Vectors and Tensors. R. Shankar Subramanian. R. Aris, Vectors, Tensors, and the Equations of Fluid Mechanics, Prentice Hall (1962).

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1 005 Vectors nd Tensors R. Shnkr Subrmnn Good Sources R. rs, Vectors, Tensors, nd the Equtons of Flud Mechncs, Prentce Hll (96). nd ppendces n () R. B. Brd, W. E. Stewrt, nd E. N. Lghtfoot, Trnsport Phenomen, John Wley, 00. () J. Slttery, Momentum, Energy, nd Mss Trnsfer n Contnu, McGrw-Hll, 97. Some Bscs We encounter physcl enttes such s poston, velocty, momentum, stress, temperture het flux, concentrton, nd mss flux n trnsport problems - there s need to descrbe them n mthemtcl terms nd mnpulte the representtons n vrous wys. Ths requres the tools of tensor nlyss. Sclrs n entty such s temperture or concentrton tht hs mgntude (nd some unts tht need not concern us rght now), but no sense of drecton, s represented by sclr. Vectors In contrst, consder the velocty of prtcle or element of flud; to descrbe t fully, we need to specfy both ts mgntude ( n some sutble unts) nd ts nstntneous sptl drecton. Other exmples re momentum, het flux, nd mss flux. These qunttes re descrbed by vectors. In books, vectors re prnted n boldfce. In ordnry wrtng, we my represent vector n dfferent wys. vvv,, or v Gbbs notton ndex notton The lst requres comment. Wheres we represent the vectorl quntty wth symbol, we often know t only v ts components n some bss set. Note tht the vector s n entty hs n nvrnt dentty ndependent of the bss set n whch we choose to represent t.

2 In ndex notton, the subscrpt s free ndex - tht s, t s llowed to tke on ny of the three vlues,,, n -dmensonl spce. Thus, relly stnds for the ordered set v, v, v. Bss Sets v ( ) The most common bss set n three-dmensonl spce s the orthogonl trd (, j, k ) correspondng to rectngulr Crtesn coordnte system. stnds for unt vector n the x- drecton nd j nd k represent unt vectors n the y nd z-drectons respectvely. Note tht ths s not unque bss set. The drectons of, j, k depend on our choce of the coordnte drectons. There s no reson for the bss set to be composed of orthogonl vectors. The only requrement s tht the three vectors chosen do not le n plne. Orthogonl sets re the most convenent, however. We fnd the components of vector n the drectons of the bse vectors by tkng nner (dot) products. v = v, v = v j, v = v k Then, v = vx + vy j+ vzk x y z You cn verfy the consstency of the bove by tkng nner products of both sdes of the equton wth the bse vectors nd recognzng tht the bse vectors re orthogonl. j = j k = k 0 The order of the vectors n the nner product s unmportnt. b = b Sclr nd Vector Felds In prctce the temperture, velocty, nd concentrton n flud vry from pont to pont (nd often wth tme). Thus, we thnk of felds - temperture feld, velocty feld, etc. In the cse of vector feld such s the velocty n flud, we need to represent the velocty t every pont n spce n the domn of nterest. The dvntge of the rectngulr Crtesn bss set (, j, k) s tht t s nvrnt s we trnslte the trd to ny pont n spce. Tht s, not only re these bse vectors of unt length, but they never chnge drecton s we move from one pont to nother, once we hve chosen our x, y, nd z drectons.

3 Vector Opertons The entty v hs n dentty of ts own. Its length nd sptl drecton re ndependent of the bss set we choose. s the vectors n the bss set chnge, the components of v chnge ccordng to stndrd rules. Vectors cn be dded; the results re new vectors. If we use component representton, we smply dd ech component. Subtrcton works n smlr mnner. Vectors lso cn be multpled, but there re two wys to do t. We defne the dot nd cross products, lso known s nner nd vector products, respectvely, s shown below. b = xbx + yby + zbz s sclr. We commonly use numercl subscrpt for the components; n ( ) ths cse, the bss set s the orthogonl trd e (), e ( ), e ( ). Let Then, = + + e e e () ( ) ( ) b = b + b+ b = b = In the bove, we usully omt mpled. =. When n ndex s repeted, summton over tht ndex s b = b Ths s clled the summton conventon = = or where s the length of s the bss set s ltered. nd s nvrnt; nvrnt mens tht the entty does not chnge b s the vector product. s mpled by the nme, t s vector; t s norml to the plne contnng nd b. (, b, b) form rght-hnded system (ths s n rbtrry conventon, but we hve to choose one or the other, so we choose rght ). The order s mportnt, for, b = b

4 tht s, b ponts opposte to b. We cn wrte b = e b () e b () e b () There s compct representton of determnnt tht helps us wrte b = ε b jk j (Note tht k s free ndex. The ctul symbol chosen for t s not mportnt; wht mtters s tht the rght sde hs one free ndex, mkng t vector) ε jk s clled the permutton symbol ε = 0 f ny two of the ndces re the sme jk =+ f, j, k form n even permutton of,, [exmple:,,] = f, j, k We cn ssgn geometrc nterpretton to nd b s θ, then b = b cos θ form n odd permutton of,, [exmple:,, ] b nd b. If the ngle between the two vectors nd the length of b s b snθ. You my lso recognze bsnθ s the re of the prllelogrm formed by nd b s two djcent sdes. Gven ths, t s strghtforwrd to see tht b c = ε b c jk j k s the volume of the prllelepped wth sdes b,,ndc Second Order Tensors. Ths s clled the trple sclr product. Note tht we dd not defne vector dvson. The closest we come s n the defnton of second-order tensors! 4

5 Imgne Insted, we wrte b =Τ =Τ b tensor (unless explctly stted otherwse we ll only be tlkng bout second-order nd shll therefore omt syng t every tme) opertes on vector to yeld nother vector. It s very useful to thnk of tensors s opertors s you ll see lter. Note the dot product bove. Usng des from vectors, we cn see how the bove equton my be wrtten n ndex notton. =Τjb j It s mportnt to note tht b Τ would be bτ j nd would be dfferent from Τ b n generl. The two underbrs n Τ now tke on cler sgnfcnce; we re referrng to doubly subscrpted entty. We cn thnk of tensor s sum of components n the sme wy s vector. For ths, we use the followng result. e Τ e =Τ Sclr () ( j) j We re not usng ndex notton here Thus, to get Τ we would fnd e( ) Τ e ( ). We cn then thnk of T s sum. Τ=Τ e e +Τ e e +Τ e e () () () ( ) () ( ) +Τ e e +Τ e e +Τ e e ( ) () ( ) ( ) ( ) ( ) +Τ e e +Τ e e +Τ e e ( ) () ( ) ( ) ( ) ( ) Wht re the qunttes e () e ( ) nd others lke them? They re clled dyds. They re bss set for representng tensors. Ech s tensor tht only hs one component n ths bss set. Note tht e e e e. () ( j) ( j) ( ) You cn see tht tensors nd mtrces hve lot n common! 5

6 In fct, we commonly wrte the components of tensor s the elements of x mtrx. Τ Τ Τ Τ Τ Τ Τ Τ Τ Nturlly, s we chnge our bss set, the components of gven tensor wll chnge, but the entty tself does not chnge. Of course, unlke vectors, we cnnot vsulze tensors we only know them by wht they do to vectors tht we feed them! good exmple of tensor n flud mechncs s the stress t pont. To completely specfy the stress vector, we not only need to specfy the pont, but lso the orentton of the re element. t gven pont, we cn orent the re n nfntely mny drectons, nd for ech orentton, the stress vector would, n generl be dfferent. Force Stress = re hs mgntude nd drecton hs mgntude nd drecton In fct, we cn show tht stress s ndeed tensor (for proof, see rs, p. 0). So, we get t = n Τ n t The symbol n represents the unt norml (vector) to the re element, nd t s the stress vector ctng on tht element. The second-order tensor Τ completely descrbes the stte of stress t pont. By conventon, t s the stress exerted by the flud nto whch n ponts on the flud djonng t. Invrnts Just s vector hs one nvrnt (ts length), tensor hs three nvrnts. They re defned s follows. 6

7 Let or j be the tensor. bbrevton I = = tr = trce{ } { } where II I II II = { } = tr Note: s the tensor j jk III = Determnnt of = Det{ } = ε jk j k = ε jk l j k s the bss set s chnged, the nvrnts do not chnge even though the components of the tensor my chnge. For more detls, consult rs, p. 6, 7 or Slttery, p. 47, 48. symmetrc tensor j s one for whch j =. Thus, there re only sx ndependent components. Stress s symmetrc tensor (except n unusul fluds). Symmetrc tensors wth rel elements re self-djont opertors, concept bout whch you cn lern more n dvnced work. skew-symmetrc tensor j s one for whch j dgonl elements must be zero ( ) j =. You cn see mmedtely tht the snce =. Skew-symmetrc tensors hve only three ndependent components. Vortcty s n exmple of skew symmetrc tensor. If we wrte skew-symmetrc tensor j n the form j We cn see tht there s vector tht cn be formed usng the elements of j. The two re relted by the followng result, whch s useful n the context of the physcl sgnfcnce of vortcty. 7

8 x = x There s specl tensor tht leves vector undsturbed. It s clled the dentty or unt tensor I. 00 I = I x = x for ny x In ndex notton, we wrte I s δ j, the Kronecker delt. Symmetrc tensors hve very specl property. Remember tht we defne tensor t pont; t tht pont, there re three specl drectons, orthogonl to ech other, ssocted wth symmetrc tensor. When the tensor opertes on vector n one of these drectons, t returns nother vector pontng n the sme (or exctly opposte) drecton! The new vector, however, wll hve dfferent length n generl. Ths mgnfcton fctor n the length s clled the prncpl vlue or egenvlue of the tensor. It cn, of course, be ether smller or lrger thn one; we cll t mgnfcton (for convenence) n both cses. Becuse there re, n generl, three drectons tht re specl, there re usully three dstnct prncpl vlues, one ssocted wth ech drecton. Even for tensors tht re not symmetrc, there re three prncpl vlues; however these need not ll be rel. Sometmes, two re complex. Even when the prncpl vlues re rel, the drectons ssocted wth them need not be orthogonl f the tensor s not symmetrc. The problem for the prncpl or egenvlues of s Therefore, x = λx λi x λ I x = 0 From lner lgebr, for non-trvl solutons of the bove system to exst, we must hve det λ I = 0 The resultng thrd degree equton for the egenvlues s λ + I λ II λ+ III = 0 8

9 nd hs three roots λ, λ, nd λ. When these roots re ech used, n turn, nd we solve for x, we obtn n egenvector tht s known only to wthn n rbtrry multplctve constnt. Commonly, the egenvector s normlzed so tht t hs unt length. From the bove, you cn see tht correspondng to symmetrc tensor, there s specl rectngulr Crtesn set of bss vectors. If we choose ths s the bss set, the tensor wll hve smple dgonl form wth the dgonl components beng the egenvlues. If you re wonderng wht hppens when two egenvlues re dentcl, t s esy to show tht ny vector n the plne norml to the thrd egenvector (correspondng to the thrd egenvlue) s cceptble s n egenvector. In other words, on tht plne, the tensor opertng on vector n ny drecton wll yeld vector n the sme drecton wth mgnfcton fctor correspondng to the repeted egenvlue. If ll three egenvlues re dentcl, then ny drecton n spce wll be cceptble s the drecton of the egenvectors. Such tensor s clled sotropc for ths reson. I s n sotropc tensor wth egenvlues equl to unty. ny sclr multple of I lso s sotropc. Fnlly, you probbly hve lredy relzed tht when n egenvlue s negtve, the tensor opertng on n egenvector n the correspondng drecton lters ts length pproprtely nd reverses the drecton. Vector Clculus If we consder sclr feld such s temperture, we fnd the rte of chnge wth dstnce n some drecton, x, by clcultng Τ x. How cn we represent the rte of chnge n three-dmensonl spce wthout specfyng prtculr drecton? We do ths v the grdent opertor. The entty Τ [we cll t grd T ] s vector feld. t gven pont n spce, the vector, Τ, ponts n the drecton of gretest chnge of T. To obtn the rte of chnge of T t tht pont n ny specfed drecton, n, we smply project Τ n tht drecton. Τ = Τ n n unt vector Surfces n spce on whch feld hs the sme vlue everywhere re contours. In the cse of temperture felds, these contours re clled sotherms. long such surfce, the temperture cnnot chnge. Therefore, the Τ vector s everywhere norml to sotherml surfces snce t must yeld vlue of zero when projected onto such surfces. It s strghtforwrd to estblsh from the defnton tht 9

10 e + e + e () ( ) ( ) x x x n rectngulr Crtesn coordnte system (,, ) x x x. Note tht s n opertor nd not vector. So, you should exercse cre n mnpultng t. The opertor s the generlzton of dervtve. We cn dfferentte vector felds n more thn one wy. Dvergence v or dv v s clled the dvergence of the vector feld v. If the rectngulr Crtesn components of v re vv, v, then, v v v v = + + x x x s you cn see, the result s sclr feld. Curl v or curl v s vector feld. s the nme mples, t mesures the rotton of the vector v. gn, n ( x, x, x ) coordntes, e e e () ( ) ( ) v = x x x v v v = ε jk v j x There re two mportnt theorems you should know. They re smply stted here wthout proof. 0

11 Dvergence Theorem If the volume V n spce s bounded by the surfce S, V dv = ds S ds V S The vector feld should be contnuous nd dfferentble. The symbol ds represents vector surfce element. If n s the unt norml to the surfce, ds = n ds The entty dv s volume element. In the theorem, the left sde s volume ntegrl nd the rght sde s n ntegrl over the surfce tht bounds the volume. Fnlly, need not be vector feld, but cn be tensor feld of ny order. The dvergence theorem, lso known s Green s trnsformton, s very useful result tht permts us to convert volume ntegrls nto surfce ntegrls. By pplyng t to n nfntesml volume, you cn vsulze the physcl sgnfcnce of the dvergence of vector feld t pont s the outwrd flow of the feld from tht pont. Stokes Theorem Ths permts the converson of ntegrls over surfce to those round boundng curve. Imgne surfce tht does not completely enclose volume, but rther s open, such s bsebll cp. Let S be the surfce nd C, the curve tht bounds t.

12 t S C If vector feld s defned everywhere necessry, nd s contnuous nd dfferentble, Stokes theorem sttes: S ( ) C ds = t ds ds : vector re element on S ds: sclr lne element on C t : unt tngent vector on C The ntegrl on the rght sde s known s the crculton of round the closed curve C. The feld pperng n the theorem cn be replced by tensor feld of ny desred order. By mgnng the surfce S to le completely on the plne of the pper s shown, you cn S C vsulze the physcl sgnfcnce of. If you mke S shrnk to n nfntesml re, the re

13 ntegrl on the left sde becomes the product of the component of norml to the plne of the pper nd the re. The lne ntegrl s stll the crculton round n nfntesml closed loop surroundng the pont. If s the velocty feld v, by mkng the boundry n nfntesml crcle of rdus ε, the rght sde cn be seen to be pproxmtely πε v, where v s the mgntude of the velocty round the loop. The left sde s pproxmtely πε ( v) n where n s the unt norml to v the plne of the pper. Therefore, ( v) n, whch becomes the nstntneous ngulr ε velocty of the flud t the pont on the plne of the pper s ε 0. Becuse there s nothng unque bout the choce of the plne of the pper, we cn see tht ( v) n fct represents the nstntneous ngulr velocty vector of flud element t gven pont, the component of whch n ny drecton s obtned by projectng n tht drecton. The Grdent of Vector Feld Just s we defned the grdent of sclr feld, t s possble to defne the grdent of vector or tensor feld. If v s vector feld, v s second-order tensor feld. The rte of chnge of v n ny drecton n s gven by v n = v n