Field calculus. C.1 Spatial derivatives

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1 Field clculus n continuum physics the bsic mthemticl objects re fields Like the geometric objects discussed in the preceding chpter, fields re clssified s sclr, vector, nd tensor fields, ech type hving different number of components A field, or field component, is simply rel-vlued function of the three sptil coordintes (nd time) A sclr field hs only one component, vector field three, nd generl tensor field of rnk r hs 3 r components n quntum physics there re lso importnt complex fields of hlf-integer rnk, clled spinor fields, but we do not need them in clssicl physics The invention of field clculus (or vector clculus, s it is mostly clled) goes bck to J Willrd Gibbs in the end of the 19th century [rowe 1996] Field clculus revolutionized the mthemticl tretment of the prtil differentil equtions tht seemed to turn up in ll brnches of physics When the fundmentl tools of clculus, differentition nd integrtion, re combined with vector lgebr, new types of differentil opertors nd new types of integrls nturlly rise n combintion with ech other, the formlism cn still become quite complicted, nd one my eventully hve to resort to index nottion to void mbiguity The mthemticl tools developed here re centrl for the whole book, but cn be smpled s the need rises Josih Willrd Gibbs ( ) Americn engineer nd theoreticl physicist n spite of his unssuming mnners, he ws towering scientist contributing to thermodynmics, electromgnetism, mthemtics nd stronomy n his lst publiction he put sttisticl mechnics on firm foundtion 1 ptil derivtives The triplet of sptil derivtives is given specil symbol, clled nbl ; : t is differentil opertor which in most respects lso behves like vector (problem 4) ome of the common rules for differentition, for exmple linerity nd the chin rule, cn be directly used in combintion with the nbl opertor, wheres others such s the product rule become more complicted (see below) 1 Although it is sometimes nturl to use the for the prtil derivtive with respect to ny vrible u, the corresponding x y z / is rrely used insted of r n the older literture the grdient, divergence nd curl re often denoted by grd r, div U r U nd curl U r U, or in continentl Europe rot U r U This nottion is now ll but obsolete opyright c Benny Lutrup

2 614 PHY OF ONTNUOU MATTER Grdient Operting on sclr field x/, nbl cretes vector field clled the grdient of, r r x ; r y ; r @y : 1 Two contour surfces defined by x/ nd x/ 1 with 1 > The grdient t is lwys orthogonl to the contour surfce nd is oriented from to 1 Here we hve for clrity suppressed the explicit dependence on the sptil coordintes The grdient is useful for clculting the difference in field vlues t two nerby points Expnding to first order in ech of the coordinte differentils, dx dx; dy; dz/, we get x dx/ @y The left hnd side is the differentil dx/ of the field, so tht this importnt rule cn be written dx/ dx rx/: (3) A sclr field is often pictured by mens of surfces of constnt vlue, x/ const, lso clled contour surfces t is now esy to see tht the grdient in given point is lwys orthogonl to the contour surfce contining this point For if both x nd x dx lie on the sme contour surfce, the differentil must vnish, dx rx/, nd since ll differentils dx re tngentil to the contour surfce, it follows from the vnishing of the dot product tht the grdient r must itself be orthogonl to the surfce ivergence otting r with vector field U x/ we obtin sclr field, clled the divergence of U, r U r x U x r y U y r z U x y : (4) f you plot vector field with positive divergence by mens of smll rrows, the rrows will hve tendency to diverge from ech other, or converge if the divergence is negtive iverging vector field U x plotted in the xy-plne ts divergence is r U 3 url Using the grdient opertor s the left-hnd component in the cross product (B7) with vector field U x/ we obtin nother vector field, clled the curl of U, r U r y U z r z U y ; r z U x r x U z ; r x U y r y U @z @x x f you plot vector field with non-vnishing curl by mens of smll rrows, the rrows will hve tendency to circulte round Product rules urling vector field U Oe z x plotted in the xy-plne ts curl is r U 2 Oe z ombintion of these opertors with products of vrious fields, leds to number of useful rules, which we list here Prentheses re put in liberlly to fcilitte understnding, even if one in prctice would use fewer ome rules re firly trivil but ll proven in problem 3 opyright c Benny Lutrup

3 FEL ALULU 615 Rules with two nbls, r r V / ; (6) r r/ ; (6b) r r V / rr V / r 2 V : (6c) imple product rules, r 1 2 / r 1 / 2 1 r 2 /; r U / r/ U r U /; r U / r/ U r U /: (7) (7b) (7c) omplex product rules r U 1 U 2 / U 2 r/u 1 U 1 r/u 2 U 1 r U 2 / U 2 r U 1 / (8) ru 1 U 2 / U 1 r U 2 / U 2 r U 1 / U 1 r/u 2 U 2 r/u 1 : (8b) The most complex rule, b c r/u b c r/u c b r/u b c/ r U ; (9) which is vlid for rbitrry vector fields, b, nd c This rule which follows from (B67) with d replced by r expresses tht ny four vectors re lwys linerly dependent in three dimensions 2 ptil integrls n physics where nothing is truly infinite or truly infinitesiml sptil integrls re best understood in the Riemnnin sense where the integrtion domin is subdivided into huge number of tiny subdomins nd the integrl is pproximted by sum over these subdomins This procedure cn be crried to ny precision for integrnds tht vry smoothly over the integrtion domin urve integrl The curve integrl of smoothly vrying vector field U x/ long n oriented curve running from to b is defined by U x/ d` lim N!1 n1 NX U x n / d`n: (1) As shown on the right, the integrl should be understood s the limit of huge sum over tiny stright-line pieces, ech represented by its vector curve element d` ner x, which is prllel with the curve nd hs length d` jd`j n rtesin coordintes the vector line element is equl to the position differentil ` d x dx; dy; dz/ b d` x U An oriented curve running from the point to b with stright-line curve element d ` ner x opyright c Benny Lutrup

4 616 PHY OF ONTNUOU MATTER d x d An oriented surfce with tiny flt surfce ptch d ner x All normls point to the sme side of the surfce The perimeter curve is oriented consistently with the surfce d dv V A volume V with its enclosing surfce, volume element dv ner x nd surfce element d urfce integrl The surfce integrl of smoothly vrying vector field U x/ over n oriented surfce is defined by, NX U x/ d lim U x n / d n : (11) N!1 n1 The integrl is the limit of huge sum over tiny flt surfce ptches, ech represented by its re vector or surfce element d ner x, which is orthogonl to the surfce nd hs re d jdj On n oriented surfce, neighboring surfce elements must consistently point to the sme side of the surfce (thereby excluding non-orientble surfces such s the Möbius bnd nd the Klein bottle) By universl convention the normls of closed surfce re lwys chosen to be directed out of the enclosed volume For resons of symbol economy we hve here used the sme letter for both the domin of integrtion nd the infinitesiml element n rtesin coordintes surfce element orthogonl to the xy-plne becomes d Oe z dxdy Volume integrl The volume integrl of smoothly vrying sclr field x/ over volume V is defined by, NX x/ dv lim x n /dv n : (12) V N!1 n1 The integrl is huge sum over tiny volume elements, ech represented by its volume dv ner x n rtesin coordintes the volume element tkes the form dv dxdydz 3 Fundmentl integrl theorems For ech of the bove types of integrl there is fundmentl mthemticl theorem relting it to n integrl of lower dimension We hve collected them here becuse they will be useful for nerly ll of the topics covered in this book Proofs re found in the following section Grdient theorem Let be n rbitrry oriented curve with endpoints nd b, nd let x/ be sclr field Then the integrl of the grdient of long equls endpoints difference of, r d` b/ /: (13) Note tht the right hnd side does not depend on the curve connecting the endpoints nd b tokes theorem Let be n rbitrry oriented surfce with the closed curve s perimeter, oriented in the sme wy s the surfce Then the surfce integrl of the curl of U over equls the curve integrl of the vector field U round the perimeter, r U d U d`: (14) The circle in the curve integrl on the right is just there to remind us tht the curve is closed Note tht the right hnd side tkes the sme vlue for ny surfce tht hs s perimeter opyright c Benny Lutrup

5 FEL ALULU 617 Guss theorem Although this theorem ws discovered severl times bout 2 yers go it is usully ttributed to Guss in 1813 Let V be n rbitrry volume bounded by the closed surfce, oriented with ll normls pointing out of the volume Then the volume integrl over V of the divergence of U equls the surfce integrl of the vector field U over, r U dv U d ; V (15) where the circled surfce integrl indictes tht the surfce is closed * 4 Proofs of the fundmentl integrl theorems n ech cse we first prove tht the theorem is dditive, mening tht if it holds for ll the prts of n integrtion domin it holds for the whole domin ince the integrls re defined s huge sums over tiny curve, surfce or volume elements, we only need to prove the theorem for such elements Grdient theorem To prove dditivity, we first divide curve into two curves, 1 nd 2, joined in point c The integrl is the sum of the two contributions, nd if it holds for ech prt it holds for the complete curve becuse the two endpoint contributions t c cncel Thus, the theorem holds in generl if it holds for the tiny stright-line element d` dx But tht hs lredy been shown in eq (3) tokes theorem Let the oriented surfce be divided into two prts 1 nd 2 with the sme orienttion by mens of curve (se the mrgin figure) f the theorem is vlid for ech prt, it will lso be vlid for the whole becuse the two boundry curves pss through the common piece with opposite orienttion, such tht the two curve integrls long will cncel onsequently, the theorem holds in generl if it holds for infinitesiml plnr surfce elements t is well-known tht every surfce cn be tringulted, mening tht it cn be pproximted by tiny djcent plnr tringles tht my be chosen to be right-ngled Without loss of generlity we my plce ny prticulr right-ngled tringle in the xy-plne with sides nd b long the xes uch tringle is described by the conditions (see the mrgin figure), x ; y b; x y b 1: (16) Using tht d Oe z dxdy, nd defining yx/ b1 x=/ nd xy/ 1 y=b/, we get r U d rx U y x; y/ r y U x x; y/ dxdy b Uy xy/; y/ U x x; / dx U d`: U y ; y/ dy b U y xy/; y/ dy Ux x; yx// U x x; / dx U x x; yx// dx b U y ; y/ dy c 2 b 2 A n oriented curve divided into two curves, 1 nd 2, joined t the point c 2 1 An oriented surfce divided into two surfces 1 nd 2 joined t the curve y b x An oriented right-ngled tringle with sides nd b long the coordinte xes opyright c Benny Lutrup

6 618 PHY OF ONTNUOU MATTER n the second step we hve integrted over x in the first term nd over y in the second n the lst step we hve collected the four terms into the curve integrl round the perimeter ccording to its orienttion V 2 1 d 2 d 1 2 V 1 Two volumes V 1 nd V 2 with enclosing surfces 1 nd 2 At common interfce the outwrd normls of the two volumes re opposite x z d z cx; y/ d y y b The shpe of the generl elementry volume Guss theorem f the volume V is divided into two volumes V 1 nd V 2 tht both stisfy this theorem, then the volume V will lso stisfy the theorem becuse the outwrd normls from V 1 nd V 2 hve opposite directions t the common interfce, nd the contributions to the surfce integrls from the common interfce therefore cncel ech other onsequently, by the definition of the volume integrl (12), the theorem holds in generl if it holds for infinitesiml volume elements rving the volume into smller pieces long the coordinte plnes, one my convince oneself tht the generl volume element cn be chosen to be rectngulr box, except ner the surfce where the box must hve curved lid (see the mrgin figure) hoosing suitble coordinte system, it is described by the conditions, x ; y b; z cx; y/; (17) where cx; y/ describes the lid For such volume we prove the theorem for specil vector field only hving z- component, U U z Oe z ntegrting over z we obtin, b cx;y/ r U dv r z U z dv r z U z dxdydz V V b Uz x; y; cx; y// U z d z U d U z x; y; / dxdy Here we used tht ll surfce elements re norml to the box nd directed outwrds, such tht d z on ll fces of the box, except for the one in the xy-plne t z where d z dxdy nd the one t z cx; y/ where d z dxdy Adding ll the little volumes together, we rrive t the theorem for the specil vector field The two other integrls over U U x Oe x nd U U y Oe y yield nlogous results, nd dding ll three concludes the proof 5 Field trnsformtions A field x/ which like the mss density tkes single vlue in ech point of spce, is clled sclr field A field U x/ tht tkes vector vlue in ech point is clled vector field, nd field T x/ tht tkes tensor vlues, tensor field Following the physics trdition, we shll in this chpter denote the rtesin coordintes by x x; y; z/ The trnsformtion lws for fields re quite similr to the ones in section B6, the only difference being tht the coordintes of the sptil position must lso trnsform For sclr, vector nd tensor fields the trnsformtion rules under pure rottions re x / x/; U x / A U x/; T x / A T x/ A > ; (18) (18b) (18c) where x A x These definitions express tht the new fields in the new position re obtined from the old fields in the old position by trnsforming them ccording to their tensor type As mentioned before, such reltions express the unique relity of physicl quntities opyright c Benny Lutrup

7 FEL ALULU 619 Problems 1 lculte the divergence nd curl of the vector field x 2 Prove the following reltions involving the nbl opertor twice (here is sclr field nd V vector field), r r V / ; r r/ ; (19) (2) r r V / rr V / r r/v : (21) Where in these reltions does it mke sense to remove the prentheses? 3 Prove ll the rules from pge 614 * 4 how tht the nbl opertor (1) trnsforms s vector, r i P j ij r j under n rbitrry rottion opyright c Benny Lutrup

8 62 PHY OF ONTNUOU MATTER opyright c Benny Lutrup