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Appendix a Relations Between Covariant and Contravariant Bases Appendix A Relations Between Covariant and Contravariant Bases The contravariant basis vector gk of the curvilinear coordinate of uk at the point P is perpendicular to the covariant bases gi and gj, as shown in Fig. A.1.This contravariant basis gk can be defined as or or a gk g  g ¼  ðA:1Þ i j oui ou j where a is the scalar factor; gk is the contravariant basis of the curvilinear coordinate of uk. Multiplying Eq. (A.1) by the covariant basis gk, the scalar factor a results in k k ðgi  gjÞ: gk ¼ aðg : gkÞ¼ad ¼ a ÂÃk ðA:2Þ ) a ¼ðgi  gjÞ : gk gi; gj; gk The scalar triple product of the covariant bases can be written as pffiffiffi a ¼ ½¼ðg1; g2; g3 g1  g2Þ : g3 ¼ g ¼ J ðA:3Þ where Jacobian J is the determinant of the covariant basis tensor G. The direction of the cross product vector in Eq. (A.1) is opposite if the dummy indices are interchanged with each other in Einstein summation convention. Therefore, the Levi-Civita permutation symbols (pseudo-tensor components) can be used in expression of the contravariant basis. ffiffiffi p k k g g ¼ J g ¼ðgi  gjÞ¼Àðgj  giÞ eijkðgi  gjÞ eijkðgi  gjÞ ðA:4Þ ) gk ¼ pffiffiffi ¼ g J where the Levi-Civita permutation symbols are defined by 8 <> þ1ifði; j; kÞ is an even permutation; eijk ¼ > À1ifði; j; kÞ is an odd permutation; : A:5 0ifi ¼ j; or i ¼ k; or j ¼ k ð Þ 1 , e ¼ ði À jÞÁðj À kÞÁðk À iÞ for i; j; k ¼ 1; 2; 3 ijk 2 H. Nguyen-Schäfer and J.-P. Schmidt, Tensor Analysis and Elementary 197 Differential Geometry for Physicists and Engineers, Mathematical Engineering 21, DOI: 10.1007/978-3-662-43444-4, Ó Springer-Verlag Berlin Heidelberg 2014 198 Appendix A: Relations Between Covariant and Contravariant Bases Fig. A.1 Covariant and g 3 u3 contravariant bases of curvilinear coordinates g 3 x 3 g 1 P g e 2 3 g 2 2 1 u u g 1 0 e e1 2 x 2 x 1 Thus, the cross product of the covariant bases gi and gj results from Eq. (A.4): ffiffiffi p k k k ðgi  gjÞ¼eijk g g ¼ eijkJg ^eijkg eijkðgi  gjÞ ) gk ¼ pffiffiffi ¼ ^eijkðg  g Þ ðA:6Þ g i j ) ^eijk ¼ðgi  gjÞgk The covariant permutation symbols in Eq. (A.6) can be defined as 8 pffiffiffi < þ g if ði; j; kÞ is an even permutation; pffiffiffi ^eijk ¼ : À g if ði; j; kÞ is an odd permutation; ðA:7Þ 0ifi ¼ j; or i ¼ k; or j ¼ k The contravariant permutation symbols in Eq. (A.6) can be defined as 8 p1ffiffi < þ g if ði; j; kÞ is an even permutation; ijk ^e ¼ p1ffiffi ðA:8Þ : À g if ði; j; kÞ is an odd permutation; 0ifi ¼ j; or i ¼ k; or j ¼ k k The covariant basis vector gk of the curvilinear coordinate of u at the point P is perpendicular to the contravariant bases gi and gj, as shown in Fig. A.1. Therefore, the cross product of the contravariant bases gi and gj can be written as i j eijk eijk ijk ðg  g Þ¼ pffiffiffiffi gk ¼ gk ^e gk g J ðA:9Þ ) ^eijk ¼ðgi  g jÞgk Thus, the covariant basis results from Eq. (A.9): ffiffiffiffi p i j i j gk ¼ eijk g ðg  g Þ¼eijkJðg  g Þ i j ðA:10Þ ¼ ^eijkðg  g Þ Obviously, there are some relations between the covariant and contravariant permutation symbols: ijk ^e ^eijk ¼ 1 ðno summationÞ A:11 ijk 2 ð Þ ^eijk ¼ ^e J ðno summationÞ Appendix A: Relations Between Covariant and Contravariant Bases 199 The tensor product of the covariant and contravariant permutation pseudo- tensors is a sixth-order tensor. 8 < þ1; ði; j; kÞ and ðl; m; nÞ even permutation; ^ijk^ ijk e epqr ¼ dpqr ¼ : À1; ði; j; kÞ and ðl; m; nÞ odd permutation; ðA:12Þ 0; otherwise The sixth-order Kronecker tensor can be written in the determinant form: di di di p q r ijk ijk j j j ^e ^epqr ¼ dpqr ¼ dp dq dr ðA:13Þ k k k dp dq dr Using the tensor contraction rules with k = r, one obtains i i i i i i dp dq dr d d d p q r dij ¼ dijr ¼ j j j ¼ j j j pq pqr dp dq dr dp dq dr dr dr dr 001 : p q r ðA 14Þ i i dp dq ^eij^e dij 1 di d j di d j ) pq ¼ pq ¼ Á j j ¼ p q À q p dp dq Further contraction of Eq. (A.14) with j = q gives iq i q i q ^e ^epq ¼ dpdq À dqdp i q i i i i ðA:15Þ ¼ dpdq À dp ¼ 2dp À dp ¼ dp for i; p ¼ 1; 2 From Eq. (A.15), the next contraction with i = p gives pq p ^e ^epq ¼ dp ðsummation over pÞ A:16 1 2 ð Þ ¼ d1 þ d2 ¼ 2 for p ¼ 1; 2 Similarly, contracting Eq. (A.13) with k = r; j = q, one has for a three- dimensional space. iq i q q i ^e ^epq ¼ dpdq À dpdq i q i i i i ðA:17Þ ¼ dpdq À dp ¼ 3dp À dp ¼ 2dp for i; p ¼ 1; 2; 3 Contracting Eq. (A.17) with i = p, one obtains pq p ^e ^epq ¼ 2dp ðsummation over pÞ 1 2 3 ¼ 2ðd1 þ d2 þ d3Þ ðA:18Þ ¼ 2ð1 þ 1 þ 1Þ¼6 for p ¼ 1; 2; 3 200 Appendix A: Relations Between Covariant and Contravariant Bases The covariant metric tensor M can be written as 2 3 g11 g12 g13 4 5 M ¼ g21 g22 g23 ðA:19Þ g31 g32 g33 where the covariant metric coefficients are defined by gij ¼ gi Á gj. The contravariant metric coefficients in the contravariant metric tensor M-1 result from inverting the covariant metric tensor M. 2 3 g11 g12 g13 MÀ1 ¼ 4 g21 g22 g23 5 ðA:20Þ g31 g32 g33 where the contravariant metric coefficients are defined by gij ¼ gi Á g j. Thus, the relation between the covariant and contravariant metric coefficients can be written as ik ik i À1 À1 g gkj ¼ gkjg ¼ dj , M M ¼ MM ¼ I ðA:21Þ ik ik In the case of i 6¼ j, all terms of g gkj equal zero. Thus, only nine terms of g gki for i = j remain in a three-dimensional space R3: ik 1k 2k 3k g gki ¼ g gk1 þ g gk2 þ g gk3 for i; k ¼ 1; 2; 3 1 2 3 i ¼ d1 þ d2 þ d3 ¼ di for i ¼ 1; 2; 3 ðA:22Þ ¼ 1 þ 1 þ 1 ¼ 3 The relation between the covariant and contravariant bases in the general curvilinear coordinates results in i ik i g :gj ¼ g gkj ¼ dj for i j i 1 2 3 ) g :gi ¼ g :g1 þ g :g2 þ g :g3 for i ¼ 1; 2; 3 ðA:23Þ ik i ¼ g gki ¼ di for i; k ¼ 1; 2; 3 ik According to Eq. (A.23), nine terms of g gki for k = 1, 2, 3 result in 8 1 1k 11 12 13 1 <> g Á g1 ¼ g gk1 ¼ g g11 þ g g21 þ g g31 ¼ d1 for i ¼ 1; g2 Á g ¼ g2kg ¼ g21g þ g22g þ g23g ¼ d2 for i ¼ 2; ðA:24Þ :> 2 k2 12 22 32 2 3 3k 31 32 33 3 g Á g3 ¼ g gk3 ¼ g g13 þ g g23 þ g g33 ¼ d3 for i ¼ 3: The scalar product of the covariant and contravariant bases gives ðiÞ ðiÞ ðiÞ g Á gðiÞ ¼ g Á gðiÞ cosð g ; gðiÞÞ qffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffi ðA:25Þ ðiiÞ p ðiÞ ¼ g Á gðiiÞ cosðg ; gðiÞÞ¼1 where the index (i) means no summation is carried out over i. Appendix A: Relations Between Covariant and Contravariant Bases 201 Equation (A.25) indicates that the product of the covariant and contravariant basis norms generally does not equal one in the curvilinear coordinates. qffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 1 gðiiÞ g 1 A:26 Á ðiiÞ ¼ ðiÞ ð Þ cosð g ; gðiÞÞ (i) In orthogonal coordinate systems, g is parallel to g(i). Therefore, Eq. (A.26) becomes qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffi ðiiÞ p ðiiÞ 1 1 g Á gðiiÞ ¼ 1 ) g ¼ pffiffiffiffiffiffiffiffi ¼ ðA:27Þ gðiiÞ hi Appendix B Physical Components of Tensors The physical component of a tensor can be defined as the tensor component on its unitary covariant basis. Therefore, the covariant basis of the general curvilinear coordinates has to be normalized. Dividing the covariant basis by its vector length, the unitary covariant basis (covariant-normalized basis) results in à gi gi à gi ¼ ¼ pffiffiffiffiffiffiffiffi ) gi ¼ 1 ðB:1aÞ jjgi gðiiÞ The covariant basis norm |gi| can be considered as a scale factor hi without summation over (i). pffiffiffiffiffiffiffiffi hi ¼ jjgi ¼ gðiiÞ ðB:1bÞ Thus, the covariant basis can be related to its unitary covariant basis by the relation ffiffiffiffiffiffiffiffi p à à gi ¼ gðiiÞgi ¼ higi ðB:2Þ The contravariant basis can be related to its unitary covariant basis using Eqs. (2.47 and B.2). i ij ij à g ¼ g gj ¼ g hjgj ðB:3Þ The contravariant second-order tensor can be written in the unitary covariant bases using Eq. (B.2). ij ij à à Ãij à à T ¼ T gigj ¼ðT hihjÞ gi gj T gi gj ðB:4Þ Thus, the physical contravariant tensor components denoted by star result in Ãij ij T hihjT ðB:5Þ The covariant second-order tensor can be written in the unitary contravariant bases using Eq.
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