2. SCALARS, VECTORS, TENSORS, AND DYADS

Transcription

1 2. SCALARS, VECTORS, TENSORS, AND DYADS This section is a eview of the popeties of scalas, vectos, and tensos. We also intoduce the concept of a dyad, which is useful in MHD. A scala is a quantity that has magnitude. It can be witten as S!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!9 (2.1) It seems self-evident that such a quantity is independent of the coodinate system in which it is measued. Howeve, we will see late in this section that this is somewhat naïve, and we will have to be moe caeful with definitions. Fo now, we say that the magnitude of a scala is independent of coodinate tansfomations that involve tanslations o otations. A vecto is a quantity that has both magnitude and diection. It is often pinted with an aow ove it (as in! V ), o in bold-face type (as in V, which is my pefeence). When handwitten, I use an undescoe (as in V, although many pefe the aow notation hee, too). It can be geometically epesented as an aow: A vecto has a tail and a head (whee the aowhead is). Its magnitude is epesented by its length. We emphasize that the vecto has an absolute oientation in space, i.e., it exists independent of any paticula coodinate system. Vectos ae theefoe coodinate fee objects, and expessions involving vectos ae tue in any coodinate system. Convesely, if an expession involving vectos is tue in one coodinate system, it is tue in all coodinate systems. (As with the scala, we will be moe caeful with ou statements in this egad late in this section.) Vectos ae added with the paallelogam ule. Geometically, 1

2 This is epesented algebaically as C = A + B. (Vecto notation was invented by J. Willad Gibbs, of statistical mechanics fame.) We define the scala poduct of two vectos A and B as A! B = ABcos" (2.1) whee A and B ae the magnitudes of A and B, and! is the angle (in adians) between them, as in the figue: The quantity S = A! B is the pojection of A on B, and vice vesa. Note that it can be negative o zeo. We will soon pove that S is a scala. It is sometimes useful to efe to a vecto V with espect to some coodinate system (x 1, x 2, x 3 ): 2

3 Hee the coodinate system is othogonal. The vectos ê 1, ê 2, and ê 3 have unit length and point in the diections of x 1, x 2 and x 3, espectively. They ae called unit basis vectos. The components of V with espect to (x 1, x 2, x 3 ) ae then defined as the scala poducts V 1 = V! ê 1!!!V 2 = V! ê 2!!!V 3 = V! ê 3!!!. (2.2a,b,c) The thee numbes ( V 1,V 2,V 3 ) also define the vecto V. (Note that not all tiplets ae components of vectos.) Of couse, a vecto can be efeed to othe coodinate systems ( x 1!, x 2!, x 3!) by means of a coodinate tansfomation. This can be expessed as x 1! = a 11 x 1 + a 12 x 2 + a 13 x 3 x 2! = a 21 x 1 + a 22 x 2 + a 23 x 3 x 3! = a 31 x 1 + a 32 x 2 + a 33 x 3 (2.3) whee the 9 numbes a ij ae independent of position; it is a linea tansfomation. Equation (2.3) can be witten as x i! = 3 " a ij x j!i = 1,2,3!!!. (2.4) j =1 We will often use the shothand notation x i! = a ij x j (2.5) with an implied summation ove the epeated index (in this case j ). This is called the Einstein summation convention. Since the epeated index j does not appea in the esult (the left hand side), it can be eplaced by any othe symbol. It is called a dummy index. The economy of the notation of (2.5) ove (2.3) is self-evident. whee Equation (2.5) is often witten as x! = A " x (2.6)! x = " x 1 x 2 x 3 $ (2.7) is called a column vecto. The tanspose of x is the ow vecto x T = ( x 1 x 2 x 3 )!!!. (2.8) The 9 numbes aanged in the aay! a 11 a 12 a 13 $ A = a 21 a 22 a 23 " a 31 a 32 a 33 (2.9) 3

4 fom a matix. (In this case the matix is 3 by 3). The dot poduct in Equation (2.6) implies summation ove the neighboing indices, as in Equation (2.5). Note that x T! A " x j a ji A! x (unless A is symmetic, i.e., a ij = a ji ). Diffeentiating Equation (2.5) with espect to x j, we find! x i "!x j = a ij = a ij jk = a ik (2.10)!x j!x k which defines the tansfomation coefficients a ik. Fo efeence, we give some matix definitions and popeties: 1. The identity matix is defined as! 1 0 0$ I = = ' (2.11) ij " The invese matix A -1, is defined by A -1! A = I. 3. If a ij ae the components of A, then a ji ae the components of A T, the tanspose of A. (If A = A T, A is symmetic.) 4. The adjoint matix is defined by A = A *T, whee (..) * is the complex conjugate; i.e., a ij = a * ji. 5. If A = A, then A is said to be self-adjoint. (This is the genealization of a symmetic matix to the case whee the components ae complex numbes.) 6. Matix multiplication is defined by A!B = A ij B jk. 7. ( A!B) T = B T! A T. 8. ( A!B) = B! A. The pototypical vecto is the position vecto = x 1 ê 1 + x 2 ê 2 + x 3 ê 3! ( x 1, x 2, x 3 )!!!!. (2.12) ( ). We say ( ) ae the components of a vecto if they tansfom like It epesents a vecto fom the oigin of coodinates to the point P x 1, x 2, x 3 that the thee numbes V 1,V 2,V 3 the components of the position vecto unde coodinate otations. Vectos ae defined by thei tansfomation popeties. Rotations look like this two-dimensional example: 4

5 We equie that the length of the position vecto, defined by l 2 = x T! x, be invaiant unde coodinate otations, i.e., l 2 = x T! x = x" T! x ". Then l 2 = x T! x = x" T! x "!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!= ( A! x) T!( A! x)!!!!!!!!!!!!!!!!!!= x T! A T ( )! ( )! A! x ( )! x!!!!!!!!!!!!!!!!!!!!= x T! A T! A so that A T! A = I, o A T = A!1. Matices with this popety ae called othogonal matices, and the otation matix A is an othogonal matix, i.e., o, since x = A!1 " x, a ij!1 = a ji (2.13)! x i " =!x j!!!. (2.14)!x j! x i " Then the components of the otation matix A have the popety a ij a ik =! x" i! x i " =! x" i!x k!x j!x k!x j! " x i =!x k!x j = jk!!!. (2.15) We now say that the thee numbes V 1, V 2 and V 3 ae the components of a vecto if they tansfom like the position vecto unde coodinate otations, i.e., V i! = a ij V j (2.16) whee the a ij ae the components of an othogonal matix. Suppose that A and B ae vectos. We now pove that the poduct defined in Equation (2.1) is a scala. To do this we must show that S! = A! " B!, the value of the poduct in the pimed coodinate system, is the same as S = A! B, the value in the unpimed system: 5

6 S! = A! " B!!!!!!!!!!= a ij a ik!!!!= a ij a ik!!!!= jk!!!!!!!= B j = S! whee the popety of othogonal matices defined in Equation (2.15) has been used. Futhe, if S = A! B is a scala, and B is a vecto, then A is also a vecto. In addition to the scala poduct of 2 vectos, we can also define the vecto poduct of 2 vectos. The esult is anothe vecto. This opeation is witten symbolically as C = A! B. The magnitude of C, C, is given by C = ABsin!, in analogy with Equation (2.1). The diection of C is pependicula to the plane defined by A and B along with the ight-hand ule. Note that it can point eithe above o below the plane, and may be zeo. In index notation, the vecto poduct is witten as C i =! ijk!!!. (2.17). The quantity! ijk is called the Levi-Civita tenso density. It will pove to be quite impotant and useful in late analysis, It has 27 components, most of which vanish. These ae defined as! 123 =! 231 =! 312 = 1!!!(even pemutation of the indices)! 132 =! 213 =! 321 = "1!(odd pemutation of the indices) $! ijk = 0!if i=j, o i=k, o j=k (2.19) We should eally pove that C so defined is a vecto, but I ll leave it as an execise fo the student. The Levi-Civita symbol satisfies the vey useful identity! ijk! lmk = " il " jm " im " jl!!!. (2.20) The expession can be used to deive a wide vaiety of fomulas and identities involving vectos and tensos. Fo example, conside the double coss poduct A! ( B! C). We wite this is some Catesian coodinate system as A! ( B! C) " ijk ( B! C) k = ijk ( klm B l C m )!!!!!!!!!!!!!!!!!!!!= ijk klm C m = ijk! A lmk j C m ( ) Even pemutation of indices!!!!!!!!!!!!!!!!!!!!= $ il $ jm $ im $ " $$ $$ l j C m = $ il B i C j $ im B j C m Fom Equation (2.20)!!!!!!!!!!!!!!!!!!!!= B i C j B j C i = B i C j ( ) C i ( B j )!!!!!!!!!!!!!!!!!!!!! B( A " C) C( A " B) (2.21) 6

7 which is commonly known as the BAC-CAB ule. The fist step in the deivation was to tanslate the vecto fomula into components in some convenient Catesian coodinate system, then tun the cank. We ecognized the fomula in the line peceding Equation (2.21) as anothe vecto fomula expessed in the same Catesian system. Howeve, if a vecto fomula is tue on one system, it is tue in all systems (even genealized, nonothogonal, cuvilinea coodinates), so we ae fee to tanslate it back into vecto notation. This is a vey poweful technique fo simplifying and manipulating vecto expessions. We define the tenso poduct of two vectos B and C as A = BC, o, A ij = B i C j!!!. (2.22) How do the 9 numbes A ij tansfom unde otations? Since B and C ae vectos, we have o A ij! = B i! C! j = a ik ( ) a jl C l ( ) = a ik a jl C l A ij! = a ik a jl A kl. (2.23) Equation (2.23) is the tenso tansfomation law. Any set of 9 numbes that tansfom like this unde otations fom the components of a tenso. The ank of the tenso is the numbe of indices. We notice that a scala is a tenso of ank zeo, a vecto is a fist ank tenso, the 3-by-3 aay just defined is a second ank tenso, etc. In geneal, a tenso tansfoms accoding to A ijkl...! = a ip a jq a k a ls...!a pqs...!!!!!!. (2.24) We can also wite A in dyadic notation: A = BC = ( B 1 ê 1 + B 2 ê 2 + B 3 ê 3 )( C 1 ê 1 + C 2 ê 2 + C 3 ê 3 )!!!!!!!!!!!!!=!!B 1 C 1 ê 1 ê 1 + B 1 C 2 ê 1 ê 2 + B 1 C 3 ê 1 ê 3!!!!!!!!!!!!!!!+B 2 C 1 ê 2 ê 1 + B 2 C 2 ê 2 ê 2 + B 2 C 3 ê 2 ê 3!!!!!!!!!!!!!!!+B 3 C 1 ê 3 ê 1 + B 3 C 2 ê 3 ê 2 + B 3 C 3 ê 3 ê 3!!!. The quantities ê i ê j ae called unit dyads. Note that (2.25) ê 1! A = B 1 C 1 ê 1 + B 1 C 2 ê 2 + B 1 C 3 ê 3 (2.26) is a vecto, while A! ê 1 = B 1 C 1 ê 1 + B 2 C 1 ê 2 + B 3 C 1 ê 3 (2.27) is a diffeent vecto. In geneal, BC! CB. We could similaly define highe ank tensos and dyads as D = AE, o D ijk = A ij E k, etc., etc. Contaction is defined as summation ove a pai of indices, e.g., D i = A ij E j. Contaction educes the ank by 2. We have also used the notation D = A! E to indicate 7

8 contaction ove neighboing indices. (Note that A! E " E! A.) notation ab The double-dot ( ) :( cd) is often used, but is ambiguous. We define A :B! A ij B ij, a scala. We now define a diffeential opeato in ou Catesian coodinate system! " ê 1 x 1 + ê 2 x 2 + ê 3 x 3 " ê i x i " ê i i!!!. (2.28) The symbol! is sometimes called nabla, and moe commonly (and by me), gad, which is shot fo gadient. So fa it is just a linea combination of patial deivatives; it needs something moe. What happens when we let it opeate on a scala function f (x 1, x 2, x 3 )? We have!f = ê 1 "f "x 1 + ê 2 "f "x 2 + ê 3 "f "x 3!!!. (2.29) What kind of a thing is!f? Conside the quantity, g = dx!"f whee dx = ê 1 dx 1 + ê 2 dx 2 + ê 3 dx 3 is a vecto defining the diffeential change in the position vecto: g = dx!"f!!!!= dx 1 f x 1 + dx 2 f x 2 + dx 3 f x 3 = df which we ecognize as the diffeential change in f, and theefoe a scala. Theefoe, by the agument given peviously, since dx! "f is a scala, and dx is a vecto, the thee quantities!f /!x 1,!f /!x 2 and!f /!x 3 fom the components of a vecto, so!f is a vecto. It measues the magnitude and diection of the ate of change of the function f at any point in space. Now fom the dyad D =!V, whee V is a vecto. Then the 9 quantities D ij =! i V j ae the components of a second ank tenso. If we contact ove the indices i and j we have D =! i V i " $ V (2.30) which is a scala. It is called the divegence of V. We can take the vecto poduct of! and V, D =! " V, o D i =! ijk " i V j!!!. (2.31) This is called the cul of V. Fo example, in Catesian coodinates, the x 1 component is by the popeties of! ijk. D 1 =! 123 "V 2 "x 3 +! 132 "V 3 "x 2 = "V 2 "x 3 "V 3 "x 2 8

9 We could also have! opeate on a tenso o dyad:!a " i k, which is a thid ank tenso. A common notation fo this is k,i (the comma denotes diffeentiation with espect to x i ). Contacting ove i and j, D k =! j k = " A (2.32) o k, j, which is the divegence of a tenso (it is a vecto). In pinciple we could define the cul of a tenso, etc., etc. So fa we have woked in Catesian coodinates. This is because they ae easy to wok with, and if a vecto expession is tue in Catesian coodinates it is tue in any coodinate system. We will now talk about cuvilinea coodinates. Cuvilinea coodinates ae still othogonal but the unit vectos ê i ae functions of x, and this complicates the computation of deivatives. Examples of othogonal cuvilinea coodinates ae cylindical and spheical coodinates. The gadient opeato is "! = ê i! (2.33) i "x i (the ode of the unit vecto and the deivative is now impotant), and any vecto V is V =! V j ê j (2.34) j whee now ê i = ê i (x). Then the tenso (o dyad)!v is "!V = ê i V j ê j i "x i j!!!!!!= ê " i V j i j "x ê j j " $!V j!ê j '!!!!!!= (( $ ê i ê j + ê i V j '!!!. (2.35) i j!x!" $ i!x!" $ i $ "Catesian" pat Exta tems if ê j =ê j (x i ) ' The fist tem is just the usual Catesian deivative. The emaining tems aise in cuvilinea coodinates. They must always be accounted fo. How can it happen that!ê j /!x i " 0? coodinates: Conside (two-dimensional) pola 9

10 The unit vectos ê and ê! emain othogonal as the position vecto otates about the z- axis, but thei spatial oientation depends on the angle!. Refeing to the figue below: fom the ightmost tiangle we have sin!" = (!ê ) " / and cos!" = [ (!ê ) ] /, so that!ê = "( 1 " cos! )ê + sin!ê. (2.36) Taking the limit!" 0 with = 1, we have!ê!" = ê "!!!!. (2.37) Similaly, fom the leftmost tiangle, sin!" = (!ê " ) / and cos!" = [ (!ê " ) " ] /, so that, in the same limit,!ê "!" = ê!!!!. (2.38) Then in these pola coodinates, 10

11 $ "!V = ê " + ê 1 " ' " ( ) êv + ê V "V!!!!!!=!!!ê ê " + ê "V ê " 1 "V!!!!!!!!!+ê ê " + ê V "ê " 1 "V!!!!!!!!!+ê ê " + ê V "ê ",!V!!!!!!=!!ê ê! + ê!v " ê"! $ 1!V!!!!!!!!!+ê " ê!" V " ' ( ( ), $ 1 ) + ê " ê "!V "!" + V ' ( ). (2.39) Expessions of the fom V!"V appea often in MHD. It is a vecto that expesses the ate of change of V in the diection of V. Then, fo pola coodinates, V V!"V = ê V + V $ V $ V 2 $ ' ( ) * + + ê V V $ + V $ V $ ' ( $ V V $ ) * +!!!. (2.40) The thid tem in each of the backets ae the new tems that aise fom the diffeentiation of the unit vectos in cuvilinea coodinates. A Digession on Non-othogonal Coodinates Of couse, thee is not need to insist that the bases ê i (x) even be othogonal. (An othogonal system has ê i! ê j = " ij.) Such systems ae called genealized cuvilinea coodinates. Then the bases (ê 1,ê 2,ê 3 ) ae not unique, because it is always possible to define equivalent, ecipocal basis vectos (ê 1,ê 2,ê 3 ) at each point in space by the pocess ê 3 = ê 1! ê 2 / J, ê 2 = ê 3! ê 1 / J!,!!!!!ê 1 = ê 2! ê 3 / J!, (2.41) whee J = ê 1! ê 2 " ê 3 is called the Jacobian. A vecto V can be equivalently expessed as o V = V i e i (2.42) V = V i e i!!!. (2.43) The V i ae called the contavaiant components of V, and the V i ae called the covaiant components. (Of couse, the vecto V, which is invaiant by definition, is neithe contavaiant o covaiant.) Ou pevious discussion of vectos, tensos, and dyads can be genealized to these non-othogonal coodinates, as long as exteme cae is taken in keeping tack of the contavaiant and covaiant components. Of paticula inteest is the genealization of 11

12 vecto diffeentiation, peviously discussed fo the special case of pola coodinates. The tenso!v can be witten as!v = ê i ê j D i V j! (2.44) whee D i V j =! i V j + V k " ik j (2.45) j is called the covaiant deivative. The quantities! ik and ae defined by ae called the Chistoffel symbols,! i ê k = " ikj ê k!!!. (2.47) j They ae the genealization of Equations (2.37) and (2.38). [We emak that the! ik ae not tensos, as they do not obey the tansfomation law, Equation (2.24).] Expessions j fo the! ik in any paticula coodinate system ae given in tems of the metic tenso components g ij = ê i! ê j, and g ij = ê i! ê j, as! ij k = g kl (" i g il + " j g li " l g ij ). (2.48) We stated peviously that if an expession is tue in Catesian coodinates, it is tue in all coodinate systems. In paticula, expessions fo genealized cuvilinea coodinates can be obtained by eveywhee eplacing the deivative! i V j with the covaiant deivative D i V j, defined by Equation (2.45). In analogy with the discussion peceding Equation (2.32), covaiant diffeentiation is often expessed in the shothand notation D i k! k..;i. Meisne, Thone and Wheele call this the comma goes to semicolon ule fo obtaining tenso expessions in genealized cuvilinea coodinates: fist get an expession in othogonal coodinates, and then change all commas to semi-colons! Genealized cuvilinea coodinates play an essential ole in the theoetical desciption of tokamak plasmas. The topic is so detailed and complex (and, fankly, difficult) that it will not be coveed futhe hee. I hope this shot intoduction will allow you to lean moe about this on you own. We now etun to Catesian coodinates. The divegence of a tenso T has been defined as! " T = i T ij. (2.49) It is a vecto whose j th component is (! " T) j = T 1 j + T 2 j + T 3 j. (2.50) x 1 x 2 x 3 Integate this expession ove all space: 12

13 o $ T (! " T) j d 3 1 j x = dx 1 dx 2 dx 3 + T 2 j + T 3 j ' * * x 1 x 2 x 3 ( ),!!!!!!!!!!!!!!!!!!!!!= * dx 2 dx 3 T 1 j + * dx 1 dx 3 T 2 j + * dx 1 dx 2 T 3 j,!!!!!!!!!!!!!!!!!!!!!= * ds 1 T 1 j + * ds 2 T 2 j + * ds 3 T 3 j,!!!!!!!!!!!!!!!!!!!!!= * ( ds " T) j,! " Td 3 x =! ds " T. (2.51) This is the genealized Gauss theoem. It is also possible to deive the following integal theoems: "!Vd 3 x =!" dsv (2.52) "!fd 3 x =!" dsf! (2.53)! " Vd 3 x =! ds " V (2.54) ds! "fd 3 x!=! dlf (2.55) S C $ ds! " Ad 3 x!=!$ A! dl!!! (2.56) S Addendum on Pseudoscalas, Pseudovectos, etc. C It seems intuitive that a physically measuable quantity shouldn t cae what coodinate system it s efeed to. We have shown that scalas, vectos, tensos, etc., things that we can associate with physical quantities, ae invaiant unde otations. This is good! Thee is anothe impotant type of coodinate tansfomation called an invesion. It is also know as paity tansfomation. Mathematically, this is given x i! = "x i. (2.58) An invesion is shown below: 13

14 The fist coodinate system is ight handed, the second coodinate system is lefthanded. Conside the position vecto. In the unpimed coodinate system it is given by = x i ê i. In the pimed (inveted) coodinate system it is given by! = x iˆ!! e i = "x i ( )("ê i ) = x i ê i = (2.59) so it is invaiant unde invesions. Such a vecto is called a pola vecto. (It is sometimes called a tue vecto.) We emak the gadient opeato! = ê i " i tansfoms like a pola vecto, since "! i = " i. Now conside the vecto C, defined by C = A! B (o C i =! ijk ) whee A and B ae pola vectos, i.e., A and B tansfom accoding to A i! = "A i and B i! = "B i. Then unde invesion the components of C tansfom accoding to C i! = " ijk A! j! = " ijk ( )( ) = " ijk = +C i!!!. (2.60) Then C! = C iˆ!! e i = C i ( ) = "C (2.61) "ê i so that C does not tansfom like a tue vecto unde coodinate invesions. Such a vecto (that changes diection in space unde coodinate invesion; see the figue) is called an axial vecto (o pseudovecto). Since it is defined as a vecto (o coss) poduct, it usually descibes some pocess involving otation. Fo example, the vecto aea ds k = dx i! dx j is a pseudovecto. Howeve, notice that if A is a pola vecto and B is an axial vecto, then C i! = "C i, and C is a pola vecto. The elementay volume is defines as dv = dx 1! dx 2 " dx 3 = ijk dx 1 dx 2 dx 3. It is easy to see that unde invesions d V! = "dv ; the volume changes sign! Such quantities ae called pseudoscalas: they ae invaiant unde otations, but change sign unde invesions. Again, it is intuitive that physical quantities should exist independent fom coodinate systems. How then to account fo the volume? Appaently it should be consideed a deived quantity, not diectly measueable. Fo example, one can measue diectly the tue vectos (i.e., lengths) dx i, but one has to compute the volume. This endes as a pseudoscala any quantity that expesses an amout of a scala quantity pe unit volume; these ae not diectly measuable. This includes the mass density! (mass/volume) and the pessue (intenal enegy/volume). (An exception is the electic chage density, which is a tue scala.) Appaently, one can measue diectly mass and length (both tue scalas), but must then infe the mass density. The following is a list of some physical vaiables that appea in MHD, and thei tansfomation popeties: Time is a scala. Tempeatue, which has units of enegy, is a scala Mass density,! = M / V, is a pseudoscala; Pessue, p =!k B T, is a pseudoscala; 14

15 Velocity, V i = d 2 x i / dt 2, is a vecto; The vecto potential A is a vecto; The magnetic flux, " A! dx, is a scala; The magnetic field, B =! " A, is a pseudovecto; The cuent density, µ 0 J =! " B, is a vecto. (Note that, since J is electic chage pe unit aea, and aea is a pseudoscala, that electic chage must also be a pseudoscala.) The Loentz foce density, f L = J! B, is a pseudovecto; The pessue foce density,!"p, is a pseudovecto; The acceleation density,!dv / dt, is a pseudoscala. Now, it s OK to expess physical elationships by using pseudovectos and pseudoscalas. What is equied is that the esulting expessions be consistent, i.e., we don t end up adding scalas and pseudoscalas, o vectos and pseudovectos. Now, on to MHD! 15