1.7 Vector Calculus 2 - Integration

Transcription

1 cio.7.7 cor alculus - Igraio.7. Ordiary Igrals o a cor A vcor ca b igrad i h ordiary way o roduc aohr vcor or aml 5 5 d 6.7. Li Igrals Discussd hr is h oio o a dii igral ivolvig a vcor ucio ha gras a scalar. L b a osiio vcor racig ou h curv bw h ois ad. L b a vcor ild. Th is a aml o a li igral. Eaml (o a Li Igral) d d d d d A aricl movs alog a ah rom h oi ( ) o ( ) whr is h sraigh li joiig h ois Fig..7.. Th aricl movs i a orc ild giv by 6 4 Wha is h work do o h aricl? d Figur.7.: a aricl movig i a orc ild olid Mchaics Par III 5 Klly

2 cio.7 oluio Th work do is 6 d 4 d d W d Th sraigh li ca b wri i h aramric orm so ha d W 6 d or W d d d I is a closd curv i.. a loo h li igral is o dod v d. No: i luid mchaics ad arodyamics wh v is h vlociy ild his igral v d is calld h circulaio o v abou..7. osrvaiv Filds I or a vcor o ca id a scalar such ha h () d is idd o h ah joiig ad () d aroud ay closd curv (.7.) I such a cas is calld a cosrvaiv vcor ild ad is is scalar oial. For aml h work do by a cosrvaiv orc ild is d d d i i d ) ( ) ( which clarly dds oly o h valus a h d-ois ad ad o o h ah ak bw hm. I ca b show ha a vcor is cosrvaiv i ad oly i curl o { Problm }. i gral o cours hr dos o is a scalar ild such ha ; his is o surrisig sic a vcor ild has hr scalar comos whras is drmid rom jus o olid Mchaics Par III 5 Klly

3 cio.7 olid Mchaics Par III Klly 5 Eaml (o a osrvaiv Forc Fild) Th graviaioal orc ild mg is a aml o a cosrvaiv vcor ild. larly o curl ad h graviaioal scalar oial is mg : ) ( ) ( mg d mg d mg W Eaml (o a osrvaiv Forc Fild) osidr h orc ild ) ( how ha i is a cosrvaiv orc ild id is scalar oial ad id h work do i movig a aricl i his ild rom ) ( o ) 4 (. oluio O has o / / / curl so h ild is cosrvaiv. To drmi h scalar oial l. Equaig coicis ad igraig lads o ) ( ) ( ) ( r q which agr i o chooss r q so ha o which may b addd a cosa. Th work do is

4 cio.7 W ( 4) ( ) Hlmholz Thory As miod a cosrvaiv vcor ild which is irroaioal i.. imlis o ad vic vrsa. imilarly i ca b show ha i o ca id a vcor a such ha a whr a is calld h vcor oial h is soloidal i.. { Problm 4}. Hlmholz showd ha a vcor ca always b rrsd i rms o a scalar oial ad a vcor oial a: Ty o cor odiio rsaio Gral a Irroaioal (cosrvaiv) o oloidal a.7.4 Doubl Igrals Th mos lmary y o wo-dimsioal igral is ha ovr a la rgio. For aml cosidr h igral ovr a rgio i h la Fig..7.. Th igral d d h givs h ara o ad jus as h o dimsioal igral o a ucio givs h ara udr h curv h igral ( ) dd givs h volum udr h (i gral curvd) surac ( ). Ths igrals ar calld doubl igrals. his dcomosiio ca b mad uiqu by rquirig ha as ; i gral i o is giv h ad a ca b obaid by solvig a umbr o dirial quaios olid Mchaics Par III 5 Klly

5 cio.7 ( ) Figur.7.: igraio ovr a rgio hag o variabls i Doubl Igrals To valua igrals o h y ( ) dd i is o covi o mak a chag o variabl. To do his o mus id a lmal surac ara i rms o h w variabls say quival o ha i h coordia sysm d d d. Th rgio ovr which h igraio aks lac is h la surac g ( ). Jus as a curv ca b rrsd by a osiio vcor o o sigl aramr (c..6.) his surac ca b rrsd by a osiio vcor wih wo aramrs ad : ( ) ( ) Paramrisig h la surac i his way o ca calcula h lm o surac d i rms o by cosidrig curvs o cosa as show i Fig..7.. Th vcors boudig h lm ar d () () d d cos d d d cos (.7.) so h ara o h lm is giv by whr J is h Jacobia o h rasormaio () () d d d dd J dd (.7.) or aml h ui circl ca b rrsd by cos si ( big i his cas h olar coordias r rscivly) olid Mchaics Par III 54 Klly

6 cio.7 J or J (.7.4) Th Jacobia is also o wri usig h oaio d d Jd d J Th igral ca ow b wri as ( ) Jdd () d d () d Figur.7.: a surac lm Eaml osidr a rgio h quarr ui-circl i h irs quadra. Th mom o iria abou h ais is did by I d d Trasorm h igral io h w coordia sysm by makig h subsiuios4 cos si. Th J cos si si cos 4 hs ar h olar coordias qual o r rscivly olid Mchaics Par III 55 Klly

7 cio.7 so I / si dd urac Igrals U o ow doubl igrals ovr a la rgio hav b cosidrd. I wha ollows cosidraio is giv o igrals ovr mor coml curvd suracs i sac such as h surac o a shr. uracs Agai a curvd surac ca b aramrizd by ow by h osiio vcor ( ) ( ) ( ) O ca gra a curv o h surac by akig ( s) ( s) so ha has osiio vcor Fig..7.4 s s) ( ) ( s A vcor ag o a a oi o is rom Eq..6. d d d ds ds ds (s) s ( ) Figur.7.4: a curvd surac May dir curvs ass hrough ad hc hr ar may dir ags wih dir corrsodig valus o d / ds d / ds. Thus h arial drivaivs / / mus also boh b agial o ad so a ormal o h surac a is giv by hir cross-roduc ad a ui ormal is olid Mchaics Par III 56 Klly

8 cio.7 / (.7.5) I som cass i is ossibl o us a o-aramric orm or h surac or aml g( ) c i which cas h ormal ca b obaid simly rom gradg / gradg. Eaml (Paramric rsaio ad h Normal o a hr) Th surac o a shr o radius a ca b aramrisd as 5 a si cos si si cos Hr lis o cos ar aralll o h la ( arallls ) whras lis o cos ar mridia lis Fig I o aks h siml rssios s / s ovr s / o obais a curv joiig ( ) ad ( ) ad assig hrough ( / / / ) as show. / Figur.7.5: a shr Th arial drivaivs wih rsc o h aramrs ar a cos a si cos si cos si si cos si so ha a si cos si si si cos 5 hs ar h shrical coordias (s.6.); olid Mchaics Par III 57 Klly

9 cio.7 olid Mchaics Par III Klly 58 ad a ui ormal o h shrical surac is cos si si cos si For aml a 4 / (his is o h curv ) o has 4 / 4 / ad as cd i is i h sam dircio as r. urac Igrals osidr ow h igral d whr is a vcor ucio ad is som curvd surac. As or h igral ovr h la rgio cos cos d d d d d oly ow d is o la ad is hr dimsioal. Th igral ca b valuad i o aramriss h surac wih ad h wris d d O way o valua his cross roduc is o us h rlaio (Lagrag s idiy Problm 5.) c b d a d b c a d c b a (.7.6) so ha (.7.7) Eaml (urac Ara o a hr) Usig h aramric orm or a shr giv abov o obais 4 si a so ha

10 cio.7 ara d a si d d 4a Flu Igrals urac igrals o ivolv h ormal o h surac as i h ollowig aml. Eaml I 4 valua d whr is h surac o h cub boudd by ; ; ad is h ui ouward ormal Fig Figur.7.6: h ui cub oluio Th igral ds o b valuad ovr h si acs. For h ac wih ad d 4 dd 4dd imilarly or h ohr iv sids whc d. Igrals o h orm d ar kow as lu igrals ad aris qui o i alicaios. For aml cosidr a marial lowig wih vlociy v i aricular h low hrough a small surac lm d wih ouward ui ormal Fig Th volum o marial lowig hrough h surac i im d is qual o h volum o h slad cylidr show which is h bas d ims h high. Th slad high is (= olid Mchaics Par III 59 Klly

11 cio.7 vlociy im) is v d ad h vrical high is h v d. Thus h ra o low is h volum lu (volum r ui im) hrough h surac lm: v d. v v d v d Figur.7.7: low hrough a surac lm Th oal (volum) lu ou o a surac is h 6 volum lu: v d (.7.8) imilarly h mass lu is giv by mass lu: v d (.7.9) For mor coml suracs o ca wri usig Eq d dd Eaml (o a Flu Igral) omu h lu igral d whr is h arabolic cylidr rrsd by ad Fig oluio Makig h subsiuios so ha by h osiio vcor h surac ca b rrsd 6 i v acs i h sam dircio as i.. oiig ouward h do roduc is osiiv ad his igral is osiiv; i o h ohr had marial is lowig i hrough h surac v ad ar i oosi dircios ad h do roduc is gaiv so h igral is gaiv olid Mchaics Par III 6 Klly

12 cio.7 Th / / ad so h igral bcoms dd Figur.7.8: lu hrough a arabolic cylidr No: i his aml h valu o h igral dds o h choic o. I o chooss isad o o would obai. Th ormal i h oosi dircio (o h ohr sid o h surac) ca b obaid by simly swichig ad sic / / / /. urac lu igrals ca also b valuad by irs covrig hm io doubl igrals ovr a la rgio. For aml i a surac has a rojcio o h la h a lm o surac d is rlad o h rojcd lm d d hrough (s Fig..7.9) cos d d dd ad so d dd olid Mchaics Par III 6 Klly

13 cio.7 Figur.7.9: rojcio o a surac lm oo a la rgio Th Normal ad urac Ara Elms I is somims covi o associa a scial vcor d wih a dirial lm o surac ara d whr d d so ha d is h vcor wih magiud d ad dircio o h ui ormal o h surac. Flu igrals ca h b wri as d d.7.6 olum Igrals Th volum igral or ril igral is a gralisaio o h doubl igral. hag o ariabl i olum Igrals For a volum igral i is o covi o mak h chag o variabls ( ) ( ). Th volum o a lm d is giv by h ril scalar roduc (Eqs ) d dd d Jd d d (.7.) whr h Jacobia is ow olid Mchaics Par III 6 Klly

14 cio.7 olid Mchaics Par III Klly 6 or J J (.7.) so ha d d J d ddydz.7.7 Igral Thorms A umbr o igral horms ad rlaios ar rsd hr (wihou roo) h mos imora o which is h divrgc horm. Ths horms ca b usd o simliy h valuaio o li doubl surac ad ril igrals. Thy ca also b usd i various roos o ohr imora rsuls. Th Divrgc Thorm osidr a arbirary diriabl vcor ild ) ( v did i som ii rgio o hysical sac. L b a volum i his sac wih a closd surac boudig h volum ad l h ouward ormal o his boudig surac b. Th divrgc horm o Gauss sas ha (i symbolic ad id oaio) i i i i d v d v d v d v div Divrgc Thorm (.7.) ad o has h ollowig usul idiis { Problm } d d d d d d u u u u curl grad ) div( (.7.) By alyig h divrgc horm o a vry small volum o ids ha d v v lim div ha is h divrgc is qual o h ouward lu r ui volum h rsul.6.8.

15 cio.7 ok s Thorm ok s horm rasorms li igrals io surac igrals ad vic vrsa. I sas ha curl d τds (.7.4) Hr is h boudary o h surac is h ui ouward ormal ad h ui ag vcor. τ d r / ds is As has b s Eq..6.4 h curl o h vlociy ild is a masur o how much a luid is roaig. Th dircio o his vcor is alog h dircio o h local ais o roaio ad is magiud masurs h local agular vlociy o h luid. ok s horm h sas ha h amou o roaio o a luid ca b masurd by igraig h agial vlociy aroud a curv (h li igral) or by igraig h amou o voriciy movig hrough a surac boudd by h sam curv. Gr s Thorm ad lad Idiis Gr s horm rlas a li igral o a doubl igral ad sas ha d dd (.7.5) d whr is a rgio i h la boudd by h curv. I vcor orm Gr s horm rads as d curl dd whr (.7.6) rom which i ca b s ha Gr s horm is a scial cas o ok s horm or h cas o a la surac (rgio) i h la. I ca also b show ha (his is Gr s irs idiy) grad d grad gradd (.7.7) No ha h rm grad is h dircioal drivaiv o i h dircio o h ouward ui ormal. This is o dod as /. Gr s irs idiy ca b rgardd as a muli-dimsioal igraio by ars comar h rul udv uv vdu wih h idiy r-wri as d d d (.7.8) olid Mchaics Par III 64 Klly

16 cio.7 or ud u d ud (.7.8) O also has h rlaio (his is Gr s scod idiy) grad grad d d (.7.9).7.8 Problms. Fid h work do i movig a aricl i a orc ild giv by 5 alog h curv rom o. (Plo h curv.). how ha h ollowig vcors ar cosrvaiv ad id hir scalar oials: (i) (ii) v (iii) u ( / ) ( / ). how ha i h curl o. 4. how ha i a h. 5. Fid h volum bah h surac ad abov h squar wih vrics ( ) ( ) ( ) ad ( ) i h la. 6. Fid h Jacobia (ad skch lis o cosa ) or h roaio cos si si cos 7. Fid a ui ormal o h circular cylidr wih aramric rrsaio ( ) a cos asi 8. Evalua d whr ad is h la surac. 9. Evalua h lu igral d whr ad is h co a a [Hi: irs aramris h surac wih.]. Prov h rlaios i (.7.). [Hi: irs wri h rssios i id oaio.]. Us h divrgc horm o show ha d whr is h volum closd by (ad is h osiio vcor).. riy h divrgc horm or v whr is h surac o h shr a.. Irr h divrgc horm (.7.) or h cas wh v is h vlociy ild. ( ). Irr also h cas o divv. olid Mchaics Par III 65 Klly

17 cio.7 4. riy ok s horm or whr is (so ha is h circl o radius i h la). 5. riy Gr s horm or h cas o wih h ui circl. Th ollowig rlaios migh b usul: si d cos d si cosd si cos d 6. Evalua d usig Gr s horm whr ad is h circl Us Gr s horm o show ha h doubl igral o h Lalacia o ovr a rgio is quival o h igral o / grad aroud h curv boudig h rgio: dd ds [Hi: L / /. Also show ha d d ds ds is a ui ormal o Fig..7.] ds d d Figur.7.: rojcio o a surac lm oo a la rgio olid Mchaics Par III 66 Klly