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 Â g ¼ Â 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 Â gjÞ: gk ¼ aðg : gkÞ¼ad ¼ a ÂÃk ðA:2Þ ) a ¼ðgi Â gjÞ : gk gi; gj; gk The scalar triple product of the covariant bases can be written as pffiffiffi a ¼ ½¼ðg1; g2; g3 g1 Â 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 Â gjÞ¼Àðgj Â giÞ

eijkðgi Â gjÞ eijkðgi Â 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 Â gjÞ¼eijk g g ¼ eijkJg êijkg

eijkðgi Â gjÞ ) gk ¼ pffiffiffi ¼ êijkðg Â g Þ ðA:6Þ g i j

) êijk ¼ðgi Â 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 êijk 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 ê 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 Â g Þ¼ pffiffiffiffi gk ¼ gk ê gk g J ðA:9Þ ) êijk ¼ðgi Â g jÞgk Thus, the covariant basis results from Eq. (A.9): ffiffiffiffi p i j i j gk ¼ eijk g ðg Â g Þ¼eijkJðg Â g Þ i j ðA:10Þ ¼ êijkðg Â g Þ Obviously, there are some relations between the covariant and contravariant permutation symbols:

ijk ê êijk ¼ 1 ðno summationÞ A:11 ijk 2 ð Þ êijk ¼ ê 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

Ã gi gi Ã 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 Ã Ã 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 Ã 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 Ã Ã Ãij Ã Ã 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. (B.3).

i j ik jl Ã Ã Ã Ã Ã T ¼ Tijg g ¼ðTijg g hkhlÞ gkgl Tijgkgl ðB:6Þ

H. Nguyen-Schäfer and J.-P. Schmidt, Tensor Analysis and Elementary 203 Differential Geometry for Physicists and Engineers, Mathematical Engineering 21, DOI: 10.1007/978-3-662-43444-4, Ó Springer-Verlag Berlin Heidelberg 2014 204 Appendix B: Physical Components of Tensors

Similarly, the physical covariant tensor components denoted by star result in

Ã ik jl Tij g g hkhlTij ðB:7Þ The mixed tensors can be written in the unitary covariant bases using Eqs. (B.2 and B.3)

i j i jk T ¼ Tj gig ¼ Tj giðg gkÞ i Ã jk Ã ¼ Tj ðhigi Þðg hkgkÞ ðB:8Þ i jk Ã Ã ¼ðTj g hihkÞgi gk i Ã Ã Ã ðTj Þ gi gk Thus, the physical mixed tensor components denoted by star result in

i Ã jk i ðTj Þ g hihkTj ðB:9Þ Analogously, the contravariant vector can be written using Eq. (B.2).

i i Ã v ¼ v gi ¼ðv hiÞgi Ãi ðB:10Þ Ãi Ã v v gi ¼ gi hi Thus, the physical component of the contravariant vector v on the unitary basis * gi is defined as ffiffiffiffiffiffiffiffi Ãi i p i v hiv ¼ gðiiÞv ðB:11Þ The contravariant basis can be normalized dividing by its vector length without summation over (i). gi gi gÃi ¼ ¼ pffiffiffiffiffiffiffiffi ðB:12Þ jjgi gðiiÞ where g(ii) is the contravariant metric coefficient that results from Eq. (A.20). Using Eq. (B.12), the covariant vector v can be written as qffiffiffiffiffiffiffiffi i ðiiÞ Ãi Ã Ãi v ¼ vig ¼ vi g g vi g ðB:13Þ Thus, the physical component of the covariant vector v results in qffiffiffiffiffiffiffiffi Ã ðiiÞ vi ¼ vi g ðB:14Þ According to Eq. (A.27), Eq. (B.14) can be rewritten in orthogonal coordinate systems: qffiffiffiffiffiffiffiffi Ã ðiiÞ 1 1 vi ¼ g vi ¼ pffiffiffiffiffiffiffiffi vi ¼ vi ðB:15Þ gðiiÞ hi Appendix B: Physical Components of Tensors 205

Using Eq. (B.3), the covariant vector can be written in

i ij v ¼ vig ¼ vig gj ¼ðv gijh ÞgÃ vÃgÃ i j j j j ðB:16Þ Ã vj i ¼ ij g g hj

* Thus, the physical contravariant vector component of v on the unitary basis gj can be deﬁned as

Ã ij vj g hjvi ðB:17Þ Furthermore, the vector v can be written in both covariant and contravariant bases.

j i v ¼ vjg ¼ v gi j i )ðvjg Þ:gk ¼ðv giÞ:gk B:18 j i ð Þ ) vjdk ¼ v gik i ) vk ¼ v gik Interchanging i with j and k with i, one obtains

j vi ¼ v gij ðB:19Þ Only in orthogonal coordinate systems, we have

2 gij ¼ 0 for i 6¼ j; gðiiÞ ¼ hi ðB:20Þ Thus, one obtains from Eq. (B.19)

j 1 2 N vi ¼ v gij ¼ v gi1 þ v gi2 þÁÁÁþv giN B:21 i i 2 ð Þ ¼ v gðiiÞ ¼ v hi Substituting Eq. (B.11) into Eq. (B.21), one obtains Eq. (B.22) that is equivalent to Eq. (B.15). Ãi i 2 v 2 Ãi vi ¼ v hi ¼ hi ¼ hiv ðB:22Þ hi Appendix C Nabla Operators

Some useful Nabla operators are listed in Cartesian and general curvilinear coordinates: 1. Gradient of an invariant f • Cartesian coordinate {xi} of of of rf ¼ e þ e þ e ðC:1Þ ox x oy y oz z • General curvilinear coordinate {ui} of of rf ¼ f gi ¼ gi ¼ gijg ðC:2Þ ;i oui oui j

2. Gradient of a vector v • General curvilinear coordinate {ui} k j k i k i rv ¼ v;i þ v Cij gkg ¼ v gkg ÀÁi ðC:3Þ j k i k i ¼ vk;i À vjCik g g ¼ vkjig g

3. Divergence of a vector v • Cartesian coordinate {xi} ov ov ov rÁv ¼ x þ y þ z ðC:4Þ ox oy oz

H. Nguyen-Schäfer and J.-P. Schmidt, Tensor Analysis and Elementary 207 Differential Geometry for Physicists and Engineers, Mathematical Engineering 21, DOI: 10.1007/978-3-662-43444-4, Ó Springer-Verlag Berlin Heidelberg 2014 208 Appendix C: Nabla Operators

• General curvilinear coordinate {ui}

i i j i r:v ¼ v ji ðv;i þ v CijÞ 1 oðJviÞ ¼ ¼ JÀ1ðJviÞ ðC:5Þ J oui ;i ki j k i rÁv ¼ vkij g ¼ðvk;i À vjCikÞg Á g 4. Gradient of a second-order tensor T • General curvilinear coordinate {ui} for a covariant second-order tensor

m m i j k rT ¼ðTij;k À CikTmj À CjkTimÞg g g ðC:6Þ i j k ¼ Tijj kg g g • General curvilinear coordinate {ui} for a contravariant second-order tensor

ij i mj j im k rT ¼ðT þ CkmT þ C T Þgigjg ;k km ðC:7Þ ij k ¼ T jkgigjg • General curvilinear coordinate {ui} for a mixed second-order tensor

i i m m i j k rT ¼ðTj;k þ CkmTj À CjkTmÞgig g : i j k ðC 8Þ ¼ Tj jkgig g

5. Divergence of a second-order tensor T • General curvilinear coordinate {ui} for a covariant second-order tensor

m m i j k r:T ¼ðTij;k À C Tmj À C TimÞg ðg :g Þ ik jk C:9 jk i ð Þ Tijj kg g • General curvilinear coordinate {ui} for a contravariant second-order tensor

ij i mj j im k rÁT ¼ðT;k þ CkmT þ CkmT Þdi gj ij i mj j im ¼ðT;i þ CimT þ CimT Þgj ðC:10Þ ij T jigj • General curvilinear coordinate {ui} for a mixed second-order tensor

i i m m i k j rÁT ¼ðTj;k þ CkmTj À CjkTmÞdi g i i m m i j ¼ðTj;i þ CimTj À Cji TmÞg ðC:11Þ i j i jk Tj jig ¼ Tj jig gk Appendix C: Nabla Operators 209

6. Curl of a vector v • Cartesian coordinate {xi}

ex ey ez o o o rÂv ¼ o o o ðC:12Þ x y z vx vy vz

The curl of v results from calculating the determinant of Eq. (C.12). ov ov ov ov ov ov rÂv ¼ z À y e þ x À z e þ y À x e ðC:13Þ oy oz x oz ox y ox oy z • General curvilinear coordinate {ui}

ijk rÂv ¼ ^e vj;igk ðC:14Þ

The contravariant permutation symbol is deﬁned by 8 < þJÀ1 if ði; j; kÞ is an even permutation; ijk À1 ^e ¼ : ÀJ if ði; j; kÞ is an odd permutation; ðC:15Þ 0ifi ¼ j; or i ¼ k; or j ¼ k

where J is the Jacobian.

7. Laplacian of an invariant f • Cartesian coordinate {xi}

o2f o2f o2f r2f Df ¼ þ þ ðC:16Þ ox2 oy2 oz2 • General curvilinear coordinate {ui}

2 k ij r f Df ¼ðf;ij À f;kCijÞg C:17 k ij ij ð Þ ¼ðvi;j À vkCijÞg vijg where the covariant vector component and its covariant derivative with respect to uk are deﬁned by

of of o2f v ¼ f ¼ ; v ¼ f ¼ ; v ¼ f ¼ ðC:18Þ i ;i oui k ;k ouk i;j ;ij ouiou j 210 Appendix C: Nabla Operators

8. Calculation rules of the Nabla operators

Div Grad f ¼rÁðrf Þ¼r2f ¼ Df ðÞLaplacian ðC:19Þ Curl Grad f ¼rÂðrf Þ¼0 ðC:20Þ Div Curl v ¼rÁðrÂvÞ¼0 ðC:21Þ DðfgÞ¼f Dg þ 2rf Árg þ gDf ðC:22Þ Curl Curl v ¼rÂðrÂvÞ ¼ rðr Á vÞÀDvðÞðCurl identity C:23Þ

The Laplacian of v in Eq. (C.23) is computed in the tensor formulation for general curvilinear coordinates.

Div Grad v ¼ Laplacian v ¼ Dv ¼rÁðrvÞ¼r2v

i i p p i jk ðC:24Þ ¼ðv À v C þ v C Þg g j;k p jk j pk i i jk v jkg gi Appendix D Essential Tensors

Derivative of the covariant basis

k gi;j ¼ Cijgk ðD:1Þ Derivative of the contravariant basis ogi gi ¼ C^i gk ¼ÀCi gk ðD:2Þ ;j ou j jk jk Derivative of the covariant metric coefﬁcient p p gij;k ¼ Cikgpj þ Cjkgpi ðD:3Þ First-kind Christoffel symbol

1 l Cijk ¼ ðgik;j þ gjk;i À gij;k Þ¼glkCij 2 ðD:4Þ l lk ) Cij ¼ g Cijk Second-kind Christoffel symbol based on the covariant basis

k k kl Cij ¼ gi;j Á g ¼ g Cijl ðD:5Þ

ouk o2xp Ck ¼ Á ¼ Ck ðD:6Þ ij oxp ouiou j ji 1 Ck ¼ gkp ðg þ g À g ÞðD:7Þ ij 2 ip;j jp;i ij;p 1 oJ oðln JÞ Ci ¼ ¼ ðD:8Þ ij J ou j ou j Second-kind Christoffel symbol based on the contravariant basis ^i i ^i Ckj ¼ÀCkj ¼ Cjk ðD:9Þ

H. Nguyen-Schäfer and J.-P. Schmidt, Tensor Analysis and Elementary 211 Differential Geometry for Physicists and Engineers, Mathematical Engineering 21, DOI: 10.1007/978-3-662-43444-4, Ó Springer-Verlag Berlin Heidelberg 2014 212 Appendix D: Essential Tensors

Covariant derivative of covariant ﬁrst-order tensors

k Tij¼ Ti;j À CijTk ¼ T;j Á gi ðD:10Þ Covariant derivative of contravariant ﬁrst-order tensors

i i i k i T j ¼ T;j þ CjkT ¼ T;j Á g ðD:11Þ

Covariant derivative of covariant and contravariant second-order tensors

m m Tijj k ¼ Tij;k À CikTmj À CjkTim D:12 ij ij i mj j im ð Þ T jk ¼ T;k þ CkmT þ CkmT Covariant derivative of mixed second-order tensors

i i i m m i Tj jk ¼ Tj;k þ CkmTj À CjkTm D:13 j j j m m j ð Þ Ti jk ¼ Ti;k þ CkmTi À CikTm Second covariant derivative of covariant ﬁrst-order tensors

m m Tikj¼ Ti;jk À Cik;jTm À CikTm;j m m n À Cij Tm;k þ Cij CmkTn ðD:14Þ m m n À CjkTi;m þ CjkCimTn Second covariant derivative of the contravariant vector

k k k p p k v jlm v l;m À v pClm þ v jlCpm ðD:15Þ where ÀÁ k k k n k n k v l;m ¼ v jl ;m v;lm þ v;mCnl þ v Cnl;m ðD:16Þ

k k n k v p v;p þ v Cnp ðD:17Þ

p p n p v jl v;l þ v Cnl ðD:18Þ Riemann–Christoffel tensor n n n m n m n Rijk Cik;j À Cij;k þ CikCmj À Cij Cmk ðD:19Þ Riemann curvature tensor n Rlijk glnRijk ðD:20Þ Appendix D: Essential Tensors 213

First-kind Ricci tensor

oCk oCk R ¼ ik À ij À Cr Ck þ Cr Ck ij o j o k ij rk ik rj u u ðD:21Þ o2ðln JÞ 1 oðJCk Þ ¼ À ij þ Cr Ck ouiou j J ouk ik rj Second-kind Ricci tensor

i ik R g Rkj j o2ðln JÞ 1 oðJCmÞ ðD:22Þ ¼ gik À kj þ Cm Cn oukou j J oum kn mj Ricci curvature ! o2ðln JÞ 1 oðJCk Þ R ¼ gij À ij þ Ck Cr ðD:23Þ ouiou j J ouk ir kj

Einstein tensor

i i 1 i ik Gj Rj À djR ¼ g Gkj 2 ðD:24Þ 1 G ¼ g Gk ¼ R À g R ij ik j ij 2 ij

i Gj ¼ 0 ðD:25Þ i First fundamental form

I ¼ Edu2 þ 2Fdudv þ Gdv2; EF ruru rurv ðD:26Þ M ¼ðgijÞ ¼ FG rvru rvrv Second fundamental form

II ¼ Ldu2 þ 2Mdudv þ Ndv2; LM ruunruvn ðD:27Þ H ¼ðhijÞ ¼ MN ruvnrvvn Gaussian curvature of a curvilinear surface

2 LN À M detðhijÞ K ¼ j1j2 ¼ 2 ¼ ðD:28Þ EG À F detðgijÞ Mean curvature of a curvilinear surface

1 EN À 2MF þ LG H ¼ ðj þ j Þ¼ ðD:29Þ 2 1 2 2ðEG À F2Þ 214 Appendix D: Essential Tensors

Unit normal vector of a curvilinear surface g Â g r Â r r Â r n ¼ 1 2 ¼ puffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu v ðD:30Þ 2 jjg1 Â g2 detðgijÞ EG À F Differential of a surface area qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dA ¼ jjg Â g dudv ¼ g g Àðg Þ2dudv q1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 11 22 12 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðD:31Þ 2 ¼ detðgijÞdudv ¼ EG À F dudv

Gauss derivative equations

k k gi;j ¼ Cijgk þ hijg3 ¼ Cijgk þ hijn D:32 k ð Þ , gi j gi;j À Cijgk ¼ hijn Weingarten’s equations

j jk ni ¼Àhi gj ¼Àðhikg Þgj ðD:33Þ Codazzi’s equations

Kij;k ¼ Kik;j )ðK11;2 ¼ K12;1 ; K21;2 ¼ K22;1ÞðD:34Þ Gauss equations

j 1 2 1 2 K ¼ det ðKi Þ¼ðK1 K2 À K2 K1 Þ 2 ðD:35Þ K11K22 À K12 R1212 ¼ 2 ¼ g11g22 À g12 g Appendix E Euclidean and Riemannian Manifolds

In the following appendix, we summarize fundamental notations and basic results from vector analysis in Euclidean and Riemannian manifolds. This section can be written informally and is intended to remind the reader of some fundamentals of vector analysis in general curvilinear coordinates. For the sake of simplicity, we abstain from being mathematically rigorous. Therefore, we recommend some literature given in References for the mathematically interested reader.

E.1 N-dimensional Euclidean Manifold

N-dimensional Euclidean manifold EN can be represented by two kinds of coordinate systems: Cartesian (orthonormal) and curvilinear (non-orthogonal) coordinate systems with N dimensions. Lines, curves, and surfaces can be considered as subsets of Euclidean manifold. Two lines or two curves can generate a ﬂat (planes) and curvilinear surface (cylindrical and spherical surfaces), respectively. Both kinds of surfaces can be embedded in Euclidean space.

E.1.1 Vector in Cartesian Coordinates

Cartesian coordinates are an orthonormal coordinate system in which the bases (i, j, k) are mutually perpendicular (orthogonal) and unitary (normalized vector length). The orthonormal bases (i, j, k) are ﬁxed in Cartesian coordinates. Any vector could be described by its components and the relating bases in Cartesian coordinates. The vector r can be written in Euclidean space E3 (three-dimensional space) in Cartesian coordinates (cf. Fig. E.1). r ¼ xi þ yj þ zk ðE:1Þ

H. Nguyen-Schäfer and J.-P. Schmidt, Tensor Analysis and Elementary 215 Differential Geometry for Physicists and Engineers, Mathematical Engineering 21, DOI: 10.1007/978-3-662-43444-4, Ó Springer-Verlag Berlin Heidelberg 2014 216 Appendix E: Euclidean and Riemannian Manifolds

Fig. E.1 Vector r in z Cartesian coordinates z

P(x,y,z) zk r zk k 0 yj y xi j y i x x (xi + yj)

Fig. E.2 Bases of the u 3 curvilinear coordinates g 3

g 3 P(u1,u2,u3) r g 1

0‘ g 2 g 2

g 1 u1 u 2 where x, y, z are the vector components in the coordinate system (x, y, z); i, j, k are the orthonormal bases of the corresponding coordinates. The vector length of r can be computed using the Pythagorean theorem as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jjr ¼ x2 þ y2 þ z2 0 ðE:2Þ

E.1.2 Vector in Curvilinear Coordinates

We consider a curvilinear coordinate system (u1, u2, u3) of Euclidean space E3, i.e., a coordinate system which is generally non-orthogonal and non-unitary (non- orthonormal basis). By abuse of notation, we denote the basis vector simply basis. In other words, the bases are not mutually perpendicular and their vector lengths are not equal to one (Klingbeil 1966; Nayak 2012). In the curvilinear 1 2 3 coordinate system (u , u , u ), there are three covariant bases g1, g2, and g3 and three contravariant bases g1, g2, and g3 at the origin 00, as shown in Fig. E.2. Generally, the origin 00 of the curvilinear coordinates could move everywhere in Euclidean space; therefore, the bases of the curvilinear coordinates only depend on Appendix E: Euclidean and Riemannian Manifolds 217 each considered origin 00. For this reason, the bases are not ﬁxed in the whole curvilinear coordinates such as in Cartesian coordinates, as displayed in Fig. E.1. The vector r of the point P(u1, u2, u3) can be written in the covariant and contravariant bases:

r ¼ u1g þ u2g þ u3g 1 2 3 E:3 1 2 3 ð Þ ¼ u1g þ u2g þ u3g where 1 2 3 u , u , u are the vector contravariant components of the coordinates (u1, u2, u3); 1 2 3 g1, g2, g3 are the covariant bases of the coordinate system (u , u , u ); 1 2 3 u1, u2, u3 are the vector covariant components of the coordinates (u , u , u ); 1 2 3 g , g , g are the contravariant bases of the coordinate system (u1, u2, u3).

The covariant basis gi can be deﬁned as the tangential vector to the i corresponding curvilinear coordinate u for i = 1, 2, 3. Both bases g1 and g2 generate a tangential surface to the curvilinear surface (u1u2) at the considered 0 origin 0 , as shown in Fig. E.2. Note that the basis g1 is not perpendicular to the 3 bases g2 and g3. However, the contravariant basis g is perpendicular to the 0 k tangential surface (g1g2) at the origin 0 . Generally, the contravariant basis g results from the cross product of the other covariant bases (gi 9 gj).

k a g ¼ gi Â gj for i; j; k ¼ 1; 2; 3 ðE:4aÞ where a is a scalar factor (scalar triple product) given in Eq. (1.43).

a ¼ðai Â ajÞÁak ÂÃ ðE:4bÞ ai; aj; ak Thus, g Â g g Â g g Â g g1 ¼ 2 3 ; g2 ¼ 3 1 ; g3 ¼ 1 2 ðE:4cÞ ½g1; g2; g3 ½g1; g2; g3 ½g1; g2; g3

E.1.3 Orthogonal and Orthonormal Coordinates

The coordinate system is called orthogonal if its bases are mutually perpendicular, as displayed in Fig. E.1. The dot product of two orthonormal bases is deﬁned as 218 Appendix E: Euclidean and Riemannian Manifolds

i : j ¼ jji Á jjj Á cos ði; jÞ p ¼ð1ÞÁð1ÞÁcos ðE:5Þ 2 ¼ 0

Thus, i Á j ¼ i Á k ¼ j Á k ¼ 0 ðE:6Þ If the length of each basis equals 1, the bases are unitary vectors. jji ¼ jjj ¼ jjk ¼ 1 ðE:7Þ If the coordinate system satisﬁes both conditions (E.6) and (E.7), it is called the orthonormal coordinate system, which exists in Cartesian coordinates. Therefore, the vector length in the orthonormal coordinate system results from

jjr 2 ¼ r Á r ¼ ðÞÁx i þ y j þ zk ðÞxi þ y j þ z k ¼ x2ði Á iÞþxyði Á jÞþxzði Á kÞ ðE:8Þ þ yxðj Á iÞþy2ðj Á jÞþyzðj Á kÞ þ zxðk Á iÞþzyðk Á jÞþz2ðk Á kÞ Due to Eqs. (E.6 and E.7), the vector length in Eq. (E.8) becomes pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jjr 2¼ x2 þ y2 þ z2 ) jjr ¼ x2 þ y2 þ z2 ðE:9Þ The cross product (called vector product) of a pair of bases of the orthonormal coordinate system is (informally) given by means of right-handed rule; that is, if the right-hand fingers move in the rotating direction from the basis j to the basis k, the thumb will point in the direction of the basis i = j 9 k. The bases (i, j, k) form a right-handed triple. 8 <> i ¼ j Â k ¼Àk Â j j ¼ k Â i ¼Ài Â k ðE:10Þ :> k ¼ i Â j ¼Àj Â i The cross product of two orthonormal bases can be defined as

jji Â j ¼ jji Á jjj Á sinði; jÞ p ¼ð1ÞÁð1ÞÁsin ðE:11Þ 2 ¼ jjk Appendix E: Euclidean and Riemannian Manifolds 219

Fig. E.3 Arc length ds of x 3 P and Q in Cartesian x 3 coordinates P ds dr Q r (C) e3 r + dr 0 x 2 e2 x 2 e1 x1

x 1

Fig. E.4 Arc length ds of u 3 P and Q in the curvilinear coordinates g 3 i P(u (t1),…) ds

r dr i Q(u (t 2),…) r + dr 0‘ g 2 (C) 1 2 u g1 u

E.1.4 Arc Length Between Two Points in a Euclidean Manifold

We consider two points P(x1, x2, x3) and Q(x1, x2, x3) in Euclidean space E3 in Cartesian and curvilinear coordinate systems, as shown in Figs. E.3 and E.4. Both 1 2 3 points P and Q have three components x , x , and x in Cartesian coordinates (e1, e2, e3). To simplify some mathematically written expressions, the coordinates x, y, and z in Cartesian coordinates can be transformed into x1, x2, and x3; the bases (i, j, k) turn to (e1, e2, e3). We now turn to the notation of the differential dr of a vector r. The differential dr can be expressed using the Einstein summation convention (Klingbeil 1966; Kay 2011):

i dr eidx for i ¼ 1; 2; 3 X3 i ðE:12Þ ¼ eidx i¼1 The Einstein summation convention used in Eq. (E.12) indicates that dr equals i the sum of ei dx by running the dummy index i from 1 to 3. 220 Appendix E: Euclidean and Riemannian Manifolds

The arc length ds between the points P and Q (cf. Fig. E.3) can be calculated by the dot product of two differentials.

ðdsÞ2 ¼ dr Á dr i j ¼ðeidx ÞÁðejdx Þ E:13 i j ð Þ ¼ðei Á ejÞdx Á dx ¼ dxi Á dxi for i ¼ 1; 2; 3: Thus, the arc length in the orthonormal coordinate system results in pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds ¼ dxi Á dxi for i ¼ 1; 2; 3: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðE:14Þ ¼ ðdx1Þ2 þðdx2Þ2 þðdx3Þ2

The points P and Q have three components u1, u2, and u3 in the curvilinear coordinate system with the basis (g1, g2, g3) in Euclidean 3-space, as displayed in Fig. E.4. The location vector r(u1,u2,u3) of the point P is a function of ui. Therefore, the differential dr of the vector r can be rewritten in a linear formulation of dui. or dr ¼ dui oui ðE:15Þ i gidu

i where gi is the covariant basis of the curvilinear coordinate u . Analogously, the arc length ds between two points of and Q in the curvilinear coordinate system can be calculated by

ðdsÞ2 ¼ dr Á dr ¼ðg duiÞÁðg du jÞ i j E:16 i j ð Þ ¼ðgi Á gjÞ du Á du i j ¼ gij dx Á dx for i; j ¼ 1; 2; 3

Therefore, q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i j ds ¼ gij dx Á dx sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Zt2 i j ðE:17Þ dx dx ) s ¼ gij Á dt dt dt t1 where t is the parameter in the curve C with the coordinate ui(t); gij is the defined as the metric coefficient of two non-orthonormal bases. Appendix E: Euclidean and Riemannian Manifolds 221

j gij ¼ gi Á gj ¼ gj Á gi ¼ gji 6¼ di ðE:18Þ

It is obvious that the symmetric metric coefﬁcients gij vanish for any i 6¼ j in the orthogonal bases because gi is perpendicular to gj; therefore, the metric tensor can be rewritten as 0ifi 6¼ j gij ¼ ðE:19Þ gii if i ¼ j

In the orthonormal bases, the metric coefﬁcients gij in Eq. (E.19) become 0ifi 6¼ j g d j ¼ ðE:20Þ ij i 1ifi ¼ j

j where di is called the Kronecker delta.

E.1.5 Bases of the Coordinates

The vector r can be rewritten in Cartesian coordinates of Euclidean space E3.

i r ¼ x ei ðE:21Þ The differential dr results from Eq. (E.21)in

i dr ¼ eidx or ðE:22Þ ¼ dxi oxi i Thus, the orthonormal bases ei of the coordinate x can be deﬁned as or e ¼ for i ¼ 1; 2; 3 ðE:23Þ i oxi Analogously, the basis of the curvilinear coordinate ui can be calculated in the curvilinear coordinate system of E3. or g ¼ for i ¼ 1; 2; 3 ðE:24Þ i oui

Substituting Eq. (E.24) into Eq. (E.18), we obtain the metric coefﬁcients gij that are generally symmetric in Euclidean space; that is, gij = gji. 222 Appendix E: Euclidean and Riemannian Manifolds

Fig. E.5 Schematic g 3 g1 visualization of the g Gram–Schmidt procedure 2/1 g3 e e3 e1 2 e1

g3/1

g g e2 e2 2 3/2

g ¼ g Á g ¼ g 6¼ d j ij i j jii or or or oxm or oxn ¼ : ¼ : oui ou j oxm oui oxn ou j oxm oxn oxm oxn ðE:25Þ ¼ ðe : e Þ¼ dn oui ou j m n oui ou j m oxk oxk ¼ for k ¼ 1; 2; 3 oui ou j According to Eqs. (E.4a and E.4b), the contravariant basis gk is perpendicular to k both covariant bases gi and gj. Additionally, the contravariant basis g is chosen such that the vector length of the contravariant basis equals the inversed vector k length of its relating covariant basis; thus, g Á gk ¼ 1. As a result, the scalar products of the covariant and contravariant bases can be written in general curvilinear coordinates (u1,…, uN). ( g Á gk ¼ gk Á g ¼ dk for i; k ¼ 1; 2; ...; N i i i E:26 k ð Þ gi Á gk ¼ gik ¼ gki 6¼ di for i; k ¼ 1; 2; ...; N

E.1.6 Orthonormalizing a Non-orthonormal Basis

The basis {gi} is non-orthonormal in the curvilinear coordinates. Using the Gram– Schmidt scheme (Grifﬁths 2005), an orthonormal basis (e1, e2, e3) can be created from the basis (g1, g2, g3). The orthonormalization procedure of the basis (g1, g2, g3) will be derived in this section; the orthonormalizing scheme is demonstrated in Fig. E.5. The Gram–Schmidt scheme for N = 3 has three orthonormalization steps:

1. Normalize the ﬁrst basis vector g1 by dividing it by its length to get the normalized basis e1.

g1 e1 ¼ jjg1 Appendix E: Euclidean and Riemannian Manifolds 223

2. Project the basis g2 onto the basis g1 to get the projection vector g2/1 on the basis g1. The normalized basis e2 results from subtracting the projection vector g2/1 from the basis g2. Then, iteratively, normalize this vector by dividing it by its length to generate the basis e2.

g g 2 À 2=1 g2 Àðg2 Á e1Þe1 e2 ¼ ¼ jjg2 Àðg2 Á e1Þe1 g2 À g2=1

3. Subtract projections along the bases of e1 and e2 from the basis g3 and normalize it to obtain the normalized basis e3.

g g g 3 À 3=1 À 3=2 g3 Àðg3 Á e1Þe1 Àðg3 Á e2Þe2 e3 ¼ ¼ jjg3 Àðg3 Á e1Þe1 Àðg3 Á e2Þe2 g3 À g3=1 À g3=2

Using the Gram–Schmidt scheme, the orthonormal basis {e1, e2, e3} results from the non-orthonormal bases {g1, g2, g3}. Generally, the orthogonal bases {e1, e2, …, eN} for the N-dimensional space can be generated from the non-orthonormal bases {g1, g2, …, gN} according to the Gram–Schmidt scheme as follows: P g j À 1 g e e j À i ¼ 1 ð j Á iÞ i ej ¼ P for j ¼ 1; 2; ...; N jÀ1 gj À i ¼ 1 ðgj Á eiÞei

E.1.7 Angle Between Two Vectors and Projected Vector Component

The angle h between two vectors a and b can be deﬁned by means of the scalar product (Fig. E.6).

a Á b g Á g aib j cos h ¼ ¼ i j jja Á jjb jja Á jjb ðE:27Þ g aib j g aib j ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiij pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiij pffiffiffiffiffiffiffiffi i j k l i j gija a Á gklb b a ai Á b bj where

jja 2 ¼ a Á a E:28 i j ij i ð Þ ¼ gija a ¼ g aiaj ¼ a ai for i; j ¼ 1; 2; ...; N 224 Appendix E: Euclidean and Riemannian Manifolds

Fig. E.6 Angle between two a(u i ) vectors and projected vector component

θ b(u i )

= θ ab a.cos in which ai, bj are the contravariant vector components; ai, bj are the covariant vector components; ij gij, g are the covariant and contravariant metric coefﬁcients of the bases. The projected component of the vector a on vector b results from its vector length and Eq. (E.27).

ab ¼ jja Á cos h g aib j g aib j ¼ jja Á ij ¼ ij jja Á jjb jjb ðE:29Þ g aib j ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiij for i; j; k; l ¼ 1; 2; ...; N k l gklb b

Examples Given two vectors a and b: pffiffiffi i a ¼ 1 Á e1 þ 3 Á e2 ¼ a gi; j b ¼ 1 Á e1 þ 0 Á e2 ¼ b gj Thus, the relating vector components are

g1 ¼ e1; g2 ¼ e2 pffiffiffi a1 ¼ 1; a2 ¼ 3 b1 ¼ 1; b2 ¼ 0

The covariant metric coefﬁcients gij in the orthonormal basis (e1, e2) can be calculated according to Eq. (E.18). g11 g12 10 ðgijÞ¼ ¼ g21 g22 01 The angle h between two vectors results from Eq. (E.27). Appendix E: Euclidean and Riemannian Manifolds 225

Fig. E.7 N tuples of Riemannian manifold RN coordinates in Riemannian manifold uN

g P 2 1 N i Pi (u ,…,u ) u1 g1 u RN

g aib j cos h ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiij pffiffiffiffiffiffiffiffiffiffiffiffiffi for i; j; k; l ¼ 1; 2 i j k l gija a Á gklb b g a1b1 þ g a1b2 þ g a2b1 þ g a2b2 ¼ 11 p12ffiffiffiffiffiffiffiffiffiffiffiffiffi p21ffiffiffiffiffiffiffiffiffiffiffiffiffi 22 i j k l gija a Á gklb b pffiffiffi pffiffiffi ð1 Á 1 Á 1Þþð0 Á 1 Á 0Þþð0 Á 3 Á 1Þþð1 Á 3 Á 0Þ 1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1 þ 0 þ 0 þ 3 Á 1 þ 0 þ 0 þ 0 2 Therefore, 1 p h ¼ cosÀ1 ¼ 2 3 The projected vector component can be calculated according to Eq. (E.29).

i j gija b ab ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi for i; j; k; l ¼ 1; 2 k l gklb b pffiffiffi pffiffiffi ð1 Á 1 Á 1Þþð0 Á 1 Á 0Þþð0 Á 3 Á 1Þþð1 Á 3 Á 0Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1 1 þ 0 þ 0 þ 0

E.2 General N-dimensional Riemannian Manifold

The concept of the Riemannian geometry is a very important fundamental brick in the modern physics of relativity and quantum field theories, theoretical elementary particles physics, and string theory. In contrast to the homogenous Euclidean manifold, the non-homogenous Riemannian manifold only contains a tuple of fiber bundles of N arbitrary curvilinear coordinates of u1, …, uN. Each of the fiber bundle is related to a point and belongs to the N-dimensional differentiable Riemannian manifold. In the case of the infinitesimally small fiber lengths in all dimensions, the fiber bundle now becomes a single point. Therefore, the tuple of fiber bundles becomes a tuple of points in the manifold. In fact, Riemannian 226 Appendix E: Euclidean and Riemannian Manifolds manifold only contains a point tuple (Riemann 2013). In turn, each point of the point tuple can move along a fiber bundle in N arbitrary directions (dimensions) in the N-dimensional Riemannian manifold. Generally, a hypersurface of the fiber bundle of curvilinear coordinates {ui} for i = 1, 2,…, N at a certain point can be defined as a differentiable (N - 1)- dimensional subspace with a codimension of one. This definition can be understood that the (N - 1)-dimensional subbundle of fibers moves along the one-dimensional remaining fiber.

E.2.1 Point Tuple in Riemannian Manifold

We now consider an N-dimensional differentiable Riemannian manifold RN that contains a tuple of points. In general, each point in the manifold locally has N curvilinear coordinates of u1,…, uN embedded at this point. Therefore, the 1 considered point Pi can be expressed in the curvilinear coordinates as Pi(u , …, uN). The notation of Riemannian manifold allows the local embedding of an N-dimensional affine tangential manifold (called affine tangential vector space) into the point Pi, as displayed in Fig. E.7. The arc length between any two points of N tuples of coordinates in the manifold does not physically change in any chosen basis. However, its components are changed in the coordinate bases that vary in the manifold. Therefore, these components must be taken into account in the transformation between different curvilinear coordinate systems in Riemannian manifold. To do that, each point in Riemannian manifold can be embedded with the individual metric coefficients gij for the relating point. Note that the metric 1 N coefficients gij of the coordinates (u , …, u ) at any point are symmetric, and they totally have N2 components in an N-dimensional manifold. That means one can embed an affine tangential manifold EN at any point in Riemannian manifold RN in which the metric coefficients gij could be only applied to this point and change from one point to another point. However, the dot product (inner product) is not valid any longer in the affine tangential manifold (Klingbeil 1966; Riemann 2013).

E.2.2 Flat and Curved Surfaces

By abuse of notation and by completely abstaining from mathematical rigorousness, we introduce the notation of flat and curved surfaces. A surface in Euclidean space is called flat if the sum of angles in any triangle ABC is equal to 180° or, alternatively, if the arc length between any two points fulfills the condition in Eq. (E.13). Therefore, the flat surface is a plane in Euclidean space. On the contrary, an arbitrary surface in a Riemannian manifold is called curved if the angular sum in an arbitrary triangle ABC is not equal to 180°, as displayed in Fig. E.8. Appendix E: Euclidean and Riemannian Manifolds 227

Fig. E.8 Flat and curved (a) α +β + γ = 180° (b) α +β + γ ≠180° surfaces A α β γ S A C α B B β γ C P

Conditions for the ﬂat and curved surfaces (Oeijord 2005): ( a þ b þ c ¼ 180 for a flat surface ðE:30Þ a þ b þ c 6¼ 180 for a curved surface

Furthermore, the surface curvature in Riemannian manifold can be used to determine the surface characteristics. Additionally, the line curvature is also applied to studying the curve and surface characteristics.

E.2.3 Arc Length Between Two Points in Riemannian Manifold

We now consider a differentiable Riemannian manifold and calculate the arc length between two points P(u1, …, uN) and Q(u1, …, uN). The arc length is an important notation in Riemannian manifold theory. The coordinates (u1, …, uN) can be considered as a function of the parameter t that varies from P(t1)toQ(t2). The arc length ds between the points P and Q thus results from ds 2 dr dr ¼ : ðE:31Þ dt dt dt where the derivative of the vector r(u1, …, uN) can be calculated as dr dðg uiÞ ¼ i dt dt ðE:32Þ i giu_ ðtÞ for i ¼ 1; 2; ...; N Substituting Eq. (E.32) into Eq. (E.31), one obtains the arc length qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i j ds ¼ eðgiu_ ÞÁðgju_ Þdt qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðE:33Þ i j ¼ e giju_ ðtÞu_ ðtÞdt for i; j ¼ 1; 2; ...; N where e (= ±1) is the functional indicator that ensures the square root always exists. 228 Appendix E: Euclidean and Riemannian Manifolds

Fig. E.9 Arc length between Riemannian surface S u 2 two points in a Riemannian a2 surface P

a1 s=PQ Q x 3 Euclidean space E3 r(u 1,u 2) u1

x 2 0

Therefore, the arc length of PQ is given by integrating Eq. (E.33) from the parameter t1 to the parameter t2.

Zt2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i j s ¼ e giju_ ðtÞu_ ðtÞ dt for i; j ¼ 1; 2; ...; N ðE:34Þ

t1 where the covariant metric coefﬁcients gij are deﬁned by

j gij ¼ gi:gj 6¼ di oxk oxk ðE:35Þ ¼ Á for k ¼ 1; 2; ...; N oui ou j We now assume that the points P(u1,u2) and Q(u1,u2) lie on the Riemannian surface S, which is embedded in Euclidean space E3. Each point on the surface only depends on two parameterized curvilinear coordinates of u1 and u2 that are called the Gaussian surface parameters, as shown in Fig. E.9. The differential dr of the vector r can be rewritten in the coordinates (u1, u2): or dr ¼ dui oui i ðE:36Þ r; idu i aidu for i ¼ 1; 2

i where ai is the tangential vector of the coordinate u on the Riemannian surface. Therefore, the arc length ds on the differentiable Riemannian parameterized surface can be computed as

ðdsÞ2 ¼ dr Á dr i j ¼ ai Á ajdu du ðE:37Þ i j aijdu du for i; j ¼ 1; 2 Appendix E: Euclidean and Riemannian Manifolds 229 whereas aij are the surface metric coefficients only at the point P in the coordinates (u1, u2) on the Riemannian curved surface S. The formulation of (ds)2 in Eq. (E.37) is called the first fundamental form for the intrinsic geometry of Riemannian manifold (Springer 2012; Lang 1999; Lee 2000; Fecko 2011). The surface metric coefficients of the covariant and contravariant components have the similar characteristics such as the metric coefficients:

j aij ¼ aji ¼ ai Á aj 6¼ di oxk oxk ðE:38aÞ ¼ : for k ¼ 1; 2; ...; N; oui ou j

j j j ai ¼ ai Á a ¼ di ðE:38bÞ

Instead of the metric coefﬁcients gij in the curvilinear Euclidean space, the surface metric coefﬁcients aij are used in the general curvilinear Riemannian manifold.

E.2.4 Tangent and Normal Vectors on the Riemannian Surface

We consider a point P(u1, u2) on a differentiable Riemannian surface that is 1 2 parameterized by u and u . Furthermore, the vectors a1 and a2 are the covariant bases of the curvilinear coordinates (u1, u2), respectively. In general, a hypersurface in an N-dimensional manifold with coordinates {ui} for i = 1, 2, …, N can be deﬁned as a differentiable (N - 1)-dimensional subspace with a codimension of 1. i The basis ai of the coordinate u can be rewritten as or ai ¼ oui ðE:39Þ r;i for i ¼ 1; 2

i The covariant basis ai is tangent to the coordinate u at the point P. Both bases a1 and a2 generate the tangential surface T tangent to the Riemannian surface S at the point P, which is deﬁned by the curvilinear coordinates of u1 and u2, as shown in Fig. E.10. The angle of two intersecting Gaussian parameterized curves ui and uj results from the dot product of the bases at the point P(ui, uj).

ai Á aj ¼ jjai Á aj cosðai; ajÞ) ai Á aj cos hij cosðai; ajÞ¼ jja Á a ðE:40Þ i j hi aij p ¼ pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 1 for hij 2 0; aðiiÞ Á aðjjÞ 2 230 Appendix E: Euclidean and Riemannian Manifolds

Fig. E.10 Tangent vectors to NP the curvilinear coordinates 1 2 tangential surface T (u , u ) 2 r,2 = a2 u nP Riemannian surface θ12 P r,1 = a1 r(u1, u 2) (S)

0 u1

Note that pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi jjai ¼ ai Á ai ¼ aðiiÞ; no summation over ðii) where aii and ajj are the vector lengths of ai and aj; aij, the surface metric coefficients. The surface metric coefficients can be defined by

j aij ¼ ai Á aj 6¼ di oxk oxk ðE:41Þ ¼ Á for k ¼ 1; 2...; N oui ou j

E.2.5 Angle Between Two Curvilinear Coordinates

We now give a concrete example of the computation of the angle between two curvilinear coordinates. Given two arbitrary basis vectors at the point P(u1, u2), we can write them with the covariant basis {ei}:

a1 ¼ 1 Á e1 þ 0 Á e2;

a2 ¼ 0 Á e1 þ 1 Á e2:

The covariant metric coefﬁcients aij can be calculated: a11 a12 10 ðaijÞ¼ ¼ a21 a22 01 The angle between two base vectors results from Eq. (E.40): Appendix E: Euclidean and Riemannian Manifolds 231

aij cos hij ¼ pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi aðiiÞ Á aðjjÞ

a12 0 ) cos h12 ¼ pffiffiffiffiffiffi pffiffiffiffiffiffi ¼ pffiffiffi pffiffiffi ¼ 0 a11 Á a22 1 Á 1 Thus, À1 a12 À1 p h12 ¼ cos pffiffiffiffiffiffi pffiffiffiffiffiffi ¼ cos ðÞ¼0 a11 Á a22 2 In this case, the curvilinear coordinates of u1 and u2 are orthogonal at the point P on the Riemannian surface S, as shown in Fig. E.10. The tangent vectors a1 and a2 generate the tangential surface T tangent to the Riemannian surface S at the point P. The normal vector NP to the tangential surface T at the point P is given by or or N ¼ Â ¼ r Â r P oui ou j ; i ; j ðE:42Þ ai Â aj for i; j ¼ 1; 2 ¼ a ak where a is the scalar factor; k a is the contravariant basis of the curvilinear coordinate of uk.

Multiplying Eq. (E.42) by the covariant basis ak, the scalar factor a results in

k k aða Á akÞ¼ad ¼ a ¼ðai Â ajÞÁ ak k ÂÃ ðE:43Þ ) a ¼ðai Â ajÞÁ ak ai; aj; ak The scalar factor a equals the scalar triple product that is given in Nayak (2012):

a ½¼ða1; a2; a3 ai Â ajÞÁ ak ¼ðak Â aiÞÁ aj ¼ðaj Â akÞÁ ai 1 1 1 2 2 2 a11 a12 a13 a31 a32 a33 a21 a22 a23

¼ a21 a22 a23 ¼ a11 a12 a13 ¼ a31 a32 a33 ðE:44Þ a a a a a a a a a q31ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi32 33 21 22 23 11 12 13

¼ det ðaijÞ J where Jacobian J is the determinant of the covariant basis tensor. The unit normal vector nP in Eq. (E.42) becomes using the Lagrange identity. 232 Appendix E: Euclidean and Riemannian Manifolds

Fig. E11 Surface area in the u 2 a2 curvilinear coordinates 2 a2du 1 2 P(u ,u ) dS a1 1 a1du 2 2 r(u1,u 2) u +du u1+du1

0 u1

ai Â aj qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiai Â aj nP ¼ ¼ ðE:45Þ ai Â aj 2 aðiiÞ Á aðjjÞ ÀðaijÞ

Note that

2 jjai ¼ ai Á ai ¼ aðiiÞ; no summation over ðii) The Lagrange identity results from the cross product of two vectors a and b.

jja Â b ¼ jja Á jjb sinða; bÞ) jja Â b 2 ¼ jja 2Á jjb 2sin2ða; bÞ ÀÁ ¼ jja 2Á jjb 2Á 1 À cos2ða; bÞ ¼ ðÞjjÁa jjb 2ÀðÞjja : jjÁb cosða; bÞ 2 ¼ ðÞjja Á jjb 2ÀðÞa Á b 2 Thus, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jja Â b ¼ jja 2Á jjb 2Àða Á bÞ2 ðE:46Þ

Equation (E.46) is called the Lagrange identity.

E.2.6 Surface Area in Curvilinear Coordinates

The surface area S in the differentiable Riemannian curvilinear surface, as displayed in Fig. E.11, can be calculated using the Lagrange identity (Nayak 2012). Appendix E: Euclidean and Riemannian Manifolds 233 ZZ o o r r i j S ¼ Â du du o i o j ZZ u u

i j ¼ ai Â aj du du ZZ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðE:47Þ 2 2 2 i j ¼ jjai : aj Àðai:ajÞ du du ZZ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 i j ¼ aðiiÞ: aðjjÞ ÀðaijÞ du du

Therefore, ZZ 1 2 S ¼ jja1 Â a2 du du for i ¼ 1; j ¼ 2 ZZ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðE:48Þ 2 1 2 ¼ a11 Á a22 Àða12Þ du du In Eq. (E.47), the vector length squared can be calculated as

2 jjai ¼ ai Á ai ¼ aðiiÞ no summation over ðiiÞ

E.3 Kronecker Delta

The Kronecker delta is very useful in tensor analysis and is deﬁned as ou j 0 for i 6¼ j d j ¼ ðE:49Þ i oui 1 for i ¼ j where ui and uj are in the same coordinate system and independent of each other. Some properties of the Kronecker delta (Kronecker tensor) are considered (Nayak 2012; Oeijord 2005). We summarize a few properties of the Kronecker delta: Property 1 Chain rule of differentiation of the Kronecker delta using the contraction rule (cf. Appendix A) ou j ou j ouk d j ¼ ¼ ¼ d jdk ðE:50Þ i oui ouk oui k i 234 Appendix E: Euclidean and Riemannian Manifolds

Property 2 Kronecker delta in Einstein summation convention

i jk i 1k i ik i Nk dja ¼ d1a þÁÁÁþdia þÁÁÁþdN a ¼ 0 þÁÁÁþ1:aik þÁÁÁþ0 ðE:51Þ ¼ aik

Property 3 Product of Kronecker deltas

i j i 1 i i i N djdk ¼ d1dk þÁÁÁþdidk þÁÁÁþdN dk i ¼ 0 þÁÁÁþ1:dk þÁÁÁþ0 ðE:52Þ i ¼ dk Note that

ðiÞ 1 2 N dðiÞ d1 ¼ d2 ¼ÁÁÁ ¼dN ¼ 1ðÞ no summation over the free index i ; However,

i 1 2 N di d1 þ d2 þÁÁÁ þdN ¼ N ðÞsummation over the dummy index i :

E.4 Levi-Civita Permutation Symbols

Levi-Civita permutation symbols in a three-dimensional space are third-order pseudo-tensors. They are a useful tool to simplify the mathematical expressions and computations (Klingbeil 1966; Nayak 2012; Kay 2011). The Levi-Civita permutation symbols can simply be deﬁned as 8 <> þ1ifði; j; kÞ is an even permutation; eijk ¼ > À1ifði; j; kÞ is an odd permutation; : E:53 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 Here, we abstain from giving an exact deﬁnition of even and odd permutations because this would go beyond the scope of this book. The reader is referred to the literature (Lee 2000; Fecko 2011). Appendix E: Euclidean and Riemannian Manifolds 235

Fig. E.12 27 Levi-Civita permutation symbols j = 1,2,3 ⎛ ⎞ ⎜ 0 1 0⎟ i = 1,2,3 ⎜−1 0 0⎟ ⎜ ⎟ ⎝ 0 0 0⎠ = ⎛ − ⎞ ijk ⎜0 0 1⎟ ⎜0 0 0 ⎟ ⎜ ⎟ ⎝1 0 0 ⎠ ⎛ ⎞ ⎜0 0 0⎟ ⎜0 0 1⎟ ⎜ ⎟ ⎝0 −1 0⎠

According to Eq. (E.53), the Levi-Civita permutation symbols can be expressed as eijk ¼ ejki ¼ ekij ðeven permutationÞ; eijk ¼ ðE:54Þ Àeikj ¼Àekji ¼Àejik ðodd permutationÞ The 27 Levi-Civita permutation symbols for a three-dimensional coordinate system are graphically displayed in Fig. E.12.

References

Fecko M (2011) Differential geometry and lie groups for physicists. Cambridge University Press, Cambridge Grifﬁths DJ (2005) Introduction to quantum mechanics, 2nd edn. Pearson Prentice Hall Inc., Englewood Cliffs Kay DC (2011) Tensor calculus. Schaum’s Outline Series, McGraw-Hill Klingbeil E (1966) Tensorrechnung für Ingenieure (in German). B.I.-Wissenschafts-verlag, Mannheim/Wien/Zürich Lang S (1999) Fundamentals of differential geometry. Springer, Berlin, New York Lee J (2000) Introduction to smooth manifolds. Springer, Berlin, New York Nayak PK (2012) Textbook of tensor calculus and differential geometry. PHI Learning, New Delhi Oeijord NK (2005) The very basics of tensors. IUniverse Inc, New York Riemann B (2013) Über die Hypothesen, welche der Geometrie zu Grunde liegen, Springer Spektrum. Springer, Berlin, Heidelberg Springer CE (2012) Tensor and vector analysis. Dover Publications Inc, Mineola