Chapter IV Vector and Tensor Analysis IV.2 Vector and Tensor Analysis September 23,

Transcription

1 hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe, 07 47

2 hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe, I. ETOR AND TENOR ANALYI I... Tenso functon th Let A n n be an n ode tenso. Then the tenso functon of a scala vaable s defned as a map n A t : n (4) The scala vaable (paamete) t can epesent tme, path, etc. Fo n 0, 0 and we have a eal-valued functon of a eal vaable: y a( t ) :I vecto functon Fo n, and we have the vecto functon of a scala vaable s a vecto-valued functon defned as a map fom the set of eal numbes to the space of vectos xt : ( t) y( t) z( t) x( t ) :I (4a) ( t ) :I (4b) 0 The vecto functon povdes a convenent method fo the defnton of cuves n space by tacng the ponts by the poston vecto t, t a,b. The change of paamete also povdes nfomaton about the poston of the pont on the cuve fo dffeent moments of tme. Ths defnton of cuves by vecto functons s equvalent to the paametc defnton of cuves: whee t [ a,b] ( t) xt y t z t [ a,b] x t,y t,z t, t [ ] x xt o y y( t) t [ a,b] (4) z z( t) ae eal valued functons. Devatve of a tenso functon The devatve of a tenso functon wth espect to a scala vaable s defned as d A t lm dt ( + ) A t t A t n n n t 0 (4) t If the lmt exsts, then the devatve s a tenso of the same ode d A ( t ) : dt n n n Repeatedly, the hghe ode devatves can be defned.

3 hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe, I... Tenso feld Let be a poston vecto whch specfes a locaton n Eucldan space E A tenso feld s defned as a map In patcula, we have s ( ) v( ) Aj n ( ) A : n scala feld (tempeatue, densty, etc) vecto feld (velocty, foce, etc) tenso feld (defomaton tenso, etc) defned at all ponts of space. A non-statonay tenso feld s defned as a map n ( ) A,t : n All consdeed functons ae assumed to be contnuous: lm A 0 ( ) A ( ) n n 0 scala feld A scala feld s defned as a eal valued functon of a vecto vaable: ϕ ( ) : ϕ ϕ ( x,y,z ) : By ths functon, a scala value s specfed fo any pont of space. A scala feld can descbe dstbuton n space of tempeatue, densty, concentaton, etc. Functon ϕ ϕ x,y : defnes a scala feld on a plane. vecto feld v( ) v(,t) A vecto feld s defned as a vecto valued functon of a vecto vaable: v( ) : P( x,y,z) + Q( x,y,z) + R( x,y,z) v j By ths functon, a vecto value s specfed at any pont. A vecto feld can be descbed by a dstbuton n space of velocty, acceleaton, foce, etc. A non-statonay scala o vecto feld ae defned as tme-dependent maps (,t ) : ϕ (,t) : v All opeatons defned fo tensos can be appled fo tenso felds pont-wse.

4 hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe, I... pace cuves onsde a bounded space cuve n wth the end ponts A and B. The gaph of the space cuve s taced by a vecto functon : ( t) t [ a,b] wth ( a ) A, b B (44) o n the tenso notatons: : x ( t ) t [ a,b] wth x a A, x b B (45) whch s just anothe fom of a tadtonal paametc defnton of the cuve: : x x( t) x x( t) t [ a,b] x x ( t) (46) Natual paametezaton ( s) wth the help of ac length s has some useful popetes. A dffeental element on the space cuve s tangent to the cuve: dx dt x ( t) dx x t dt (47) d d ds ds T d T ds (48) dt ds dt dt Types of some patcula space cuves: s a closed cuve f ( a) ( b) o x ( a) x ( b) s a smooth cuve f x ( t) [ a,b] (devatves ae contnuous) s a pece-wse smooth f, whee ae smooth cuves s a smple cuve f x ( t) x ( t ) f t t (w/o self-ntesecton)

5 hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe, 07 5 I..4. Level cuves and sufaces f : be a scala feld, then equaton Let f c o f ( x,x) c descbes the cuves n the plane xx called the level cuves: {( x,x ) f ( x,x ) c, c } Let ϕ : be a scala feld, then equaton ϕ ( ) c o ( x,x,x ) descbes the suface n ϕ c called the level suface: {( x,x,x ) ϕ ( x,x,x ) c, c } Fo dffeent values of the constant c, we obtan the famles of un-ntesected level cuves and sufaces (why do level cuves not ntesect?). I..5. Opeato nabla, gadent and dectonal devatve nabla : s a dffeental vecto opeato defned as,, tenso notaton (49) x shot-hand tenso notaton ometmes opeato nabla s also called the Hamlton opeato. gadent Opeato nabla appled to a scala-valued vecto functon ϕ yelds a vecto called the gadent of the scala feld ϕ ϕ, ϕ, ϕ ϕ + ϕ + ϕ (50) x x x whch s a vecto othogonal to the level suface of the scala feld (o to the level cuve n the -D case). ϕ ϕ x ϕ shot-hand tenso notaton

6 hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe, 07 5 Dectonal devatve of the scala feld Let functon ϕ : defnes a scala feld, and let s be a unt vecto, s. ecto hs wth a small h > 0 s an ncement n the decton s. Then a devatve of functon defned as ϕ ( ) n the decton s at the pont n space s ϕ ϕ D s ϕ ( + hs) ϕ( ) lm s h 0 h (5) It detemnes the ate of change of the scala feld decton s. ϕ ( ) at the pont n the Usng the dffeentaton ules fo the multvaable functons and the defnton of the opeato nabla, let us wte the othe epesentatons of the dectonal devatve: ϕ s ϕ x x + ϕ + ϕ x s x s x s ϕ ϕ ϕ cos, + cos, + cos, x x x ( s ) ( s ) ( s ) ϕ ϕ ϕ s+ s+ s x x x ϕ s ϕ s (5) x Theefoe, the devatve of ϕ n any decton s equal to the pojecton of the gadent ϕ onto ths decton: ϕ s ϕ s ϕ cos ( ϕ, s ) (5) It follows fom ths equaton that the maxmum value of the dectonal devatve s acheved n the decton of the gadent of the scala feld at ths pont. o we can conclude that the gadent of the scala feld s a vecto whch has a decton of the geatest ncease and ts magntude s equal to the dectonal devatve n ths decton. The opposte decton ϕ coesponds to the decton of the geatest decease. Fom equaton (5) yelds also that dϕ ϕ d. If n s the unt nomal vecto to the level suface of ϕ, then ϕ ϕ n (54) n Gadent of scala feld s othogonal to level suface.

7 hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe, 07 5 Dectonal devatve of the vecto feld a : defne a vecto feld, and let s be a unt vecto, Let functon s. ecto hs wth a small h > 0 s an ncement n the decton s. Then the devatve of functon a n a decton s at the pont of space s defned as D s a s a a ( + hs) a lm h 0 h (55) povded that the lmt exsts. It detemnes the ate of change of the vecto feld a at the pont n the decton s : s a + + x s x s x s cos, + cos, + cos, x x x ( s ) ( s ) ( s ) s+ s+ s x x x ( sa ) (56) In tenso notaton, the dectonal devatve s wtten as a s (57) a s x Dectonal devatve of the nd ode tenso feld The dectonal devatve of the nd ode tenso feld s the natual genealzaton of the dectonal devatves of the scala and vecto felds n tems of tensos gven by equatons (5) and (57). A : defne a nd ode tenso feld, and let s be a unt Let functon 9 9 vecto, s. ecto hs wth a small h > 0 s an ncement n the decton. Then a devatve of tenso functon A space s defned as n a decton s at the pont of s DA s A A ( + hs) A ( ) lm s h 0 h (58) povded that the lmt exsts. onsde the components of the dectonal devatve A s A A A x A A A + + s + s + s x s x s x s x x x o as a contucton of the d ode tenso A s A s (59) j x j

8 hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe, I..6. Flux onsde a vecto feld a. Let us defne a flux of a vecto feld though the suface as a suface ntegal of the vecto functon a : flux though the suface flux a d (60) Let be a suface and let a unt vecto n detemne the unt decton to suface (fo the decton of the nomal vecto, we wll agee to tae the exteo decton fo closed sufaces, and one of two dectons fo non-closed sufaces and stc to ths decton when changng poston on the suface). The dot poduct a n a cos n, a n s the magntude of the pojecton of vecto a on the nomal decton n. ubdvde suface nto subsufaces wth the aeas whch can be assumed to be flat and be chaactezed by the nomal vecto wth the magntude equal to the aea : n. Then the suface ntegal can be defned as a lmt of the sums a d lm a 0 (6) Usng the dffeental elaton d n d, we can expess a d ( an )d and ( n ) ( n ) ( n ) ax cos, ax cos, ax cos, + + d ( n ) ( n ) ( n ) ax cos, d + ax cos, d + ax cos, d a dx dx + a dx dx + a dx dx (6) x x x Then the flux of the vecto feld though the suface can be wtten n the tadtonal fom of the suface ntegal: ( x ) x x a d an d a dx dx + a dx dx + a dx dx (6) The flux of the vecto feld though the closed suface s denoted by a d an d Useful fact: the flux though any closed suface n the constant vecto feld a a s zeo (consevaton law): 0 a 0 d 0

9 hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe, Flux of the nd ode tenso Let functon A : defne a nd ode tenso feld and let n be a 9 9 vaable nomal vecto to the smooth suface. Then the flux of the nd ode tenso feld though the suface s defned as a vecto f wth the components f A n d (64) Fo example, the flux of the stess tenso p n the elastc medum though the suface defnes the total stess vecto P actng on the suface whch has the components: P p n d I..7. Dvegence Dvegence of the vecto feld a at the pont of space s defned as a lmt of the aveaged flux though the suface of the abtay volume contanng pont : dva lm 0 a d (65) Use a paallelepped fo the abtay volume wth one cone located at the pont ( x,x,x), sdes x and faces pependcula to the coodnate axes, and x x x wth the sufaces x x, (see pctue). Then dva lm 0 a d an d lm 0 a lm 0 a n lm 0 ( ) x x,x,x x,x,x... lm a + a + 0 lm x x x 0 lm x x x 0 a x + x,x,x a x,x,x x x +... x x x ( ) a x + x,x,x a x,x,x +... x

10 hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe, lm x x x 0 lm x x x 0 ( ) ax x + x,x,x ax x,x,x +... x ( ) ax x + x,x,x ax x,x,x +... x + + x x x x a (66) Physcal meanng of the dvegence [Z-59, K-45]: dv a a Fo ncompessble flud, the dvegence of the velocty vecto feld dv v 0 Dvegence of the nd ode tenso feld s defned as a lmt ( dva ) j lm 0 A n d jm m (67) It can be shown that the dvegence of the nd ode tenso s a vecto wth the components ( dva ) j A jm (68) x m

11 hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe, I..8. ul The othe physcal chaactestc of a tenso feld s gven by the cul of the tenso feld. The cul of the vecto feld a at the pont of space s defned as a lmt of the aveaged flux though the suface of the abtay volume contanng pont cul a n a d lm 0 (69) ompae wth dvegence (equaton (66)): dva na d lm 0 a Applcaton of the opeato nabla yelds a smla epesentaton of the cul cul a a (70) Physcal meanng: cul a measues how fast vecto feld otates. Fo otatonal flud, the cul of the velocty vecto feld s zeo cul v 0 Useful fomulas: cul a a x x x a a a (7) ( cula ) j j whee ndces, j, ae a cyclc pemutaton of the numbes,, Wth the help of altenatng unt tenso, t can be wtten as cula εj j (7) In the vecto fom: cul a + j+

12 hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe, I..9. OPERATOR NABLA AND RELATED DIFFERENTIAL OPERATOR nabla ( ) x x gad ϕ n ϕd lm 0 ϕ ( ϕ ) ϕ ϕ ϕ x ϕ x dv a na d lm 0 a dva x + + n a d cul a lm a ( cula) 0 ε j j j j cul a a x x x a a a x x x + j+ ϕ ϕ ϕ ϕ dv gad + + ϕ ϕ ϕ ϕ, ϕ : Laplacan of scala feld ϕ δ ϕ ϕ ϕ j j shot-hand notaton fo Laplacan notatons fo the Laplacan opeato

13 hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe, Let a, b, F : be vecto felds, ϕ,u : be scala felds, c [B&T, p.68] ϕ gad ϕ a a dv a cul a ( a ) ϕ ( a ) gad dv a dv gad ϕ ϕ ϕ Laplacan opeato dv cul a 0 vanshes dentcally ϕ cul gad ϕ 0 vanshes dentcally ( a ) cul cul a. ( ϕ + ψ ) ϕ + ψ. ( ϕ+ ψ) ϕ+ ψ gad ϕ+ ψ gadϕ+ gadψ. ( a+ b) ( a) + ( b ) gad dv a+ b gad dv a+ gad dv b 7. ( ( a+ b) ) ( a) + ( b ) cul cul ( a+ b) cul cul ( a) + cul cul ( b ) ( ϕψ ) ψ ϕ + ϕ ψ 0... ( a+ b) a+ b ( ca) c a dv( ca) ( a+ b) a+ b. ( ) ( ) ( ) a a a dv a+ b dva+ dvb cdva cul a+ b cula+ culb ( a ϕ ) ( a ) cul ( cula+ gadϕ ) cul ( cula) + ( ϕa) ϕ a a ϕ + ( a b) b ( a) a ( b ) ( ϕa) ϕ( a) ϕ a + gad ϕψ ψ gadϕ + ϕ gadψ dv ϕa ϕdva+ a gadϕ dv a b b cula a culb cul ϕa ϕcula+ gadϕ a cul cula gad dva a Fo composte functons ϕ f ( ) and f dϕ df a, the chan ule s appled 4. ϕ f ( ) f ( ) 5. f ( f ) d a a a ( ) df 6. a f ( ) ( f ) a ( ) da df m ( a) εjεlm xj x a a l dϕ gad ϕ f gad f df da dv f gad f df da cul f gad f df

14 hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe, atesan coodnates ( x,y,z ) ylndcal coodnates (,,z) phecal coodnates (, φ, ) x cos x cosφsn y sn y snφsn z z z cos x + y y tan x x + y + z y tanφ, x z z cos x + y + z Bass vectos (,0,0) ( 0,,0) ( 0,0,) e cos + j sn e cos φ sn + j sn φ sn + cos j e - sn + j cos e -sn + j cosφ e z eφ cosφcos + jsnφcos sn, Lne elements dx,dy,dz d,d,dz d, snd φ, d Dffeental aeas da x dydz da y dxdz da z dxdy da d dz da sndφd da ddz da φ sndφ d da z d d da ρdφd ρ φ Dffeental volume Ac length d dxdydz ds dx + dy + dz d dd dz ds d + d + dz d sndφd d ds d + sn dφ + d

15 hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe, 07 6 scala feld u ( ) u( x,y,z ) u(,,z) u(, φ, ) Gadent u u u u u,, y z u u u u,, z u u u u,, sn φ u u u u u u u u u + j+ e + e + e z e + eφ + e y z z sn φ Laplacan u u u u y z u + + u u + u u + + z u u + u u + + sn sn φ sn vecto feld F( ) ( F x,f y,f z) ( F,F,Fz) ( F,F φ,f ) F Fx cos + Fy sn F Fx cosφsn + Fy snφsn + Fz cos F Fx sn + Fy cos Fφ Fx snφ + Fy cosφ F z Fz x + y z F F cosφcos F snφcos F sn F F cos F sn x x F F sn F cos F y + y z Fz z F F cosφsn F snφ + F cosφcos φ F F snφsn + F cosφ + F snφcos F F cos F sn φ Dvegence dv F F F F x y Fz + + y z F F z z ( F ) + + ( F ) Fφ + + sn φ + ( sn ) F sn cul F F j e e ez e e sne y z z sn φ F F F F F Fz F F sn F x y z F z F y + y z F F z e Fφ sn F e + sn φ z + φ φ F F z j F Fz + + z x z + + Fy F x + y F F + e φ + e ( F ) F e z F sn φ + ( Fφ ) sn e

16 hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe, 07 6 Execses: ) Nave-toes equatons (govenng equaton fo flud moton): ρ + ( ) P+ ρ g + µ t Usng tenso notatons show that fo ncompessble, otatonal, steady flow ths smplfes to Benoull s equaton: ρ P+ ρg ) Maxwell s equatons [B&T, 46] Electomagnetc feld n a medum of dalectc constant ε, magnetc pemeablty µ, and conductvty σ ( ρ 0, no fee chadge nconductng medum, and j σ E (Ohm s law) ) s descbed by a system of Maxwelll s equatons: dv E 0 Gauss s law dv H 0 Gauss s law fo magnetsm µ H cul E Faaday s law c t ε E 4πσ cul H + E Maxwell-Ampee law c t c whee E E (,t) s the electc feld and (,t) H H s the magnetc feld. how that the followng wave equaton can be deved fom Maxwell s equatons: E E + c t c µε 4πµσ E t Hnt: apply opeato cul to the last two equaton.

17 hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe, 07 6 I..0. Lne Integal onsde a vecto-valued functon a, whee vectos belong to the space cuve : ( t) t [ a,b] wth ( a ) A, b B (69) We wll consde a lne ntegal whch symbolcally s wtten as: a d (70) Let us see how ths ntegal s defned n ts physcal sense. et up a patton P n of the cuve nto a dscete set of n ponts: { } P a,,,..., b n 0 n and defne the ncement Defne the nom of patton as the bggest ncement n the patton: P max Denote the values of functon Fom a dot poduct n a at the ponts of patton whch has a physcal sense of the wo pefomed by the foce a ove path. Then the lne ntegal s defned smlaly to defnton of the defnte ntegal as a lmt of the Remann sum: whch physcally expesses the wo done by the foce a cuve. alculaton of the lne ntegal. Let the vecto functon a a have the followng specfcaton: j P x,x,x + Q x,x,x + R x,x,x j along the space (7) P x (7) Let the paametezaton of the cuve be defned by x t + x t + x t ( t) (7) dffeentaton of ths equaton yelds: d t x t o d ( t) x ( t) dt dx (74) dt a a a n a d lm a n Pn 0 P x,x,x + P x,x,x + P x,x,x ( t) x fom whch follows that d dx (75)

18 hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe, Then the lne ntegal (5) can be tansfomed to a d P x,x,x P x,x,x P x,x,x d + + P x,x,x d P x,x,x d P x,x,x d + + P x,x,x dx + P x,x,x dx + P x,x,x dx P x,x,x dx + P x,x,x dx + P x,x,x dx ths s the tadtonal fom wthout paentheses. Then applyng dx x dt, b P( x,x,x) x P( x,x,x) x P( x,x,x) x dt (76) a + + Example Fnd a d fo vecto functon a p,q,x p,q along the space cuve t cos t + sn t + t t fom 0 to π : Identfy: x t cos t x ( t) snt x ( t) t x t snt x t cos t x t Then, usng equaton (56), one ends up wth a d π [ ] p snt + q cos t + t dt 0 t p cos t + q snt + π 0 π p+ p π p If the cuve s defned wth the natual paametezaton, then d d s ds T ds whee T s a unt vecto tangent to the cuve. dt ds dt dt Then d T ds ( s) Recall also ds dx + dx + dx x + x + x dt

19 hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe, The lne ntegal then s calculated accodng to a d b a P x,x,x T+ P x,x,x T+ P x,x,x T x + x + x dt b at x + x + x dt (77) a P x,x,x P x,x,x P x,x,x d + + Theefoe, the wo s pefomed only by the tangental component of the foce. onsevatve vecto felds If a s a gadent feld of some scala feld a ϕ (78) (n ths case functon ϕ ( ) s called a potental functon fo the gadent feld a ), then a lnea ntegal along the cuve connectng two ponts and s equal to the dffeence between values of the scala functon at these end ponts: a d ϕ d dϕ ϕ ϕ (79) It means that the same esult wll occu fo any cuve connectng ponts and, and the lne ntegal s sad to be ndependent of path. Of patcula nteest ae the lnea ntegals along the closed cuves denoted by a d It s obvous that fo the gadent feld a d ϕ d 0 That s why f a vecto feld s a gadent feld of some scala feld t s sad to be consevatve. We have fo a consevatve feld that dϕ ϕ d a d ( P P P) d ( P d P d P d ) Pdx + Pdx + Pdx (80) Theefoe, Pdx + Pdx + Pdx s an exact dffeental. o, the lne ntegal P dx + P dx + P dx s ndependent of path, f Pdx + Pdx + Pdx dϕ s an exact dffeental. Test fo path ndependence Recall fom calculus that the dffeental fom Pdx + Pdx + Pdxs an exact dffeental f and only f P P It s called the test fo path ndependence of a lnea ntegal n space. (8)

20 hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe, I... olume ntegal onsde a scala feld ϕ ( ) and let be a volume. ubdvde the volume nto subvolumes and constuct an ntegal of the functon ϕ ( ) ove the volume as a lmt ϕ ( ) d lm ϕ 0 (8) whee s an abtay pont n the subvolume. I... The Dvegence Theoem (the Gauss-Ostogadsy Theoem o the Dvegence Theoem) Let be a volume bounded by a closed suface. Then flux of the vecto feld a though the suface s equal to the ntegal of the dvegence of the vecto feld ove the volume : a d dvad (8) Poof: We wll show that equaton (8) s appoxmately vald wth any degee of accuacy,.e. that fo any ε > 0. a d - dvad < ε ubdvde volume nto such that a d - dv a <δ that s possble accodng to the defnton of the dvegence as a lmt (65). Multply ths nequalty by a d - dv a <δ then summaton ove all yelds a a d - dv <δ In ths esult, the suface ntegal only ove the exteo suface s left. All nteo sufaces have to be the boundaes of some adjacent volumes m and. Havng the opposte nomal vectos, n n m, the suface ntegals ove cancel each othe n the summaton: a d + a d m and m + an m d and and 0

21 hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe, Let wth 0, then accodng to the volume ntegal defnton (8), lm dva 0 and dva d a d - dvad <δ. ε hoose δ, then a d - dvad < ε fo any specfed ε > 0. Recall x x x dva + + a n a cos n, a x n Then the othe foms of the Dvegence Theoem (Gauss-Ostogadsy theoem) can be wtten as a d dvad an d dvad n a a d dv d x x x a cos ( n, ) + a cos ( n, ) + a cos ( n, ) d + + d x x x Applcaton: The Dvegence Theoem has a geat mpotance n mathematcal modelng n engneeng. In devaton of the govenng equatons fo physcal pocesses n the contnuous meda, the consevaton laws ae appled to the contol volume yeldng an equaton whch contans both suface ntegals and volume ntegals. Applcaton of the Dvegence Theoem educes all ntegals to the volume ntegal whch allows combnaton of all tems n one volume ntegal and concludes wth the patal dffeental equaton whch govens the physcal pocess unde consdeaton (see example of devaton of the Heat Equaton n the ecton III..).

22 hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe, 07 68