Vector Analysis MATH 332
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Dr. Cleo Wunsch
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Date Created: 10/15/15
Ivan Avramidi MATH 332 Vector Analysis Formulas MATH 332 Vector Analysis Formulas Vector Algebra Line parallel to A h z0 i Ait 17 2y7 3Z i70i t A 0 911 2j7 3k Ai 17 Plane orthogonal to N el 39 6739 6 el gtlt ej eijkek Identities Agtlt BXCBAC7CAB A Aiei A 39 B AiBi Tensors ABBA 61776711 lAl2AAAlAi 5113 i j k FigAj A1 A X B eijkAJBkei A1 A2 A3 B1 B2 B3 A X EijkAjBk 8 Ejik Eikj Ekji A gtlt A 0 8m 57M 8W A7B7C A 39 gtlt EijkAiBjOk 8m 8M 8m 0 A7B7C BCA C7A7B Sum Eijk5ik Eijk5jk 0 Ivan Avrarnidi MATH 332 Vector Analysis Formulas EijkAjAk EijkAiAk EijkAiAj 0 i j ij H 6776176 761 5ijk5ijk eijkeml 626 ign g g gym 7 ign g g i ign gl i ign y g mnk i m n m n EijkE i 67 67 mjk 26m EleE 7 l EijkEWk 6 Vector Functions Position Rziei iyjzk Velocity V 7 dB 7 dt Acceleration dv dZR a dt dtz Arc Length d5 lvldt t slttgt Mm dT to Speed d5 V dt Tangent T L M Curvature 1 i dT ilvxal M E M3 Radius of curvature Principal Normal 7 1 dT 7 klvl dt Binorrnal B T gtlt N Torsion 1 dN WM 39E Scalar and Vector Fields Partial derivatives 7 i l 7 3x1 61 6m 32 g 33 dz Nabla Del Operator Gradient grad f Vf 615139 Vif 31f Directional derivative df dR dz E E gmdf E5 Flow curves d i E d5 6 Ivan Avrarnidi MATH 332 Vector Analysis Formulas ds V F 569179 9ze Divergence 1 p Pee ez V X F 6p 69 62 P Fp pF9 F1 CllVFVF6iE 9 192 92 J PltF 2 9 z Curl curl F V gtlt F eijkaijei i j k Spherical Coordinates am 6y 32 dsz drz r2dltp2 72 sin2 ltpdl92 F1 F2 F3 7 2 curl 8117MBij dV 7 r s1nltpdr dltpd6 Laplacian Vf e767 ewlaw e969 f r rsm ltp 2 2 2 2 1 A V V V 6161 6maz v1767T2FT6WSinltpFW r rsrnltp Vector identities 1 59F9 wwov 1 e7 rew rsin gee VVXF0 0 VXFm a aw 9 WM mg fVg 1 F Fr mm 7 2 VltfFl Vf 39 F fVF Af 5570 57f m5w51nlt 5wf VXltfFVfxFfVXF 1 2 2259f df r Sln ltp VfltltPl d ltpVltP VR 3 Line Surface and Volume Integrals V gtlt R 0 Line Integrals F VR F 17 dB 2 r r FdR Femamt dt Cylindrical Coordinates c a dt 17 d5 dpz p2d62 dz E 6 dt dV p dp d6 d2 Potentials 1 mm W epap eepae e161 f F W 3 WW F dR 1011010 Ivan Avramidi MATH 332 Vector Analysis Formulas Q PVltpdR Q 7 MP F VgtltG gt Gzyz 1 Ftxtytz gtltRt dt 0 Unit Normal to a surface R Ruv i 6MB gtlt QER WMR gtlt 3va to a surface fyz C Vf n 7 W 11 Surface Element dS 3MB gtlt QER du d1 d8 WMR gtlt 9va du do For a surface given by 2 fz7y7 a E z S 57 MW 11 311296 dxdy 2 2 dS 16mf 6yf dydx COSV Flux through S SFdSSFndS 17 MW Fn 15mf25yf2dydx 0 MW Divergence Theorem DVFdVSFdS Green7s Theorem 0 D Stokes7 Theorem DVXFdSCFdR Ivan Avramidi MATH 332 Vector and Tensor Analysis Formulas 1 MATH 332 Vector and Tensor Analysis Vector Algebra Scalar Product Commutativity A B B A Magnitude lAlZ A A Vector Product Anti commutativity A gtlt B 7B gtlt A A gtlt A 0 Scalar Triple Product A gtlt B C Cyclic symmetry AxBCBXCACXAB Vector Triple Product AXBXCBAC7CABl Kronecker Symbol Symmetry Unity Matrix 1 0 0 I 0 1 0 0 0 1 Transformation of Rectangular Coordinates r 04mm new Summation from 1 to 3 is assumed over all repeated indices Vector Form 1 1 95951 952 7 95139 952 3 3 Transformation matrix 0411 0412 0413 04 Uzik 0421 0422 0423 0431 0432 0433 Matrix Form of the Transformation x 0w x0 Orthonormal Basis ik k 17 23 Orientation of Orthonormal Basis right handed if i1 gtlt i2 i3 1 left handed if i1 gtlt i2 i3 71 Ivan Avrarnidi MATH 332 Vector and Tensor Analysis Formulas Transformation of Orthonormal Basis i i ljiOtjklk lkiOtjklj 04M cos1j1k Orthogonality condition OtikOLjk 61739 Dim04W 39 Matrix Form of Orthogonality Condition aaTI T means transposition of a matrix replacement of rows by columns Proper transformation no change of orientation CletO jk 1 Improper transformation changes orientation Clef047710 1 Cartesian Vectors Afikik7 AkAik Scalar Product in Cartesian Components lAlZ AiAl Vector Product i1 i2 i3 A X B det A1 A2 A3 B1 B2 B3 Scalar Triple Product A1 A2 A3 A X 39 C det B1 B2 B3 01 02 Ca LeviCivita Alternating Symbol 1 if ijk 127 231 3172 8 71 if ijk 217 321 1372 0 otherwise Eijk 5W Anti symmetry Eijk Ejik 781k igk Cyclic symmetry Eijk 87M 8 Orthonormal basis ijgtlt ik8jklil ij gtlt i1 8 Vector Product in Cartesian Components A X EijkAjBk Scalar Triple Product A X C EijkAiBjOk Tensor Notation Ivan Avramidi MATH 332 Vector and Tensor Analysis Formulas 8m 8M 8m 0 Sum Eijk5z k Eijk5jk 0 EijkAjAk EijkAiAk EijkAiAj 0 aijke m 66 ignog g smug 7 orogag 7 5mm 7 ogoyog eijkemk 7 6m 7 W671 1 Cartesian Tensors tensors in rectangular coordinates Scalar 0 tensor p Vector 1 tensor Al 2 tensor AM n tensor Ahmin Transformation Laws P P A OtikOLjlAkl A A ilmin 0 1 171 39 39 39O injn jlen Matrix Form of the Transformation Laws Vector A1 A A2 A3 A 04A Matrix Form of a 2 tensor A11 A12 A13 A A21 A22 A23 A31 A32 A33 A aAaT Stress Tensor pik Stress 10 10ika m unit normal Moment of lnertia Tensor N M 2 m kimoxgo 7 doggy j1 Angular Momentum Li wk angular velocity Deformation Tensor 1 fl 3 duk 6u1 ulk T 2 an 3 6x143 ui displacement vector Rate of Deformation Tensor 1 3111 30k 0139 i k 2 an 3 ol velocity vector eld Isotropic Terzsors built from 6 only no pre ferred directions Isotropic 2 tensor Aik 105m Ivan Avramidi MATH 332 Vector and Tensor Analysis Formulas 4 Isotropic 4 tensor Aiklm P5ik5im P5ii5km A5z m5ki Tensor Algebra Tensor Product OM Ain Oijkl AijBkl Contraction Djkl Oiijkl Trace A AM Inner Product 7 Dij ijlekl Symmetric Tensors Sij Si 511 512 Sis 12 22 23 13 23 SSS Anti symmetric Tensors A 7A7 0 A12 A13 AM A12 0 A23 A13 A23 0 Contraction of antisymmetric tensor Symmet rizat ion Antisymmetrization 1 TM Tik TM Decomposition of 2 Tensor Tm Tik Tm Duality Equivalence of antisymmetric 2 tensor to an ax ial vector 1 Ar EijkAjlm Aij EijkAk 2 Al A237 A2 A317 A3 A127 Principal Axes Eigenvalues Am characteristic values and Eigenvectors n T characteristic or principal di rections Tani Mn lnml 1 Characteristic Equation ClefTM 7 0 T11 7 A T12 T13 det T21 T22 7 A T23 0 T31 T32 T33 7 A A3711A212A7130 Invariants of a 2 Tensor k 11 Tn39T11 T22T33 T22 T23 T11 T12 I d t d t 2 e T32 T33 e T21 T22 Ivan Avrarnidi MATH 332 Vector and Tensor Analysis Formulas detlt 3 detTk det T1 1 T1 3 T3 1 T33 T1 1 T12 T2 1 T22 T3 1 T32 T1 3 T23 T33 The eigenvalues AT T 123 of a symmetric 2 tensor are real A symmetric 2 tensor has three orthogonal prin cipal axes n r 1 23 no no 5m In the principal axes a symmetric 2 tensor Ti has diagonal matrix Ti A1 0 0 0 A2 0 0 0 A3 Decomposition in terms of orthonormal eigenvec tors n and eigenvalues Am 3 Ti Z AUWEMTLEJ 71 Traceless Tensors Deviators i 1 Tm Tm 5ikT where T Ti trace Decomposition Curvilinear Coordinates ql39 fir 21273 Cartesian Coordinates 95139 951M Radius vector 111 mg i k 961Qi1 962q i2 96361 i3 Basis tangent vectors to coordinate curves for meaqi Transformation of basis 39 ek o ke k ejiOtjek J Uzik04ij OtikOtkj Orientation right handed if e1 gtlt e2 eg gt 0 and left handed if e1 gtlt e2 e3 lt 0 Reciprocal Basis ek ekgtlt e1 61 X 6239 63 where jkl 123 231 312 e7 Contravariant Components A Covariant components A Alei Ivan Avrarnidi MATH 332 Vector and Tensor Analysis Formulas Transformation of components A ai kAk A Mink Metric Tensor 3r 3r M l k gikei ek67qi 67qu 9 e 393 Symmetry 9m 9M7 9M 9 9mng Determinant of the Metric Tensor G detgik7 detgik g The matrix 9 is the inverse matrix of the matrix am given by gik ilykdet G where M is a 2 gtlt 2 matrix obtained from the 3 gtlt 3 matrix 917 by removing the t th row and the k th column Relations between components Displacement Line Element Arc Length 3r 3r 2 77 d5 7dr dr aqi aqk dqi qu dsz 9m dqi qu Volume Element ldv xadql dqqug G detgik Tensors in Curvilinear Coordinate System Contravariant components A Covariant components AM Mixed components Aik Aik Relations 7 r L i r L 7 Wm Aik gmAnk gknAm etc Orthogonal Coordinate System 3r 3r 7 0 if 39 k dq qu 7 1 Z 7g Basis ei ek eiek0 ifz39y k l i hi7 l le l le l hi Metric 91 0 if 239 7 k 91439 h no summation g 0 if 239 7 k ii 1 i V g no summation Metric Coef cients Ivan Avrarnidi MATH 332 Vector and Tensor Analysis Formulas 6 2 a 2 6 2 1 pcos ltp hi 39 39 39 R d v aqz aqz aqz a ius ector Relation between covariant and contravariant com ponents 2psinltp7 232 r pcosltpi1 psinltpi2 zi3 Orthonormal basis Ai hlei no summation ep cosltpi1 sinltpi2 e Orthonormal Basis Weisin i1cosltpi2 ez i3 6139 Line Element 1 6139 e ei q 2 2 2 2 2 6r d5 7dp p do d2 dqi Volume Element Line Element lets h ltdq1gt2 h ltqu hi dqs Volume Element dV pdp dltp dz Metric Coef cients hp1 h p hz1 dV hlhghg dql dqz qu Cartesian Coordinates Spherical Coordinates Line Element r207 0 0 7n 0 ltplt27r 2 2 d5 ltdz1gt2 cm ltsz r M z tame V T 9 2 3 Volume Element tan p Q 1 dV d l dig dig Metric Coef cients 1 7quotsinl9cosltp7 2 7 sinl9sinltp7 3 r cos 6 hl hz ha 1 Radius Vector r rsin0cosltpi1 rsin0sinltpi2 rcosl9i3 Cylindrical Coordinates P20 0 ltplt27n p96 9c 7 tambi 1 Orthonormal Basis T sinl9cosltpi1 sin0sinltpi2cos0i3 e9 cosl9cosltpi1 cos0sinltpi2 isin ig Ivan Avrarnidi MATH 332 Vector and Tensor Analysis Formulas e 7sinltp11oosltp12 Line Element dsz dr2 r2d62 r2 sinZ 6dltp2 Volume Element dV rzsin dr dl9dltp Metric Coef cients Vector And Tensor Analysis Functions of Single variable Product Rules d do dA WA EA H77 E dt d dB dA i B A 71x B 7 dt dt dt d dB dA ampB gtltAEgtltAB X Trajectory rt xkt ik a S t S b Velocity V e 7 dt Unit Tangent Vector dr dl E u i E 7 dt Speed 7 dr 7 d5 lVl a i a Acceleration 7 dv 7 dgr 3 7 E 7 E Arc Length Line Element d5 lvldt Fields in Cartesian Coordinates Partial derivatives 6 l Nabla Del Operator V ik 3k i151 i252 i353 Gradient grad f Vf ik 5kf grad fi aif Directional Derivative gradf u gradf Flow Lines dr E dil i dig i dig E 7 E 7 E F Divergence divFVF6iFi Ivan Avramidi MATH 332 Vector and Tensor Analysis Formulas 9 curl i1 i2 i3 curlF 61 62 63 F1 F2 F3 V 1 curl gigk6ij T curl F v gtlt F il eljkaij WW 1 T T I39 dF 7 dr dF dr Laplacian V F T AdivgradV2VV Aaiai6 6 6 Vgtlt Frgtlt FVrF Vector identities Flelds 1n Orthogonal Coordlnate Sys V X V X F AF VV F tern in orthonormal basis ei curl curl F iAF grad div F Vector Components VXVltp07 V V gtlt F 0 1 1 6 1 6 1 6 gm F1671 62726qu egaquf ng WM fV9 VltfF W F NF 1 VgtltfFVfgtltFfVgtltF dIVF W6qlh2h3F1 VF gtlt GGVgtlt FiFVgtlt G 6 6 ih3h1F2 h1h2F3 d 2 3 WW 1W 69 69 dltp dF v F 7 V P T a a F curl F e1 hzhs Emmy i M 112112 V gtlt Fltp Vltp gtlt 7 dltp 1 6 6 W 3 Wm WW WWW Ivan Avrarnidi MATH 332 Vector and Tensor Analysis Formulas 1 6 6 Wm lwlth2F2gt lwlth1FIgtl 1 1 7 i 2 Af i T2670quot 37f T2 Sin669s1n669f l 3 hghg 3 A 7 7 1 f h1h2h36q1 hl aql maif 130213 3021133 aql hl W W hl aql Integrals Parametrization of a curve C a lt t lt b Cylindrical Coordinates 1 rad e 3 e 76 ezaz g f p pf WP Wf f Line Integrals d l dt FdrbFzt dt 0 a 1 1 diVF 76pFJ 73wa 3ze P P Circulation of vector eld along a closed contour 1 ep pew ez curlF 7 3p 3 dz p FJ pFw Fz C 1 1 Af 5pP5pf jaifJfagf p p Parametrization of a surface S Spherical Coordinates r 117107 a S u 3 b7 0 S 1 S d 1 1 grad f erarf eeiaef ew iawf Unlt Normal r rs1n0 aur gtlt avr n 1 1 laur gtlt am div F 7267MB famine F9 r rs1n0 For a surface gwen by La F rsinl W W frC l the Unit Normal is e T69 rs1n0e QT 69 3 7 Vf lVfl 1 curl F T Sln F TF9 rsinl9Fw Ivan Avramidi MATH 332 Vector and Tensor Analysis Formulas 11 Surface Element dS ndS dur gtlt dur dude dS Bur gtlt Surf dude For a surface given by 2 f 967 y aSQch y196 Sy yzm the Surface Element is wWWme Surface Integral of a scalar eld d b ltpdSltpruv laurx Surf dude S c a 17 MW LthWlD a yl 1 emf dzf dy dx Flux of a Vector Field F through the surface S FdSFndS S S Line lntegral of a Gradient grad ltp dr Q 7 ltMP U Circulation of a Gradient along a Closed contour fgradowdr 0 C Gauss Divergence Theorem dideVFdS D 30 3D is a Closed surface7 which is the boundary of the solid region D Green7s Theorem 61F2 7 62F1d1d2 fltF1d1 ng g S BS 3S is a Closed plane curve7 which is the boundary of the region S in the sag plane Stokes7 Theorem curlF dSfF dr S as 3S is a Closed space curve7 which is the boundary of the surface S Flux of a curl through a closed surface S 0 curlF dS0 Tensor Fields Flux of a Tensor Field rmw iww S S Ivan Avramidi MATH 332 Vector and Tensor Analysis Formulas 12 Divergence of a Tensor in Cartesian Coordi nates 51TH akTik Directional Derivative in Cartesian Coordi nates dle dxj 7 7 3T d5 d5 7 1k Analog of curl of an antisymmetric 2 tensor 35ijk6iAjk 51A23 62AM 33A12 9141 62A2 6343 MATH 332 Vector Analysis Tensors l MATH 332 Vector Analysis Tensors Ivan Avramidi New Memico Tech Cartesian Coordinate System First of all7 let us introduce a Cartesian coordinate system in three dimensional Euclidean space We will denote the coordinates by 1 zzy 3z 1 and the unit vectors in the direction of positive axes called the basis vectors by E 1l7 e2j 3k Index Notation This can be denoted simply by xi and ej where ij 17 27 3 For the indices one usually uses the lowercase Latin letters i7j7 k l7 m7 it etc do not confuse with ijk If you run out of letters7 you can use any other letters The convention is though that the indices are denoted by small versus capital Latin versus Greek letters7 and take values 123 Greek indices are used in four dimensional space time in special relativity7 where they take values 01237 with x0 t denoting time Kronecker Delta Symbol The scalar products of the basis vectors are 17 if i j ei ei 0 if i797 3 One says that they form an orthonormal system This can be written in a compact form by de ning so called Kronecker symbol 6i i 1 if i j 6 0 if 279739 4 This can also be represented by the unit 3 gtlt 3 matrix 1 0 0 67 0 1 0 5 0 0 1 MATH 332 Vector Analysis Tensors 2 Then e 39 6739 6H Scalars Physical quantities like mass energy volume temperature den sity etc that can be described by one number are called scalars This number does not depend on the coordinate system it is an invariant Vectors Vectors are physical quantities like velocity position displace ment force acceleration electric eld magnetic eld etc that are described by three numbers Tensors A tensor is a geometric object that requires for its full description more than just one number as scalar and even more than three numbers as a vector Examples of tensors include stress tensor strain tensor inertia tensor energy momentum tensor tensor of the electromagnetic eld metric tensor curvature tensor etc Tensor Components These numbers are called the components of the tensor The components of a tensor are labeled by indices for example 5M Eijkv T77 B 739 Um R Mk 7 A tensor whose all components are zero is called a zero tensor Types of Tensors The tenswors with upper indices are called contravari ant and the ones with lower indices are called covariant If a tensor has both types of indices then it is of mixed type The total number of indices is called the rank of the tensor A tensor that has p upper indices and q lower indices 1 I Tilipj1qu is called a tensor of type p q So a scalar is a tensor of rank 0 A vector is a tensor of rank 1 Transformation Law The actual numerical values of the components of a tensor do depend on the coordinate system If one changes the coordinate system for example rotates it then the components of a tensor will change If one goes from the Cartesian coordinate system to a curvilinear coordinate MATH 332 Vector Analysis Tensors 3 system for example a system of spherical or cylindrical coordinates then the components of a tensor will also change It is this transformation law of the components of the tensor that makes a collection of numbers a tensor We will not give the formal de nition of a tensor rather we give here a very short review of tensor analysis in Cartesian coordinates along with some very useful formulas and rules that enable one to deal with tensors Metric Tensor ln Cartesian coordinates the square of the distance be tween two in nitesimally close points in space one with coordinates xi and another with coordinates d is 6182 61902 dz2 dz2 ZWT 9 i1 This can be written in the following form Mo d52 i1j1 The distance between in nitesimally close points determines a tensor of rank 2 so called metric tensor gij In general coordinate system one has 3 3 d52 Z Z gigd dx 11 i1 j1 Thus the covariant components of the metric tensor in Cartesian coordinates are given by Kronecker delta symbol 9o 5o 12 The contravariant components of the metric tensor are de ned by 9M 91771 13 In Cartesian coordinates 9 5o 14 MATH 332 Vector Analysis Tensors 4 Tensor Equations 0 In any tensor equation an index can appear only once single indeco or twice repeated indem For example Ail is impossible o A single index can be either covariant in the whole equation or con travariant in the whole equation It cannot be contravariant in one term and covariant in another term For example A7 Bij is wrong 0 The repeated indices always appear in pairs one covariant and another contravariant For example Allj o A pair of repeated indices cannot appear more than once For example Alll is wrong Einstein Summation Convention ln tensor analysis one always en counters the sums over the indices that appear twice in an equation For examle in the formula for the distance above the indices i and j appear twice and there is summation over i and j running from 1 to 3 Accord ing to the standard convention called Einstein summation convention one has agreed to sum oveiquot repeated indices and omit the summation signs For example 3 3 i1 j1 3 ABquot Z ABquot 16 i1 Ti 17 i1 3 leik Zleik 18 i1 3 3 own Z Z own 19 i1 j1 3 3 3 EijkAjkBi i1 j1 k1 MATH 332 Vector Analysis Tensors 5 Raising and Lowering Indices The metric tensor can be used to raise and lower indices of tensors For example if A1 are contravariant components of a vector then its covariant components are A 6in 21 Conversely Al WAJ 22 This operations called raising and lowering indices can be applied to any tensor If one applies it to the metric tensor one gets 6 776 23 By raising and lowering indices any tensor can be put in a covariant or contravariant form One has to be careful though with the order of indices For example I I I 1 Al SJMilk 7 A SJMil 24 Remark ln Cartesian coordinates the covariant and contravariant com ponents are equal since the metric tensor is given by the Kronecker symbol Therefore in this case it does not make any di erence and all indices can be placed down for example Properties and Identities of Kronecker Delta Symbol 6 6 25 6 6 a 6 26 63 3 27 639A A 28 6 AB AiB A B Addition One can add tensors of the same type The result is a tensor of the same type Multiplication By Scalars One can multiply tensors by scalars The result is a tensor of the same type MATH 332 Vector Analysis Tensors 6 Tensor Multiplication If one multiplies a tensor of rank 7 with a tensor of rank k one gets a new tensor of rank rk More precisely if one multiplies a tensor of type pq with a tensor of type r s then one gets a new tensor of type p r q s For example AiBj OH Tanl39j Rmnl j Note one just multiplies the components of the tensors without any sum mation Contraction Given a tensor of type p q that is of rank 7 p q one may select a pair of indices of which one should be an upper index and another an lower index and replace them by two identical repeated indices summation over the latter being implied by the summation convention This process is called contraction As a result one gets a new tensor of type p71q 7 1 of rank 7 7 2 p q 7 2 For example All Rijkn Oikk 31 Clearly 6 6116 6 3 32 Symmetrization and Antisymmetrization A tensor A of rank 2 is said to be symmetric if AU Aji 33 and anti symmetric or skew symmetric if AU Aa39i 34 Any tensor A of second rank can be decomposed Au AW AW 35 into its symmetric 1 AW 5Aia39 Aji 36 and anti symmetric parts 1 Aw Aij A 37 2 MATH 332 Vector Analysis Tensors 7 One can also symmetrize a tensor over three indices 1 Bait 6Bz jk BM Bkij Bikj 37 Bkji 38 Correspondingly the anti symmetrization over three indices is de ned by 1 Bijk BM Bkij BM BM Bkji 39 BM 6 What one does here is one sums over all possible permutations of indices and changes sign if the permutation is odd Contraction of Symmetric and Antisymmetric Tensors Let A be a symmetric tensor and BH an antisymmetric Then AijBij AjiBij AjiBji AijBijv and therefore H AME 0 41 Levi Civita Symbol Levi Civita symbol 67 is de ned by 1 if 2 k is an even permutation of 1 23 677 71 if 2 k is an odd permutation of 1 23 42 0 otherwise If one raises the indices then one sees that in Cartesian coordinates one obtains the same symbol so that Eijk Eijk The Levi Civita symbol de nes a tensor of rank 3 called a Levi Civita tensor This is another very important tensor that is purely geometric in nature It describes not the distances but the volume in three dimensional space The volume of a parallelepiped based on three displacement vectors Al B7 Ck is V 5jkAi Bi 0k 44 MATH 332 Vector Analysis Tensors 8 Properties and Identities of LeviCivita Symbol The Levi Civita symbol de nes a completely antisymmetric tensor The following properties immediately follow from its de nition 857 Ejik Eikj Ekji 4 on 8m 57M 8W 4 1 8H 871739 877439 0 Eijk j Eijk Eijk j 0 EijkAjAk EijkAl Ak EijkAl Aj 0 4 00 4 VVVVVV AAAAAA Eijk 817 50 eijkeml 66g y g 51 7 arm ign g g 6mm 76mm 7 smug 7 ignay g 52 eijkemk 7 26 7 6m 7 6 53 cijkcmjk 26quot 54 gigkg 6 55 Vector Operations in Tensor Notation Tensor notation is very use ful in vector analysis7 in particular when manipulating the multiple vector products and vector identities First of all7 the scalar and vector products of two vectors A and B are given by A B AiBi 56 and A X EijkAjBk The triple product of three vectors A7 B and C is then given by ABCAB gtlt Cei7kAiBJOk 58 MATH 332 Vector Analysis Tensors 9 Note that the position of indices up versus down in Cartesian coordinates is not important However it is still more clear when you see one index up and the same index down then you should immediately notice that this is a contraction and there is a summation over this index from 1 to 3 We repeat once again that the name of such repeated indices is not important they are dummy indices one can rename them to any other letter if needed make sure that there are no other indices with that name in the given tensor equationl By using the properties of Levi Civita symbol and Kronecker symbol one can derive now all vector identities For example A gtlt B gtlt C gtlt D EMA gtlt B7C gtlt Dk cijkcjmnAmBnckququ 6 6 6 em AmBnOqu 658mm 7 fcpm AmBnOqu cqmnAmBnOiDq icp mAmBnOpD 59 D7A7Bl0i7lC7A7BlDi 60 In other words we have just proven the following vector identity AxBXCxDDABC7CABD 61 do not forget that the triple product is a scalar
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