Chapter I Vector Analysis

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1 . Chpte I Vecto nlss

2 . Vecto lgeb j It s well-nown tht n vecto cn be wtten s Vectos obe the followng lgebc ules: scl s ) ( j v v cos ) ( e Commuttv ) ( ssoctve C C ) ( ) ( v j ) ( ) ( ) ( ) (

3 (v) he lw of cosnes C Let C C C cos C commuttve not () ple Poduct C C C ) ( C cos volume of pllelepped ule cb bc ) ( C C C (v) j

4 Poston, Dsplcement, & Septon Vectos he poston vecto of pont n 3-D s epessed n Ctesn coodntes s j P(,,) he nfntesml dsplcement vecto s dl d dj d souce pont s the pont whee n e. chge s locted. feld pont s the pont t whch ou e clcultng electc o mgnetc fled. Souce pont he septon vecto fom the souce pont to the feld pont s defned s d feld pont

5 Dffeentl Clculus df d df tht s f s chnged b d, the functon chnges b df wth d Odn Devtves It s nown tht df d popotonlt fcto. s the In othe wods we s tht the devtve s smll f the functon ves slowl wth nd lge f the functon ves pdl wth. df Geometcll we s tht s the slope of the gph f() vs.. d Gdent (Dectonl Devtve) Let be scl functon of 3-vbles,.e., (,, ) d d d d () hs tells us how ves s we go smll dstnce (d, d, d) w fom the pont (,, ). Let us ewte Eq. () s

6 d o d dl wth j j d d j d () s clled the gdent of the functon. he smbol j s clled the del opeto. Le n vecto the gdent hs mgntude nd decton. Now Eq. (3) cn be wtten s d dl cos Fo fed dl, d s mmum when, tht s d s mmum when we move n the sme decton s. (3) (4) (5)

7 he gdent ponts n the decton of mmum ncese of the functon. he mgntude gves the slope long ths mml decton. s dected noml to the level sufce of though the pont beng consdeed,.e., s pependcul to the sufce =constnt. Emple Fnd the gdent of the poston vecto. Souton In Ctesn coodntes the mgntude of the poston vecto s j j hs mens tht the dstnce fom the ogn nceses most pdl n the dl decton.

8 It s lso defned s the net te of flow pe unt volume,.e., denst the souce Emple Fnd the dvegence of the vectos Souton: Snce nd,, j Dvegence he dvegence of vecto s defned s j j he dvegence s mesue of how much the vecto speds out (dvege) fom the pont n queston. 3

9 he Cul he cul of vecto s defned s j he cul of vecto s mesue of how much the vecto cul ound the pont n queston. Emple Fnd the cul of the vectos Souton: j j nd j j

10 j j he Del Opeto Opetons ) ( ) ( ) ( v ) (

11 Integl Clculus Lne Integl he lne ntegl s epessed s b dl whee s vecto functon nd s n nfntesml dsplcement vecto long pth fom pont to pont b. If the pth foms closed loop, ccle s put on the ntegl sgn,.e., dl If the lne ntegl s ndependent on the pth followed, the vecto consevtve dl s clled

12 Emple Let j fnd b dl fom pont =(,,) to pont b=(,,) long the sold pth (pth ) nd long the dshed pth (pth ). Souton: dl d dj d b we hve fo the fst pth b b dl d b d d d d c c b c d d Now long the pth c =, nd long the pth cb =. hs leds to b dl d 4 d

13 Fo the second pth we hve = nd ths gves b dl d d d d d d Note tht the two esults e dffeent,.e., the vecto s not consevtve. Sufce Integl S ds he sufce ntegl s epessed s ds whee s vecto functon nd s n nfntesml element of e. gn f the sufce s closed we put ccle on the ntegl sgn, tht s ds ds he decton of s pependcul to the sufce n dected outwd fo closed sufces nd bt fo open sufces.

14 Emple Let ds j 3 Fnd S ove the 5-sdes of cube of sde, s shown n the fgue (ecludng the bottom). Souton: Fo the top sde ds top dd dd ds 3 dd top ut on the top sde = ds d 4 d top Fo the ght sde ds ght ddj ds dd d d ght ds left ddj ds dd left Fo the font sde ds dd ds dd font ut on the top sde = ds 4 d 6 d font Fo the bc sde ds dd ds dd bc ut on the top sde = ds bc dd Fo the left sde d d dd

15 Volume Integl he volume ntegl s epessed s V d whee s vecto functon nd d s n nfntesml element of volume. he Fundmentl heoem of Clculus bdf d d f ( b) f ( ) he Fundmentl heoem of Gdent whee =(,,) be scl functon of thee vbles, then b dl ( b) ( ) Snce the ght sde of the lst equton depends onl on the end ponts nd not on the pth followed we conclude tht

16 b Cooll : dl s ndependent on the pth followed fom to b. Cooll dl Emple: Let Chec the fundmentl theoem of gdent b tng two pths fom pont (,,) to pont b (,,). Souton: he fst pth s -steps: step () long the -s nd then up step (). Now dl d dj d nd j b b Fo the st pth dl d dj () () () () d d b () dl u fo step () =, nd fo step () = Fo the nd pth b dl d d d d d d 3 d ( ) 4 4

17 he Fundmentl heoem of Dvegence v d S ds It sttes tht the ntegl of dvegence ove volume s equl to the vlue of the functon t the bound. In nothe wold, the dvegence theoem sttes tht the outwd flu of vecto feld though sufce s equl to the tple ntegl of the dvegence on the egon nsde the sufce. Emple: Chec the dvegence theoem usng the vecto j Ove the unt cube stuted t the ogn. Soluton: v d ddd

18 o fnd S the fces: ds ds top we hve to clculte the ntegl ove ll dd dd ut fo the top sde =, so we hve top ds ds d bottom d dd ut fo the top sde =, so we hve ds ght ds dd bottom dd dd ut fo the ght sde =, so we hve ds d d 3 4 ght dd

19 ds dd left ut fo the left sde =, so we hve ds d 3 ds font ds bc S ds Stoes' heoem S dd dd ds left dl d d d 3 d Snce the bound lne fo n closed sufce shn down to pont, then S ds 3

20 Emple: Chec the Stoes' theoem usng the vecto 3 j 4 Ove the sque sufce shown. Soluton: 4 S ds dd ds 4 dd ut on the sufce =, so we hve S ds 4 dd 3 4

21 dl bottom dj ght d top dj d left long the bottom sde ==, so we hve dj bottom 3 d long the top sde =, = so we hve dj 3 d top long the ght sde =, = so we hve d 4 d 3 4 ght long the left sde =, = so we hve d 4 d left dl

22 Integton b Pts d d It s nown tht fg f dg d b Integtng both sdes we get fgd f d g d d d g df d Usng the fundmentl theoem of clculus we get b f dg d d fg b b g df d d Emple: Evlute the ntegl e d Soluton: It s nown tht e d e d e d e b d d dg d b df d

23 Sphecl coodntes (,,) Cuvlne Coodntes : s the dstnce fom the ogn (fom to ) : the pol ngle, s the ngle between nd the -s (fom to ) : the muthl ngle s the ngle between the pojecton of to the - plne nd the -s (fom to ) he elton between the Ctesn coodntes nd the sphecl coodntes cn be wtten s sn cos sn sn sncos he unt vectos ssocted wth the sphecl coodntes e elted to the coespondng unt vectos n the Ctesn coodntes s sn cos sn snj cos cos cos cos snj sn sn cosj

24 he nfntesml dsplcement vecto n sphecl coodntes s epessed s dl d d sn d he volume element s epessed s d dl dl dl sn ddd Fo the sufce elements we hve ds dl dl dd s constnt ds sn dl dl dd s constnt ds3 dl dl sn dd s constnt o fnd the volume of sphee of dus R we hve V d R sn d d d 4 3 R 3

25 o fnd the gdent n sphecl coodntes let = (,, ) so ) ( sn d d d dl d () d d d d Equtng the bove two equtons we get, sn sn o sn Smll, one cn fnd the dvegence nd the cul n sphecl coodntes

26 sn sn sn sn sn sn sn sn sn he Lplcn s defned s

27 Clndcl coodntes (,, ) : s the dstnce fom the -s (fom to ) : the muthl ngle s the ngle between nd the -s (fom to ) : the dstnce fom the - plne (fom - to ) he elton between the Ctesn coodntes nd the clndcl coodntes cn be wtten s cos sn he unt vectos ssocted wth the clndcl coodntes e elted to the coespondng unt vectos n the Ctesn coodntes s cos sn j sn cos j

28 he nfntesml dsplcement vecto n clndcl coodntes s epessed s dl d d d he volume element s epessed s d dl dl dl ddd Fo the sufce elements we hve ds dl dl d d ds dl dl dd ds dl dl d d 3 s constnt s constnt s constnt

29 he Del he Dvegence he Cul sn he Lplcn

30 he Dc Delt Functon Consde the functon Now sn nd d d ds S ut fom the dvegence theoem we now tht 4 sn d d S v ds d He we hve contdcton. he poblem s the pont =, whee the vecto blows up. He we hve contdcton. he poblem s the pont =, whee the vecto blows up. So we wte 3 4

31 Whee () s clled the Dc delt functon wth the followng popetes: 3 F d d F Usng theses popetes we hve v d 4 d 4 v 3 s epected

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