pecil Vect Clculus essin Engineeing Electmgnetics I Pfess et. cill J.
pecil Vect Clculus essin f Engineeing Electmgnetics I. imple cmputtin f cul diegence nd gdient f ect. [peicl Cdinte stem] Cul Diegence Gdient [Ceful tis is gdient f ect.] ct ] [ ] [ ] [ ] [
. Integtin f ect e cdinte. π d wee mied cdinte sstem In tis mied cdinte sstem te cefficient is functin f nd. efeing t te figue elw s cnges cnges. m te figue π d [ ] π [ ] π d π d π d d
. Css pduct f ects: B B B Bt in clindicl cdinte sstem B B But [ ] B B B B B B B B [ ] [ B B ] [ ] B B B B s speicl B B c clindicl e-epess te ect fm speicl t clindicl cdinte sstem. ẑ c ŷ 5
6 c c c s c s B B B B
. Veif tke s Teem f te functin e te unded sufces elw. lng te cntu nd tke s Teem dl c s ds Nte te cul is eluted e te sufce unded ζ ug H cnentin. N te cnstints e impsed n te sufce. lng Cntu ζ dl dl dl dl ζ ζ ζ ζ ζ dl 7
8 d d d l d l d is LWY psitie cnentin signs e incpted in te limits f integtin. d dl d dl d dl d l d Te cntu ζ lies in te plne. Tus dl d d d d d d dl ζ ζ Oe ufce B te H d pints in ŷ diectin dd d dd dd ds
Oe ufce B te H d d d pints dill inwd in clindicl cdinte sstem. llwing te tw ends f te clinde t e slnted sligtl inwd suc tt te pints f te lf clinde lies etween << it is cle te H d d plne ndd d plne Tnsfm fm Ctesin t clindicl cdinte sstem. Ctesin Clindicl ẑ ẑ [ ] 9
{ [ ] [ ] } { [ ] [ ] } dd { [ ] [ ] } dd ds π π π π [ ] π d [ { } dd [ ] π dd Oe ufce imil t integtins e nd. B guments gien tee we m wite d dd d dd d dd d dd d 5 dd ds dd dd dd suf. dd dd Oe ufce imil t integtins e pst sufces. Teefe suf.
d dd d dd d dd d nd wee n is diected in te ŷ like diectins. T detemine n cnside te tw ects psitining pints - nd t pint -. Teefe will gie us te desied diectin f n [ ] n [ ] n [ ] ecll tt: d [ ] dd nd dd d d [ ] d [ ] n n d n [ ] m stndd gemet it is es t sw tt n [ ] dd
uppse we wnt t sw te lst integl fm clculus. Cnside pjecting d nt te plne s swn elw. ~ d n d dd d d dd n n [ ] dd [ ] [ ] Cnsequentl d dd [ ] n [ ]
Questin: If te integnd ws functin f w must cnge s nd cnges in de tt ne emins n? Cnside te pictue gin. m te pictue it is es t see tt is independent f. Tt is lding cnstnt nd ing etween nd des nt cnge. On te te nd ing cnges. m simple equtin f line f ll n Teefe if ppeed in te integnd we wuld e t eplce it efe we pefm te integtin.
5. Veifing te Diegence Teem f te functin in te lume V nd e te sufce unding V swn in te figue elw. NOTE: d is te OUTWD NOML sufce element eltie t V Diegence Teem ds dv d V d d d d Oe Vlume V 5 d 5 V dv V dv d [ Oe ufce Tp d d d d
In tis plem d pints in te diectin. Let d nd. T detemine n cnside te tw ects psitining pints - nd t pint -. Teefe Teefe will gie us te desied diectin f n. n n [ ] [ ] Cnside pjecting d nt te plne s swn in te figues e nd elw. 5
6 Pjecting d nt ~ d ields dd dd n nd d dd d dd n dd d dd d n d n n n n ~ ~ n Cnside te pictue gin.
7 m te pictue it is es t see tt is independent f. Tt is lding cnstnt nd ing etween nd des NOT cnge. On te te nd ing cnges. m simple equtin f line f ll n tt is Tus [ n n n d d dd dd d Oe ufce dd d
d n [ ] [ ] [ dd ] Oe ufce d d dd B nlg t [ ] d [ ] [ dd ] Oe ufce d d dd But d t ck Oe ufce 5 8
9 5 5 [ 5 5 dd dd d dd d d Cnsequentl V dv d d d d d d 5 5
6. Elute te ect e te cil illustted elw wee nd. Te cil etends fm L t L. Empl clindicl cdinte sstem. dl d d d dl [ dl dl ] [ d d d ] But cnstnt d d d dl c dl [ ] d [ ] L n cil L d But
L L L L L L c cil n d d d dl
7. Elute te ect lng te cil illustted elw wee nd. Te cil etends fm L t L. Empl clindicl cdinte sstem. c dl d d d d d L [ ] [ d d ] L dl L L d
8. plne unded te fist ctnt psses tug pints P P nd P. Detemine te unit ect nml t te sufce f te plne. Cnside te tw ects wic spn te plne; nd. Ten te css pduct te ect nml t te plne t pint is n n plne s tw sides Te unit ect is ten n ± Lk t pictue f signs n n nd e gien numes. ince te sufce is plne n is te sme t ec pint n te sufce!
5 9. ecnside plem 8. Elute te ect field psg nml tug te plne unded te fist ctnt psg tug pints nd. Nte nd e numes. Cse tt sufce nml t pint in te like diectin. m plem n ds guntees tt nl nml cmpnent f eltie t is eluted d d Nte tt nd eist in te integnd. Nw let us pject te sufce element d nt te - plne nd integte e te pjected imge in tis plne. efe t te figues elw.
6
Pjecting d nt d ields d n d d [ ] [ ] dd d dd d m te figue e te limits f integtin f e nd. t n te nge f is. T cmplete u nlsis we must detemine w is elted t nd in de tt we emin n te pln sufce descied. Tw specil cses cn e esil tined in te nd plnes; equtins f lines. ince te sufce is pln nd e linel elted t ec te. Teefe f K cnstnt c Let K. In de tt te tw equtins f line e stisfied in tis epessin we fce c 7
8 f Tis epessin cectl descies u plne. ince nd e eing ied in te integtin must pppitel cnge in de t emin n te sufce. Teefe Nw let us cmine ll u tems. n ufce n d d dd dd d d Te eminde f tis plem is simple integl mecnics nd is left f te student t sle.