whee k is n noneo constnt. ol : Conside the intel fo n it test function f δ k d Let k, so tht δ k δ k k, d k d positie : the intetion uns fom - to k netie : the intetion uns fom to - f δ k d ± f / k δ d f k k f, o δ k sees the sme pupose s δ nd δ δ. k Po..45 d δ δ d Let θ e the step function :, if > θ, if < how tht dθ d δ.5. he thee-dimensionl Dic Delt Function he enelied D delt function δ δ δ δ whee is the position ecto. It is eo eewhee ecept t,,, whee it lows up. Its olume intel is: ll spce δ d δ δ δ ddd 64 s in the -D cse, the intel with delt function picks out the lue of the function t the loction of the spike. ll spce f δ d f 65

17 .5. he thee-dimensionl Dic Delt Function II We found tht the dieence of is eo eewhee ecept t the oiin, nd et its intel oe n olume continin the oiin is constnt of 4π. he Dic delt function cn e defined s: Moe enell, / 4πδ 4 πδ whee is the seption ecto -. Note tht the diffeentition hee is with espect to, while is held constnt. 4 πδ 66.6 he heo of Vecto Fields.6. he Helmholt heoem o wht etent is ecto function F detemined its dieence nd cul? he dieence of F is specified scl function D, F D nd the cul of F is specified ecto function C, F C with F C Cn ou detemine the function F? Helmholt theoem untees tht the field F is uniquel detemined the dieence nd cul with ppopite ound conditions Potentils simple emple If the cul of ecto field F nishes eewhee, then F cn e witten s the dient of scl potentil V: Homewok # F F V conentionl Polems:.7,.9,.4,.45,.48 If the dieence of ecto field F nishes eewhee, then F cn e epessed s the cul of ecto potentil : F F 68 69