Review of Vector Algebra

Transcription

1 apt HPTE EIEW OF ETO LGE vw of cto lgba.. cto.. Dfto of a cto Dfto: vcto s a uatt tat posss bot magtu a cto a obs t paalllogam law of ao. ommutatv: D D Ut vcto:.. Scala Pouct Dot Pouct cos W a t magtu of t vctos a. s t agl btw t vctos a w t a aag tal to tal. cos s t pocto of vcto to vcto. If / a a otogoal to ac ot a ommutatv: Eampl: F Wo o b a foc F ug a ftsmal splacmt S S - -

2 HPTE EIEW OF ETO LGE... cto Pouct oss Pouct s W ê s t ut vcto omal to t pla cotag a. Dcto s m accog to t gt-a ul. a of t paalllogam ê If t two vctos a paalll tat s f o t. cto pouct s ot commutatv.... Howv pplcato ampl: Momt about O: M O F F O - -

3 HPTE EIEW OF ETO LGE Scala Tpl Pouct: s t volum of t paalllpp fom b t o-coplaa vctos a...5. cto Tpl Pouct: m W m a scala paamts. Poof: [ Tus vcto s s t pla of vctos a wl t vcto s s t pla of vctos a. Tfo:

4 HPTE EIEW OF ETO LGE. cto alculus If a a fftabl vcto fuctos of a scala a U t U U U U ; a scalas. [ [ [ [ - 4 -

5 HPTE EIEW OF ETO LGE Patal Dvatvs of ctos If a a two vctos pg o mo ta o scala vaabl sa a fo ampl t lm } { } { f as cotuous patal vatvs of t sco o at last.

6 HPTE EIEW OF ETO LGE.4 cto Spac a ass st of S of all -tupls [ of al/compl umbs s call a la vcto spac a ts lmts a call vctos. W ot [ b t smbol a t umbs of a call compots of. T vcto spacs S S a S av smpl gomtc tptatos. To pctu S fo stac w pst t vcto b [ b t l sgmt spac avg ts tal pot at t og a ts pot at t pot wt cooat. s ot cssal possss t mso of lgt. T s a stcto btw t tpl wc w call a vcto a t tpl [ wc psts a pot. W o fft tgs wt tm. W ca a two tpls [ a [ togt fo stac but w ctal o ot a two pots togt. O t ot a w spa of t stac btw two pots but ot t stac btw two vctos..4.. Lal Ipc ass a Dmso st of vctos combato of P P s tu w. P S s lal pt f a ol f t ol la p - 6 -

7 HPTE EIEW OF ETO LGE.4.. La Dpc st of vctos satsf t uato of P P. S s lal pt f t st scalas P ot all o to If t a lal pt t at last o of t s ca b pss as a la combato of t ots. Fo stac Suppos t t follows tat [ P P W sa tat a vcto spac S s -msoal f t cotas a st of lal pt vctos but ot + lal pt vctos..4.. ass bass fo a gv vcto spac vcto o S s a st of lal pt vctos. S ca b pa tms of tm pss as a la combato of tm. [.4.4. Uuss Qusto: Is t psso of uu? [ Poof : If also av aot pstato t fom of Subtactg fom w av - 7 -

8 HPTE EIEW OF ETO LGE c c c. Sc a la pt w ca sa c c c Tfo t psso s uu Otogoal a otoomal Dfto: T bass a otogoal f fo. Dfto: T bass a otoomal f fo a fo. Eampl: os t vcto sts. T ; ; Tfo vcto st s la pt a t ca b us as a st of bas vctos. Howv sc tus t a ot otogoal. Nomal t vcto: ut vcto t cto of - 8 -

9 HPTE EIEW OF ETO LGE uvla ooats.5.. atsa ooat Sstm ctagula cooat sstm X Y cooats a t cospog ut bas vcto wc a otoomal. ; ; Y W ; ;. I ot wos a t compots of vcto a t a t poctos of o X Y as spctvl. î ĵ

10 HPTE EIEW OF ETO LGE.5. uvla ooats I vcto spac S t-msoal spac w f a vcto aaltcall as a o st of t umbs tat a uu spct to a cos bas.. [. Lt cooat a ow as t gal cooats of a pot t ot cssal to posss t mso of lgt. I ot wos t a ot cssa t compot of t posto vcto scbg t pot. os a atsa cooat spac X Y cos t fucto f b cost cost cost If ts fuctos a sgl valu a ca b solv uul fo b latos ; ; a also f ts fuctos av cotuous vatvs t ca b a cuvla cooat of P P. T sufacs: a cooat sufacs a ac pa of ts sufacs tact cuvs call cooat cuvs o ls. as as as If t cooat sufacs tact at gt agls t t cuvla cooat sstm s otogoal. - -

11 HPTE EIEW OF ETO LGE.5. Spac uv If s t posto vcto og og O of a cooat sstm a a pot P t s gv b:. P T spac cuv s f b. S ê N s a vcto t cto of t tagt to. O ê ê T If s ta as t ac lgt S masu fom som f pots o t s a ut tagt vcto s ot b S ê T. T at at wc ê T cags wt spct to S s a masu of t cuvatu of a s gv b T a omal to t cuv at t pot. If ê N s t ut vcto t cto of ts omal s T t N K w K s call cuvatu of at t spcf pot a s t S K aus of cuvatu at tat pot. ut vcto ê ppcula to ê T a ê N suc as cuv. s call t bomal to t T N T ctos T N a fom a local gt-a ctagula cooat sstm at a spcf pots of. - -

12 HPTE EIEW OF ETO LGE.5.4 Dfto of Scal factos a Ut ctos Lt b t posto vcto of a pot P a cuvla cooat sstm... tagt to -cuv as at P at wc a a costat s. T a ut vcto s gv b: / o ê. Smlal / / ; ; a call t scal factos a a call ut vctos t casg cto of a. Not: T scal factos lat t fftal stac to t fftal of t cooats. Ts scal factos vag fom pot to pot a tus a gal fucto of posto. Sc w av S s s s W s s s ot lmtal stac alog a as spctvl. T fftal of ac lgt S ca b m fom I gal S M N m m m m S [log -cuv as a a costats tus ê tfo ac lgt alog at pot P s s. Smlal s a s. - -

13 HPTE EIEW OF ETO LGE Fo otogoal sstms: Tfo S spac cuv ca b pst b paamtc uatos suc as a. spac sufac ca b pst b a two paamt fucto faml a volum spac ca b pst b a t-paamtc faml a. Kpg a o of t s costat gats a sufac. Kpg a two of t s costat gats a spac cuv. Eampl: atsa sstm XY: X costat a sufac paalll to Y- pla lcal sstm : costat a sufac of a cl Spcal sstm: costat a sufac of a sp =costat =costa t =costa t atsa sstm lcal sstm Spcal sstm vw: uv spac Sufac spac olum spac - -

14 HPTE EIEW OF ETO LGE.6 uvla Sufacs Lt pst a spac sufac lt u v b a gv pot o t sufac. u psts a spac cuv gv b u.. a l paalll to t -as. v psts aot spac cuv gv b v.. a l paalll to t -as. at u v s a tagt to t cuv u at u v s a tagt to t cuv v s a vcto omal to t pla cotag t two tagts. uv s t ut vcto omal to t gv sufac at u v =v =u Elmtal/fftal stac alog t cuv v s s alog t cuv v S o t sufac S. It bcoms Smlal lmtal/fftal stac alog t cuv u s s It bcam S S alog t cuv o t sufac v

15 HPTE EIEW OF ETO LGE Elmtal/fftal aa at pot u v s gv b S S s ct out fom t sufac. Smlal ê ê a a - 5 -

16 HPTE EIEW OF ETO LGE Dtmato of Ut ctos a Scal Factos ê ; ê ; ê.. Tfo: } {

17 HPTE EIEW OF ETO LGE lcal ooat Sstm pot P spac s gv b p o p wt bas vcto gv b. s cos ; cos s s cos cos cos s s s cos s cos s Tfo: ; ; - Dcto: s cos s cos s cos - Dcto: lcal sstm

18 HPTE EIEW OF ETO LGE cos s cos s cos s - Dcto: Summa: ; cos s s cos Tasfomato latosp o cos s s cos ; cos s s cos Dvatvs of t ut vctos: ; ;

19 HPTE EIEW OF ETO LGE Eampl: If t s t posto vcto of a patcl clcal cooats obta psso fo vloct vcto a acclato vcto a at tat pot. Sc t Tfo Smlal a [ [

20 HPTE EIEW OF ETO LGE Scal factos a ut vctos Spcal cooat sstm O O s s cos s s cos s cos s s cos O cos cos cos s s s s cos - -

21 HPTE EIEW OF ETO LGE.8 Fuctos of cto to Dscb Pscal Poblms Tp of fuctos scala as a fucto of a scala fo ampl: T vcto as a fucto of a scala fo ampl: t scala as a fucto of a vcto fo ampl: T T vcto as a fucto of a vcto fo ampl: Gal scpto: t a t Scala fl: scala uatt gv as a fucto of cooat spac a tm t s call scala fl. Fo ampls: p p t p t a T T t T t cto fl: vcto uatt gv as a fucto of cooat spac a tm t s call vcto fl. Fo ampls: t t a M M t M t I gal a fl ots a go tougout wc a uatt s f as a fucto of locato wt t go a tm. If t uatt s pt of tm t fl s sta o statoa. - -

22 HPTE EIEW OF ETO LGE.9 Gat Gat s a vcto gat b t fftato of a scala fucto Lt W f to t spatal vaato of a patculat cto as a ctoal vatv a gal ts vatv s fft fft ctos. smpl but usful pstato of ctoal vatv s touc toug goupg togt t patal vatvs of alog t cooat as bass as t compots of a vcto call t gat of. os t cag of ov t ct stac lm [?. f Fom t total fftal fomula of t calculus t fst o fftal wll b g Os tms s s s Sc S s s s Now touc a vcto [ ot b otogoal cooat sstm wt ut vcto. t [ [ s s s S Sc S S S tfo s S s s a mamum w S t sam cto. I ot wos a s t magtu of t cag. t cuvla ês s a mamum... w a ê s a s t cto of mamum cags of T cto of t gat to a lvl of a scala fl s omal to t sufac at a gv pot.. s a vcto omal to. I t - -

23 HPTE EIEW OF ETO LGE pla S bcaus s a costat. S mas a S a otogoal. Gat s a vcto compos of patal vatvs of a scala t cooat ctos o alog t otogoal bass of a vcto spac. Gal oto of gat: I a scala fl of t spatal vaato of ca b calculat b S. Suppos a scala fl s a fucto of a vcto.. w s a vcto. T: o. W a t compots of vcto. If s a fucto of mo ta o vctos o s a fucto of sval sts of pt vaabls.. t t T t Itgal fto of gat: lm [ S / lm [ / S t w s a ftsmal abta volum a S s t sufac of t volum cos. s lmtal aa o t sufac. a ê s a ut vcto potg outwa omal to t sufac. - -

24 HPTE EIEW OF ETO LGE ota-vaat cto a ovaat cto otavaat compots of a vcto Fo a posto vcto os t bass a pag a vcto tms of ts bass a obta. ~ ~ ~ o ~ ~ ~ ~ ~ ~ a call t cota-vaat compot of. Sc s a tagt to t t cooat as t bass a t tagt spac. ~ ; ~ ; ~. ovaat compots of vcto s a gat to t -t cooat sufac. I ot wos s a gat to t sufac cost. w ê s a vcto omal to t cost sufac. Smlal a.

25 HPTE EIEW OF ETO LGE If w cos t bass a vcto ca b pa tms of ts bass as o. W ; ; a call t covaat compot of vcto W ; ;

26 HPTE EIEW OF ETO LGE Dvgc of a cto Fl Dfto: T vgc of a vcto at a pot s t t outflow fflu of t vcto fl p ut volum closg t pot. Lt b a lmtal volum wt a sufac S. lmt aa o t sufac S s. If s a vcto at a pot ts vcto fl t: [ lm Dv S W s t outflow of toug a ê s a ut vcto potg outwa a omal to t sufac. S s t t outflow fflu fom t sufac. If t vloct vcto t s t volum flu fom t pot.. t at at wc flu volum s lavg a pot p ut volum. If t vcto compos of st tms t vloct vcto t s t mass outflow fflu fom t sufac. ; Dv atsa sstm: [

27 HPTE EIEW OF ETO LGE lcal sstm: Tm Tm Tm Tfo [ I gal fom: [

28 HPTE EIEW OF ETO LGE Poof : Gal fto of vgc fom vcto algba Fom t fto of covaat vcto ; ; os t fst tm: Sc s a otogoal bas vcto t Sc [ Smlal tm a tm ca b calculat. Tfo: } {

29 HPTE EIEW OF ETO LGE Poof : Gal fto of Dvgc of a vcto fl usg tgal fom Lt b fto Dv lm [ S os a volum lmt a cuvla spac aou t pot P wt t as t g of t volum. Outflow t DI of at pot P s gv b sufac as P as sufac as Nt outflow fom t sufac wc s omal to -as wt a stac P s gv b fom t pot Sufac =. Smlal flow fom t sufac wc s omal to -as wt a stac pot P s gv b fom t Sufac =. T t outflow fflu alog t cto wll b: Sufac - Sufac [ - 9 -

30 HPTE EIEW OF ETO LGE - - Smlal t t outflow fflu alog t a cto wll b: a Tfo [ [ lm lm [ lm [ S

31 HPTE EIEW OF ETO LGE - -. T ul of a cto Fl ul ; ; os t fst tm Sc } { [ Smlal } { } { Tfo

32 HPTE EIEW OF ETO LGE - - } { } { } { O

33 HPTE EIEW OF ETO LGE - -. Som latos Ivolvg t cto Opato s a vcto opato a ot a vcto. Tus t s cssa to pst t os wc appas wt spct to t ot tms. Fo ampl: Som tts of tst: a scala vaabls a a vcto vaabls: Poof: smpl paso: [ T vcto a scala t tts a f tscall - tat s wtout fc to a spcal cooat sstm. fcato of t abov uatos a o cooat sstm g atsa s uvalt to vfcato of all cooat sstm.

34 HPTE EIEW OF ETO LGE Dtmato of Laplaca uato os a scala vaabl.. ; ; [ } { } {

35 HPTE EIEW OF ETO LGE.4 Gauss Dvgc Tom S call tat: Dv lm [ ca b appomat as: volum. S o fo a lmt cotol S Now cos a ft cotol volum spac subv to ma small lmtal subvolums. Suppos fo all t sub volum a valuat a summ: N N S N lm lm S volumtgalb fto N + + lm S N T flow of toug t commo facs of aact volums cacl bcaus t flow toug o fac uals t outflow toug t ot. Tus f w ow sum t t outflow of of all t sub-volums ol facs o t sufac closg t go wll cotbut to t summato. Stat tgal fom t abov statmt bcoms: lm N S S Tus Gauss vgc tom stats: S - 5 -

36 HPTE EIEW OF ETO LGE.5 Stos Tom Fo a cuv a t-msoal spac lt us assum t s a fucto f f f vw o. Lt us ma N sub-vsos btw t two pots P a P. T: f l lm N N ac l f l If w spcf l as t ac lgt paamt S t f ca b paamtcall pst foms of t ac lgt S. p l S.. f l f s s s s S L tgal fo a vcto valu fucto l T l s bfo w lt l as t ac lgt paamt S t l [ s s s S S S S S Sc S S S S S S S S - 6 -

37 HPTE EIEW OF ETO LGE ompot of ul a cto ê..? Fom t tgal fto: lm [ S Tus: usg t tt S S lm [ lm [ fo t gt a w ca wt: S S lm [ lm [ To valuat ts tgal w popos a volum lmts as a cl ot cssal of ccula coss scto wt ts as paalll to ê. Tus lm [ S S lm top S s S bottom ê ê ê ê T - 7 -

38 HPTE EIEW OF ETO LGE t t top a bottom ê a ê a paalll tfo. s S S lm lm [ t at t s of t volum tfo s S t lm assumg t s costat alog t as o t s sufac T t bottom t bottom t s S t s s s lm lm lm lm W v S to a lag umb of t sufac gos sa N of tm Gv t -t go t S t S S [ g t N-uatos fo = to N t lft a s bcoms t sufac tgal as t poto bcom ft t as sow blow: N galb fto sufac N S t lm [ N S S lm S S ê ê ê

39 HPTE EIEW OF ETO LGE os sufac go a wc av t poto D of t bouas commo. t a pot o t sgmts D ot tat t S fo a a oppostl otat wl t s uul f. So t cotbuto fom t D poto of b actl cacls t cotbuto of. Smlal w av cacllato fom all t s cpt fo sgmts alog t boua cuv suc as wc a ot sa. D Stat tgal fom: N lm S S Tfo S S - 9 -