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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 ∂r ∂r α gk g  g ¼  ðA:1Þ i j ∂ui ∂uj where α is the scalar factor; gk is the contravariant basis of the curvilinear coordi- nate of uk. g3 u3 g3 x3 g1 P g e 2 3 g2 u2 u1 g 0 e 1 e1 2 x2 x1 Fig. A.1 Covariant and contravariant bases of curvilinear coordinates © Springer-Verlag Berlin Heidelberg 2017 313 H. Nguyen-Scha¨fer, J.-P. Schmidt, Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers, Mathematical Engineering, DOI 10.1007/978-3-662-48497-5 314 Appendix A: Relations Between Covariant and Contravariant Bases Multiplying Eq. (A.1) by the covariant basis gk, the scalar factor α results in ÀÁ  Á ¼ α k Á ¼ αδk ¼ α gi gj gk g gk k hi ðA:2Þ ) α ¼  Á ; ; gi gj gk gi gj gk The scalar triple product of the covariant bases can be written as pffiffiffi α ¼ ½¼; ; ðÞÁ ¼ ¼ ð : Þ g1 g2 g3 g1 g2 g3 g J A 3 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. There- fore, the Levi-Civita permutation symbols (pseudo-tensor components) can be used in expression of the contravariant basis. pffiffiffi g gk ¼ J gk ¼ g  g ¼À g  g i j j i ε  ε  ðA:4Þ ijk gi gj ijk gi gj ) gk ¼ pffiffiffi ¼ g J where the Levi-Civita permutation symbols are defined by 8 < þ1ifðÞi, j, k is an even permutation; ε ¼ À ðÞ ; ijk : 1if i, j, k is an odd permutation 0ifi ¼ j,ori ¼ k; or j ¼ k ðA:5Þ 1 , ε ¼ ðÞÁi À j ðÞÁj À k ðÞk À i for i, j, k ¼ 1, 2, 3 ijk 2 Thus, the cross product of the covariant bases gi and gj results from Eq. (A.4): pffiffiffiffi g  g ¼ ε g gk ¼ ε Jgk ^ε gk i j ijk ijk ijk ε  ijk gi gj ) k ¼ ¼ ^ε ijk  ðA:6Þ g pffiffiffi gi gj g ) ^ε ¼  Á ijk gi gj gk The covariant permutation symbols in Eq. (A.6) can be defined as Appendix A: Relations Between Covariant and Contravariant Bases 315 8 pffiffiffi < þ ðÞ ; pffiffiffig if i, j, k is an even permutation ^ε ¼ À ðÞ ; ð : Þ ijk : g if i, j, k is an odd permutation A 7 0ifi ¼ j,ori ¼ k; or j ¼ k The contravariant permutation symbols in Eq. (A.6) can be defined as 8 > 1 > þ pffiffiffi if ðÞi, j, k is an even permutation; < g ^ε ijk ¼ 1 ð : Þ > Àpffiffiffi ðÞ ; A 8 > if i, j, k is an odd permutation :> g 0ifi ¼ j,ori 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 ε ε ðÞ¼gi  gj pijkffiffiffiffig ¼ ijkg ^ε ijkg g k J k k ðA:9Þ ) ^ε ijk ¼ ðÞÁgi  gj gk Thus, the covariant basis results from Eq. (A.9): pffiffiffi ÀÁÀÁ g ¼ ε g gi  gj ¼ ε J gi  gj k ijk ÀÁ ijk ð : Þ i j A 10 ¼ ^ε ijk g  g Obviously, there are some relations between the covariant and contravariant per- mutation symbols: ^ε ijk^ε ¼ ðÞ ijk 1 no summation ð : Þ ijk 2 A 11 ^ε ijk ¼ ^ε J ðÞno summation The tensor product of the covariant and contravariant permutation pseudo-tensors is a sixth-order tensor. 8 < þ1; ðÞi, j, k and ðÞl; m; n even permutation ^ε ijk^ε ¼ δijk ¼ À ; ðÞðÞ; ; ð : Þ pqr pqr : 1 i, j, k and l m n odd permutation A 12 0; otherwise The sixth-order Kronecker tensor can be written in the determinant form: 316 Appendix A: Relations Between Covariant and Contravariant Bases δ i δ i δ i p q r ^ε ijk^ε ¼ δijk ¼ δ j δ j δ j ð : Þ pqr pqr p q r A 13 δ k δ k δ k p q r Using the tensor contraction rules with k ¼ r, one obtains δ i δ i δ i δ i δ i δ i p q r p q r δij ¼ δijr ¼ δ j δ j δ j ¼ δ j δ j δ j pq pqr p q r p q r δ r δ r δ r p q r 001 ðA:14Þ δ i δ i ) ^ε ij^ε ¼ δij ¼ : p q ¼ δ i δ j À δ i δ j pq pq 1 δ j δ j p q q p p q Further contraction of Eq. (A.14) with j ¼ q gives iq i q i q ^ε ^ε pq ¼ δ δ À δ δ p q q p ðA:15Þ ¼ δ i δ q À δ i ¼ δ i À δ i ¼ δ i ¼ p q p 2 p p p for i, p 1, 2 From Eq. (A.15), the next contraction with i ¼ p gives pq p ^ε ^ε pq ¼ δ ðÞsummation over p p ðA:16Þ ¼ δ1 þ δ2 ¼ ¼ 1 2 2forp 1, 2 Similarly, contracting Eq. (A.13) with k ¼ r; j ¼ q, one has for a three-dimensional space. iq i q q i ^ε ^ε pq ¼ δ δ À δ δ p q p q ðA:17Þ ¼ δ i δ q À δ i ¼ δ i À δ i ¼ δ i ¼ p q p 3 p p 2 p for i, p 1, 2, 3 Contracting Eq. (A.17) with i ¼ p, one obtains ^ε pq^ε ¼ 2δ p ðÞsummation over p pq ÀÁp ¼ δ1 þ δ2 þ δ3 ð : Þ 2 1 2 3 A 18 ¼ 21ðÞ¼þ 1 þ 1 6forp ¼ 1, 2, 3 The covariant metric tensor M can be written as 2 3 g11 g12 g13 ¼ 4 5 ð : Þ M g21 g22 g23 A 19 g31 g32 g33 ¼ Á where the covariant metric coefficients are defined by gij gi gj. Appendix A: Relations Between Covariant and Contravariant Bases 317 À The contravariant metric coefficients in the contravariant metric tensor M 1 result from inverting the covariant metric tensor M. 2 3 g11 g12 g13 À M 1 ¼ 4 g21 g22 g23 5 ðA:20Þ g31 g32 g33 where the contravariant metric coefficients are defined by gij ¼ gi Á gj. Thus, the relation between the covariant and contravariant metric coefficients can be written as ik ¼ ik ¼ δ i , À1 ¼ À1 ¼ ð : Þ g gkj gkjg j M M MM I A 21 ik ik In case of i 6¼ j, all terms of g gkj equal zero. Thus, only nine terms of g gki for i ¼ j remain in a three-dimensional space R3: ik ¼ 1k þ 2k þ 3k ¼ g gki g gk1 g gk2 g gk3 for i, k 1, 2, 3 ¼ δ1 þ δ2 þ δ3 ¼ δ i ¼ ð : Þ 1 2 3 i for i 1, 2, 3 A 22 ¼ 1 þ 1 þ 1 ¼ 3 The relation between the covariant and contravariant bases in the general curvilin- ear coordinates results in i Á ¼ ik ¼ δ i; ) g gj g gkj j for i j i Á ¼ 1 Á þ 2 Á þ 3 Á ¼ ð : Þ g gi g g1 g g2 g g3 for i 1, 2, 3 A 23 ¼ ik ¼ δ i ¼ g gki i for i, k 1, 2, 3 ik According to Eq. (A.23), nine terms of g gki for k ¼ 1,2,3 result in 8 < 1 Á ¼ 1k ¼ 11 þ 12 þ 13 ¼ δ1 ¼ ; g g1 g gk1 g g11 g g21 g g31 1 for i 1 2 Á ¼ 2k ¼ 21 þ 22 þ 23 ¼ δ2 ¼ ; ð : Þ : g g2 g gk2 g g12 g g22 g g32 2 for i 2 A 24 3 Á ¼ 3k ¼ 31 þ 32 þ 33 ¼ δ3 ¼ : g g3 g gk3 g g13 g g23 g g33 3 for i 3 The scalar product of the covariant and contravariant bases gives gðÞi Á g ¼ gðÞi Á g cos gðÞi , g ðÞi ðÞi ðÞi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi ðA:25Þ ¼ ðÞii Á ðÞi ¼ g gðÞii cos g , gðÞi 1 where the index (i) means no summation is carried out over i. Equation (A.25) indicates that the product of the covariant and contravariant basis norms generally does not equal 1 in the curvilinear coordinates. 318 Appendix A: Relations Between Covariant and Contravariant Bases qffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 1 ðÞii Á ¼ ð : Þ g gðÞii 1 A 26 ðÞi ; cos g gðÞi (i) In orthogonal coordinate systems, g is parallel to g(i). Therefore, Eq. (A.26) becomes qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 1 1 ðÞii Á ¼ ) ðÞii ¼ ¼ ð : Þ g gðÞii 1 g pffiffiffiffiffiffiffiffi A 27 gðÞii 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 g g * ¼ i ¼ pffiffiffiffiffiffiffiffii ) * ¼ ð : Þ gi jj gi 1 B 1a gi gðÞii The covariant basis norm |gi| can be considered as a scale factor hi without summation over (i).
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