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 Â Á ¼ α k Á ¼ αδk ¼ α gi gj gk g gk k  hi ðA:2Þ ) α ¼ Â Á  ; ; gi gj gk gi gj gk

The scalar triple product of the covariant bases can be written as pffiffiffi α ¼ ½Š¼; ; 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). pffiffiffiffiffiffiffiffi ¼ jj¼ ð : Þ hi gi gðÞii B 1b

Thus, the covariant basis can be related to its unitary covariant basis by the relation pffiffiffiffiffiffiffiffi ¼ * ¼ * ð : Þ gi gðÞii gi higi B 2

The contravariant basis can be related to its unitary covariant basis using Eqs. (2.47) and (B.2).

i ¼ ij ¼ ij * ð : Þ g g gj g hjgj B 3

The contravariant second-order tensor can be written in the unitary covariant bases using Eq. (B.2).

© Springer-Verlag Berlin Heidelberg 2017 319 H. Nguyen-Scha¨fer, J.-P. Schmidt, Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers, Mathematical Engineering, DOI 10.1007/978-3-662-48497-5 320 Appendix B: Physical Components of Tensors ÀÁ ¼ ij ¼ ij * *  *ij * * ð : Þ T T gigj T hihj gi gj T gi gj B 4

Thus, the physical contravariant tensor components denoted by star result in

*ij ij T  hihjT ðB:5Þ

The covariant second-order tensor can be written in the unitary contravariant bases using Eq. (B.3). ÀÁ ¼ i j ¼ ik jl * *  * * * ð : Þ T Tijg g Tijg g hkhl gkgl Tijgkgl B 6

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

*  ik jl ð : Þ Tij g g hkhlTij B 7

The mixed tensors can be written in the unitary covariant bases using Eqs. (B.2) and (B.3) ÀÁ T ¼ T ig gj ¼ T ig gjkg j i j ÀÁi ÀÁk ¼ T i h g* gjkh g* j i i k k ð : Þ ¼ i jk * * B 8 Tj g hihk gi gk  *  i * * Tj gi gk

Thus, the physical mixed tensor components denoted by star result in  * i  jk i ð : Þ Tj g hihkTj B 9

Analogously, the contravariant vector can be written using Eq. (B.2).

¼ i ¼ ðÞi * v v gi v hi gi v*i ðB:10Þ  *i * ¼ v gi gi hi

* Thus, the physical component of the contravariant vector v on the unitary basis gi is defined as pffiffiffiffiffiffiffiffi *i  i ¼ i ð : Þ v hiv gðÞii v B 11

The contravariant basis can be normalized dividing by its vector length without summation over (i). Appendix B: Physical Components of Tensors 321

gi gi g*i ¼ ¼ pffiffiffiffiffiffiffiffi ðB:12Þ jjgi gðÞii 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 ¼ i ¼ ðÞii *i  * *i ð : Þ v vig vi g g vi g B 13

Thus, the physical component of the covariant vector v results in qffiffiffiffiffiffiffiffi * ¼ ðÞii ð : Þ vi vi g B 14

According to Eq. (A.27), Eq. (B.14) can be rewritten in orthogonal coordinate systems: qffiffiffiffiffiffiffiffi 1 1 * ¼ ðÞii ¼ ¼ ð : Þ vi g vi pffiffiffiffiffiffiffiffi vi vi B 15 gðÞii hi

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

v ¼ v gi ¼ v gijg ÀÁi i j ¼ v gijh g*  v*g* i j j j j ðB:16Þ v* ¼ j i ij g g hj

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

*  ij ð : Þ vj g hjvi B 17

Furthermore, the vector v can be written in both covariant and contravariant bases.

¼ j ¼ i v ÀÁvjg v gi ) v gj Á g ¼ ðÞÁvig g j k i k ðB:18Þ ) δ j ¼ i vj k v gik ) ¼ i vk v gik

Interchanging i with j and k with i, one obtains

¼ j ð : Þ vi v gij B 19

Only in orthogonal coordinate systems, we have 322 Appendix B: Physical Components of Tensors

¼ 6¼ ; ¼ 2 ð : Þ gij 0fori j gðÞii hi B 20

Thus, one obtains from Eq. (B.19)

v ¼ vjg ¼ v1g þ v2g þ ...þ vNg i ij i1 i2 iN ð : Þ ¼ i ¼ i 2 B 21 v gðÞii v hi

Substituting Eq. (B.11) into Eq. (B.21), one obtains Eq. (B.22) that is equivalent to Eq. (B.15).  *i ¼ i 2 ¼ v 2 ¼ *i ð : Þ vi v hi hi hiv B 22 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}

∂f ∂f ∂f ∇f ¼ e þ e þ e ðC:1Þ ∂x x ∂y y ∂z z

• General curvilinear coordinate {ui}

∂f ∂f ∇f ¼ f gi ¼ gi ¼ gijg j ðC:2Þ , i ∂ui ∂ui j

2. Gradient of a vector v • General curvilinear coordinate {ui}  ∇ ¼ k þ jΓ k i ¼ k i v v,i v ij gkg v igkg  ðC:3Þ ¼ À Γ j k i ¼ j k i vk,i vj ik g g vk ig g

3. Divergence of a vector v • Cartesian coordinate {xi}

© Springer-Verlag Berlin Heidelberg 2017 323 H. Nguyen-Scha¨fer, J.-P. Schmidt, Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers, Mathematical Engineering, DOI 10.1007/978-3-662-48497-5 324 Appendix C: Nabla Operators

∂v ∂v ∂v ∇ Á v ¼ x þ y þ z ðC:4Þ ∂x ∂y ∂z

• General curvilinear coordinate {ui}  ∇ Á ¼ ij  i þ jΓ i v v i v,i v ij ÀÁpffiffiffi 1 ∂ gvi 1 ÀÁpffiffiffi ¼ pffiffiffi ¼ pffiffiffi gvi ðC:5Þ g ∂ui g , i  ∇ Á ¼ j ki ¼ À Γ j ki v vkig vk, i vj ik g

4. Gradient of a second-order tensor T • General curvilinear coordinate {ui} for a covariant second-order tensor  m m i j k ∇T ¼ Tij, k À Γ Tmj À Γ Tim g g g ik jk ðC:6Þ i j k ¼ Tijj kg g g

• General curvilinear coordinate {ui} for a contravariant second-order tensor  ∇ ¼ ij þ Γ i mj þ Γ j im k T T T T gigjg ,k km km ðC:7Þ ¼ ijj k T kgigjg

• General curvilinear coordinate {ui} for a mixed second-order tensor  ∇ ¼ i þ Γ i m À Γ m i j k T Tj,k kmTj jk Tm gig g ðC:8Þ ¼ ij j k Tj kgig g

5. Divergence of a second-order tensor T • General curvilinear coordinate {ui} for a covariant second-order tensor ÀÁ m m i j k ∇ Á T ¼ Tij, k À Γ Tmj À Γ Tim g g Á g ik jk ðC:9Þ jk i  Tijj kg g

• General curvilinear coordinate {ui} for a contravariant second-order tensor  ∇ Á T ¼ Tij þ Γ i Tmj þ Γ j Tim δ kg , k km km i j ¼ ij þ Γ i mj þ Γ j im ðC:10Þ T,i imT imT gj  ijj T igj Appendix C: Nabla Operators 325

• General curvilinear coordinate {ui} for a mixed second-order tensor  ∇ Á T ¼ T i þ Γ i T m À Γ mT i δ kgj j,k km j jk m i ¼ i þ Γ i m À Γ m i j ðC:11Þ Tj,i imTj ji Tm g  ij j ¼ ij jk Tj ig Tj ig gk

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

ex ey ez

∇ Â v ¼ ∂ ∂ ∂ ðC:12Þ ∂x ∂y ∂z vx vy vz

The curl of v results from calculating the determinant of Eq. (C.12).  ∂v ∂v ∂v ∂v ∂v ∂v ∇ Â v ¼ z À y e þ x À z e þ y À x e ðC:13Þ ∂y ∂z x ∂z ∂x y ∂x ∂y z

• General curvilinear coordinate {ui}

∇ Â ¼ ^ε ijk ð : Þ v vj,igk C 14

The contravariant permutation symbol is defined by 8 > 1 > þpffiffiffi if ðÞi, j, k is an even permutation; < g ^ε ijk ¼ 1 ð : Þ > À pffiffiffi ðÞ ; C 15 > if i, j, k is an odd permutation :> g 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}

∂2f ∂2f ∂2f ∇2f  Δf ¼ þ þ ðC:16Þ ∂x2 ∂y2 ∂z2

• General curvilinear coordinate {ui}  ∇2  Δ ¼ À Γ k ij f f f ,ij f , k ij g  ðC:17Þ ¼ À Γ k ij  ij vi,j vk ij g vijg 326 Appendix C: Nabla Operators

where the covariant vector component and its derivative with respect to uk are defined by

∂f ∂f ∂2f v ¼ f ¼ ; v ¼ f ¼ ; v ¼ f ¼ ðC:18Þ i ,i ∂ui k , k ∂uk i, j , ij ∂ui∂uj

8. Calculation rules of the Nabla operators

Div Grad f ¼ ∇ Á ðÞ¼∇f ∇2f ¼ Δf ðÞLaplacian ðC:19Þ Curl Grad f ¼ ∇ Â ðÞ¼∇f 0 ðC:20Þ Div Curl v ¼ ∇ Á ðÞ¼∇ Â v 0 ðC:21Þ ΔðÞ¼fg f Δg þ 2∇f Á ∇g þ gΔf ðC:22Þ Curl Curl v ¼ ∇ Â ðÞ¼∇ Â v ∇∇ðÞÀÁ v Δv ðÞðCurl 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 Δv ¼ ∇ Á ðÞ∇ ¼ ∇2  v v p ðC:24Þ ¼ vi À vi Γ þ vp Γ i gjkg j,k p jk j pk i  i jk v jkg gi

9. Essential Vector and Nabla Identities Let f and g be arbitrary scalars in R; A, B, and C arbitrary vectors in RN.

Products of vectors

A  B ¼ÀB  A A Á ðÞ¼B  C B Á ðÞ¼C  A C Á ðÞA  B : even permutation A Á ðÞ¼ÀB  C C Á ðÞ¼ÀB  A B Á ðÞA  C : odd permutation A  ðÞ¼B  C BAðÞÀÁ C CAðÞÁ B

Gradient ∇()

∇ðÞ¼f þ g ∇f þ ∇g 2 RN ∇ðÞ¼fg f ∇g þ g∇f 2 RN ∇ðÞ¼A Á B ðÞA Á ∇ B þ ðÞB Á ∇ A þ A Â ðÞþ∇ Â B B Â ðÞ2∇ Â A RN ∇f  ∇ f 2 RN : 1st order tensor, vector ∇A  ∇ A 2 RN Â RN : 2nd order tensor Appendix C: Nabla Operators 327

Divergence ∇Á()

∇ Á ðÞ¼A þ B ∇A þ ∇B 2 R ∇ Á ðÞ¼f A f ∇ Á Α þ A Á ∇f 2 R ∇ Á ðÞ¼A Â B B Á ðÞÀ∇ Â A A Á ðÞ2∇ Â B R 1 ðÞA Á ∇ A ¼ ∇A2 À A Â ðÞ2∇ Â A RN 2

Curl ∇Â()

∇ Â ðÞ¼A þ B ðÞþ∇ Â A ðÞ2∇ Â B RN ∇ Â ðÞ¼f A f ðÞÀ∇ Â Α A Â ∇f 2 RN ∇ Â ðÞ¼A Â B AðÞÀ∇ Á B ðÞA Á ∇ B À BðÞþ∇ Á A ðÞB Á ∇ A 2 RN ∇ Â ðÞ¼∇ Â A ∇∇ðÞÀÁ A ∇2A 2 RN ðÞA Á ∇ A ¼ 1∇A2 À A Â ðÞ2∇ Â A RN 2

Laplacian Δ(),∇2()

Δf  ∇2f ¼ ðÞ∇ Á ∇ f ¼ ∇ Á ðÞ∇f : Laplacian f 2 R ΔðÞfg ∇2ðÞ¼fg f ∇2g þ 2∇f Á ∇g þ g∇2f 2 R ΔA ¼ ∇2A  ∇∇ðÞÀÁ A ∇  ðÞ∇  A : Laplacian A 2 RN ΔðÞf A ∇2ðÞ¼f A A∇2f þ 2ðÞ∇f Á ∇ A þ f ∇2A 2 RN ΔðÞA Á B ∇2ðÞ¼A Á B A Á ∇2B À B Á ∇2A þ 2∇ Á ½Š2ðÞB Á ∇ A þ B  ðÞ∇  A R

Differentiations

∇ Â ðÞ∇f ∇ Â ðÞ¼∇ f 0 ∇ Á ðÞ¼∇ Â A 0 ∇ Á ðÞf ∇g ¼ f ∇2g þ ∇f Á ∇g 2 R ∇ Á ðÞ¼f ∇g À g∇f f ∇2g À g∇2f 2 R

10. Summary of essential Nabla operators

Operator Laplacian Operand ( ) Grad ∇( ) Div ∇Á( ) Curl ∇Â() Δ() Function f 2 R (0th Vector 2 RN – – Scalar 2 R order tensor) Vector v 2 RN (1st 2nd order tensor Scalar Vector Vector order tensor) 2 RN Â RN 2 R 2 RN 2 RN Tensor T 2 RN Â RN 3rd order tensor Vector –– (2nd order tensor) 2 RN Â RN Â RN 2 RN Order of results One order higher One order Order Order lower unchanged unchanged Appendix D: Essential Tensors

Derivative of the covariant basis

¼ Γ k ð : Þ gi,j ijgk D 1

Derivative of the contravariant basis

i ∂g i g i ¼  Γ^ gk ¼ÀΓ i gk ðD:2Þ ,j ∂uj jk jk

Derivative of the covariant metric coefficient

¼ Γ p þ Γ p ð : Þ gij, k ikgpj jkgpi D 3

First-kind Christoffel symbol  Γ ¼ 1 g þ g À g ¼ g Γ l ) Γ l ¼ glkΓ ðD:4Þ ijk 2 ik, j jk, i ij, k lk ij ij ijk

Second-kind Christoffel symbol based on the covariant basis

Γ k ¼ Á k ¼ klΓ ð : Þ ij gi, j g g ijl D 5 ∂uk ∂2xp Γ k ¼ Á ¼ Γ k ðD:6Þ ij ∂ p ∂ i∂ j ji x u u Γ k ¼ gkp1 g þ g À g ðD:7Þ ij 2 ip,j jp,i ij,p

© Springer-Verlag Berlin Heidelberg 2017 329 H. Nguyen-Scha¨fer, J.-P. Schmidt, Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers, Mathematical Engineering, DOI 10.1007/978-3-662-48497-5 330 Appendix D: Essential Tensors

1 ∂J ∂ðÞlnJ Γ i ¼ ¼ ij ∂ j ∂ j J u pffiffiffi u ÀÁpffiffiffi ð : Þ 1 ∂ g ∂ ln g D 8 ¼ pffiffiffi ¼ g ∂uj ∂uj

Second-kind Christoffel symbol based on the contravariant basis

Γ^ i ¼ÀΓ i ¼ Γ^ i ð : Þ kj kj jk D 9

Covariant derivative of covariant first-order tensors

¼ À Γ k ¼ Á ð : Þ Tij Ti,j ijTk T,j gi D 10

Covariant derivative of contravariant first-order tensors

i ¼ i þ Γ i k ¼ Á i ð : Þ T j T, j jkT T,j g D 11

Covariant derivative of covariant and contravariant second-order tensors

T j ¼ T À Γ mT À Γ mT ij k ij,k ik mj jk im ð : Þ ijj ¼ ij þ Γ i mj þ Γ j im D 12 T k T, k kmT kmT

Covariant derivative of mixed second-order tensors

i i i m m i T jk ¼ T þ Γ T À Γ T j j, k km j jk m ðD:13Þ jj ¼ j þ Γ j m À Γ m j Ti k Ti, k kmTi ik Tm

Second covariant derivative of covariant first-order tensors

¼ À Γ m À Γ m Tikj Ti, jk ik, jTm ik Tm,j ÀΓ m þ Γ mΓ n ð : Þ ij Tm, k ij mkTn D 14 ÀΓ m þ Γ mΓ n jk Ti, m jk imTn

Second covariant derivative of the contravariant vector

kj  kj À k Γ p þ pj Γ k ð : Þ v lm v l,m v p lm v l pm D 15 where ÀÁ kj ¼ kj  k þ n Γ k þ nΓ k ð : Þ v l, m v l ,m v, lm v, m nl v nl, m D 16 Appendix D: Essential Tensors 331

k  k þ nΓ k ð : Þ v p v, p v np D 17 pj  p þ nΓ p ð : Þ v l v, l v nl D 18

Riemann-Christoffel tensor

n  Γ n À Γ n þ Γ mΓ n À Γ mΓ n ð : Þ Rijk ik, j ij,k ik mj ij mk D 19

Riemann curvature tensor

 n ð : Þ Rlijk gnlRijk D 20

First-kind Ricci tensor

∂Γ k ∂Γ k R ¼ ik À ij À Γ rΓ k þ Γ r Γ k ij ∂uj ∂uk ij rk ik rj ÀÁ pffiffiffi ð : Þ 2 pffiffiffi ∂ Γ k D 21 ∂ ln g 1 g ij ¼ À pffiffiffi þ Γ r Γ k ∂ui∂uj g ∂uk ik rj

Second-kind Ricci tensor

i  ik Rj g Rkj  0 ÀÁ pffiffiffi 1 2 pffiffiffi ∂ Γ m ∂ ln g 1 g kj ðD:22Þ ¼ gik@ À pffiffiffi þ Γ m Γ n A ∂uk∂uj g ∂um kn mj

Ricci curvature  0 ÀÁ pffiffiffi 1 2 pffiffiffi ∂ Γ k ∂ ln g 1 g ij R ¼ gij@ À pffiffiffi þ Γ k Γ r A ðD:23Þ ∂ui∂uj g ∂uk ir kj

Einstein tensor

G i  R i À 1 δ iR ¼ gikG j j 2 j kj ¼ k ¼ À 1 ð : Þ Gij gikGj Rij gijR D 24 2

i ¼ ð : Þ Gj 0 D 25 i

First fundamental form 332 Appendix D: Essential Tensors

¼ 2 þ þ 2; I Edu2FdudvGdv Á Á ð : Þ ¼  EF¼ ru ru ru rv D 26 M gij FG rv Á ru rv Á rv

Second fundamental form

II ¼ Ldu2 þ 2Mdudv þ Ndv2; ÀÁ LM ruu Á nruv Á n ðD:27Þ H ¼ hij  ¼ MN ruv Á nrvv Á n

Gaussian curvature of a curvilinear surface ÀÁ À 2 LN M dethij K ¼ κ1κ2 ¼ ¼ ðD:28Þ EG À F2 det gij

Mean curvature of a curvilinear surface À þ 1 EN ÀÁ2MF LG H ¼ ðÞ¼κ1 þ κ2 ðD:29Þ 2 2 EG À F2

Unit normal vector of a curvilinear surface

g  g ru  rv ru  rv n ¼ 1 2 ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðD:30Þ jj 2 g1 g2 EG À F det gij

Differential of a surface area qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ jj ¼ À ðÞ2 dA rg1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig2 dudv g11g22 g12 dudv  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðD:31Þ ¼ ¼ À 2 det gij dudv EG F dudv

Gauss derivative equations

g ¼ Γ kg þ h g ¼ Γ kg þ h n i, j ij k ij 3 ij k ij ð : Þ ,  À Γ k ¼ D 32 gi j gi,j ijgk hijn

Weingarten’s equations ÀÁ ¼À j ¼À jk ð : Þ ni hi gj hikg gj D 33

Codazzi’s equations Appendix D: Essential Tensors 333

Kij, k ¼ Kik, j ) ðÞðK11,2 ¼ K12,1 ; K21,2 ¼ K22,1 D:34Þ

Gauss equations  ÀÁ ¼ j ¼ 1 2 À 1 2 K det Ki K1K2 K2K1 À 2 ðD:35Þ ¼ K11K22 K12 ¼ R1212 À 2 g11g22 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 coordi- nate systems, Cartesian (orthonormal) und 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).

© Springer-Verlag Berlin Heidelberg 2017 335 H. Nguyen-Scha¨fer, J.-P. Schmidt, Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers, Mathematical Engineering, DOI 10.1007/978-3-662-48497-5 336 Appendix E: Euclidean and Riemannian Manifolds

z z

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

x (xi + yj)

Fig. E.1 Vector r in Cartesian coordinates

r ¼ xi þ yj þ zk ðE:1Þ 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 [1, 2]. In the curvilinear coordinate system (u1,u2,u3), there are 1 2 3 three covariant bases g1, g2, and g3 and three contravariant bases g , g , and g 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 Appendix E: Euclidean and Riemannian Manifolds 337 coordinates only depend on 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:

¼ 1 þ 2 þ 3 r u g1 u g2 u g3 ð : Þ 1 2 3 E 3 ¼ u1g þ u2g þ u3g where u1, u2, u3 are the contravariant vector 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 covariant vector components of the coordinates (u ,u ,u ); g1, g2, g3 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 ¼ 1, 2, 3. Both bases g1 and g2 generate a tangential surface to the curvilinear surface (u1u2) at the considered origin 00,as shown in Fig. E.2. Note that the basis g1 is not perpendicular to the bases g2 and g3. 3 However, the contravariant basis g is perpendicular to the tangential surface (g1g2) at the origin 00. Generally, the contravariant basis gk results from the cross product of the other covariant bases (gi  gj).

α k ¼ Â ¼ ð : Þ g gi gj for i, j, k 1, 2, 3 E 4a where α is a scalar factor (scalar triple product) given in Eq. (1.6).  α ¼ Â Á gi gj gk hi ðE:4bÞ  ; ; gi gj gk

Thus,    1 ¼ g2 g3 ; 2 ¼ g3 g1 ; 3 ¼ g1 g2 ð : Þ g ½Š; ; g ½Š; ; g ½Š; ; E 4c g1 g2 g3 g1 g2 g3 g1 g2 g3 338 Appendix E: Euclidean and Riemannian Manifolds

u3

g3

g3 P(u1,u2,u3) r g1

0‘ g2 g2 g u1 1 u2

Fig. E.2 Bases of the curvilinear coordinates

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

i Á j ¼ jji Á jjj Á cos ðÞi; j π ¼ ðÞÁ1 ðÞÁ1 cos 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 Appendix E: Euclidean and Riemannian Manifolds 339

jjr 2 ¼ r Á r ¼ ðÞx i þ y j þ z k :ðÞx i þ y j þ z k ¼ x2ðÞþi Á i xyðÞþi Á j xzðÞi Á k ðE:8Þ þyxðÞþj Á i y2ðÞþj Á j yzðÞj Á k þzxðÞþk Á i zyðÞþk Á j z2ðÞk Á k

Due to Eqs. (E.6) and (E.7) the vector length in Eq. (E.8) becomes pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jjr 2 ¼ 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; i.e., 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 ¼ j  k. The bases (i, j, k) form a right-handed triple. 8 < i ¼ j  k ¼Àk  j ¼  ¼À  ð : Þ : j k i i k E 10 k ¼ i  j ¼Àj  i

The magnitude of the cross product of two orthonormal bases is calculated as

i  j ¼ jji Á jjj Á sin ðÞi; j k  π ðE:11Þ ) jji  j ¼ jji Á jjj Á sin ðÞi; j jjk ¼ 1 Á 1 Á sin jjk ¼ jjk 2

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 [1, 3]:

i dr  eidx for i ¼ 1, 2, 3 X3 i ðE:12Þ ¼ eidx i¼1 340 Appendix E: Euclidean and Riemannian Manifolds

The Einstein summation convention used in Eq. (E.12) indicates that dr equals the i sum of ei dx by running the dummy index i from 1 to 3. The arc length ds between the points P and Q (cf. Fig. E.3) can be calculated by the dot product of two differentials.

ðÞ2 ¼ Á ds dr dr ÀÁ ¼ ðÞÁe dxi e dxj ÀÁi j ðE:13Þ i j ¼ ei Á ej dx dx ¼ dxidxi for i ¼ 1, 2, 3: Thus, the arc length in the orthonormal coordinate system results in pffiffiffiffiffiffiffiffiffiffiffiffi ds ¼ qdxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiidxi for i ¼ 1, 2, 3 ðE:14Þ ¼ ðÞdx1 2 þ ðÞdx2 2 þ ðÞdx3 2

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.

∂r dr ¼ dui ∂ui ðE:15Þ  i gidu i where gi is the covariant basis of the curvilinear coordinate u . x3 x3

P ds dr Q r (C) e3 r + dr 0 x2 e2 x2 e1 x1

x1 Fig. E.3 Arc length ds of P and Q in Cartesian coordinates Appendix E: Euclidean and Riemannian Manifolds 341

u3

g 3 P(ui(t ),…) 1 ds r dr ui t Q( ( 2),…) r + dr 0‘ g 2 (C) 1 2 u g1 u

Fig. E.4 Arc length ds of P and Q in the curvilinear coordinates

Analogously, the arc length ds between two points of P and Q in the curvilinear coordinate system can be calculated by

ðÞds 2 ¼ dr Á dr ÀÁ ¼ i Á j gidu gjdu  ðE:16Þ ¼ Á i j gi gj du du ¼ i j ¼ gij du du for i, j 1, 2, 3

Therefore, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

¼ i j ds gij du du sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðt2 i j ðE:17Þ du du ) s ¼ g dt ij dt dt t1 where t is the parameter in the curve C with the coordinate ui(t); gij is defined as the metric coefficient of two non-orthonormal bases: 342 Appendix E: Euclidean and Riemannian Manifolds

¼ Á ¼ Á ¼ 6¼ δ j ð : Þ gij gi gj gj gi gji i E 18

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

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

δj where i 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 ∂r ðE:22Þ ¼ dxi ∂xi

i Thus, the orthonormal bases ei of the coordinate x can be defined as

∂r e ¼ for i ¼ 1, 2, 3 ðE:23Þ i ∂xi

Analogously, the basis of the curvilinear coordinate ui can be calculated in the curvilinear coordinate system of E3. Appendix E: Euclidean and Riemannian Manifolds 343

∂r g ¼ for i ¼ 1, 2, 3 ðE:24Þ i ∂ui

Substituting Eq. (E.24) into Eq. (E.18) we obtain the metric coefficients gij that are generally symmetric in Euclidean space; i.e., gij ¼ gji.

g ¼ g Á g ¼ g 6¼ δ j ij i j ji i  ∂ ∂ ∂ ∂ m ∂ ∂ n ¼ r Á r ¼ r x Á r x ∂ui ∂uj ∂xm ∂ui ∂xn ∂uj ð : Þ ∂xm ∂xn ∂xm ∂xn E 25 ¼ ðÞ¼e Á e δ n ∂ui ∂uj m n ∂ui ∂uj m ∂xk ∂xk ¼ for k ¼ 1, 2, 3 ∂ui ∂uj

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: ¼ length of its relating covariant basis; thus,g gk 1. As a result, the scalar products of the covariant and contravariant bases can be written in general curvilinear coordinates (u1,...,uN).  Á k ¼ k Á ¼ δ k ¼ ... gi g g gi i for i, k 1, 2, , N ð : Þ Á ¼ ¼ 6¼ δ k ¼ ... E 26 gi gk gik gki i for i, k 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 [4], 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; and the orthonormalizing scheme is demonstrated in Fig. E.5. 344 Appendix E: Euclidean and Riemannian Manifolds

g 3 g1

g2/1 g3 e e3 e1 2 e1

g3/1

g g3/2 e2 e2 2

Fig. E.5 Schematic visualization of the Gram-Schmidt procedure

The Gram-Schmidt scheme for N ¼ 3 has three orthonormalization steps:

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

¼ g1 e1 jj g1

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 À g À ðÞ: ¼ 2 2=1 ¼ g2 g2 e1 e1 e2 jjÀ ðÞ: À g2 g2 e1 e1 g2 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 À g À g À ðÞ: À ðÞ: ¼ 3 3=1 3=2 ¼ g3 g3 e1 e1 g3 e2 e2 e3 jjÀ ðÞ: À ðÞ: À À g3 g3 e1 e1 g3 e2 e2 g3 g3=1 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: Appendix E: Euclidean and Riemannian Manifolds 345

jXÀ 1  À Á gj gj ei ei ¼ i ¼ 1 ¼ ... ej À  for j 1, 2, , N Xj 1 À Á gj gj ei ei i ¼ 1

E.1.7 Angle Between Two Vectors and Projected Vector Component

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

a Á b g Á g aibj cos θ ¼ ¼ i j jja Á jjb jja Á jjb i j i j ðE:27Þ gija b gija b ¼ qffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi i j Á k l i Á j gija a gklb b a ai b bj where

jja 2 ¼ a Á a ðE:28Þ ¼ i j ¼ ij ¼ i ¼ ... gija a g aiaj a ai for i, j 1, 2, , N 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 θ g aibj g aibj ¼ jjÁ ij ¼ ij a jjÁ jj jj a b b ðE:29Þ g aibj ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiij for i, j, k, l ¼ 1, 2, ..., N k l gklb b 346 Appendix E: Euclidean and Riemannian Manifolds

a(ui )

θ b(ui )

ab = a.cosθ

Fig. E.6 Angle between two vectors and projected vector component

Examples Given two vectors a and b: pffiffiffi ¼ Á þ Á ¼ i ; a 1 e1 3 e2 a gi ¼ Á þ Á ¼ j b 1 e1 0 e2 b gj

Thus, the relating vector components are

¼ ; ¼ g1 e1 g2 ep2 ffiffiffi 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 calcu- lated according to Eq. (E.18).   ¼ g11 g12 ¼ 10 gij g21 g22 01

The angle θ between two vectors results from Eq. (E.27).

g aibj cos θ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiij qffiffiffiffiffiffiffiffiffiffiffiffiffi f or i, j, k, l ¼ 1, 2 i j: k l gija a gklb b g a1b1 þ g a1b2 þ g a2b1 þ g a2b2 ¼ 11 q12ffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi21 22 i j: k l gija a gklb b ÀÁpffiffiffi ÀÁpffiffiffi ðÞþ1 Á 1 Á 1 ðÞþ0 Á 1 Á 0 0 Á 3 Á 1 þ 1 Á 3 Á 0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1 1 þ 0 þ 0 þ 3 : 1 þ 0 þ 0 þ 0 2

Therefore, Appendix E: Euclidean and Riemannian Manifolds 347  1 π θ ¼ cos À1 ¼ 2 3

The projected vector component can be calculated according to Eq. (E.29).

i j gija b ab ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffi for i, j, k, l ¼ 1, 2 g bkbl kl ÀÁpffiffiffi ÀÁpffiffiffi ðÞþ1 Á 1 Á 1 ðÞþ0 Á 1 Á 0 0 Á 3 Á 1 þ 1 Á 3 Á 0 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 nonhomogenous 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 Rie- mannian manifold. In case of the infinitesimally small fiber lengths in all dimen- sions, the fiber bundle now becomes a single point. Therefore, the tuple of fiber bundles becomes a tuple of points on the manifold. In fact, Riemannian manifold only contains a point tuple [10]. 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 ¼ 1, 2,...,N at a certain point can be defined as a differentiable (NÀ1)-dimensional subspace with a codimension of one. This definition can be understood that the (NÀ1)-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 on the manifold locally has N curvilinear coordinates of u1,...,uN embedded at this point. Therefore, the consid- 1 N ered point Pi can be expressed in the curvilinear coordinates as Pi(u ,...,u ). 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 348 Appendix E: Euclidean and Riemannian Manifolds

Riemannian manifold RN

uN gN

g2 N Pi P (u1,…,u ) 1 i u 2 g1 u RN

Fig. E.7 Bundle of N coordinates at Pi in Riemannian manifold displayed in Fig. E.7. The arc length between any two points of N tuples of coordinates on the manifold does not physically change in any chosen basis. However, its components are changed in the coordinate bases that vary on the manifold. Therefore, these components must be taken into account in the transfor- mation 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 coefficients gij of the coordinates (u1,...,uN) 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 [1, 10].

E.2.2 Flat and Curved Surfaces

By abuse of notation and by completely abstaining from mathematical rigorous- ness, 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 349

α +β + γ = 180° α +β + γ ≠ 180°

A α A β γ S C α B B β γ C P

⊂ 3 (b): curved surface S ⊂ R3 (a): flat surface P E Fig. E.8 (a) Flat surfaces P  E3 and (b) curved surfaces S  R3

Conditions for the flat and curved surfaces [5]:  α þ β þ γ ¼ 180 for a flat surface ðE:30Þ α þ β þ γ 6¼ 180 for a curved surface

Furthermore, the surface curvature in Riemannian manifold can be used to deter- mine 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) in the curve C (see Fig. E.9). The arc length is an important notation in Riemannian manifold theory. The coordinates 1 N (u ,...,u ) can be considered as a function of the parameter t that varies from P(t1) to Q(t2). The arc length ds between the points P and Q thus results from  ds 2 dr dr ¼ Á ðE:31Þ dt dt dt where the derivative of the vector r(u1,...,uN) can be calculated as

ðÞi dr ¼ d giu dt dt ðE:32Þ  _ iðÞ ¼ ... giu t for i 1, 2, , N

Substituting Eq. (E.32) into Eq. (E.31), one obtains the arc length 350 Appendix E: Euclidean and Riemannian Manifolds

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ εðÞÁ_ i _ j ds giu gju dt qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðE:33Þ ¼ ε _ iðÞ_ jðÞ ¼ ... giju t u t dt for i, j 1, 2, , N where ε (¼Æ1) is the functional indicator that ensures the square root always exists. Therefore, the arc length of PQ is given by integrating Eq. (E.33) from the parameter t1 to the parameter t2.

ðt2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ε _ iðÞ_ jðÞ ¼ ... ð : Þ s giju t u t dt for i, j 1, 2, , N E 34

t1 where the covariant metric coefficients gij are defined by

¼ Á 6¼ δ j gij gi gj i ð : Þ ∂xk ∂xk E 35 ¼ Á for k ¼ 1, 2, ..., N ∂ui ∂uj

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.

Riemannian surface S u2 a2 P Q a1 (C) x3 Euclidean space E3 r(u1,u2) u1

x2 0

x1

Fig. E.9 Arc length between two points in a Riemannian surface Appendix E: Euclidean and Riemannian Manifolds 351

The differential dr of the vector r can be rewritten in the coordinates (u1,u2):

∂r dr ¼ dui ∂ui 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

ðÞds 2 ¼ dr Á dr i j ð : Þ ¼ ai Á ajdu du E 37 i j  aijdu du for i, j ¼ 1, 2 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 [6–9]. The surface metric coefficients of the covariant and contravariant components have the similar characteristics such as the metric coefficients:

¼ ¼ Á 6¼ δ j aij aji ai aj i ð : Þ ∂xk ∂xk E 38a ¼ Á for k ¼ 1, 2, ..., N; ∂ui ∂uj j ¼ Á j ¼ δ j ð : Þ ai ai a i E 38b

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 param- 1 2 eterized 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 ¼ 1, 2,...,N can be defined as a differentiable (NÀ1)-dimensional subspace with a codimension of 1. i The basis ai of the coordinate u can be rewritten as

∂r a ¼ i ∂ui ðE:39Þ  r, i for i ¼ 1, 2 352 Appendix E: Euclidean and Riemannian Manifolds

NP tangential surface T u2 r,2 = a2 nP Riemannian surface θ12 P r,1 = a1 r(u1, u2) (S)

0 u1

Fig. E.10 Tangent vectors to the curvilinear coordinates (u1,u2)

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). ÀÁ

ai Á aj ¼ jjai Á aj cos ai; aj ) ÀÁa Á a cos θ  cos a ; a ¼ i j ij i j jjÁ ð : Þ ai aj hi E 40 aij π ¼ pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi  1forθij 2 0, aðÞii Á aðÞjj 2

Note thatpffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi jjai ¼ ai Á ai ¼ aðÞii , no summation over (ii)

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

¼ Á 6¼ δ j aij ai aj i ð : Þ ∂xk ∂xk E 41 ¼ Á for k ¼ 1, 2, ..., N ∂ui ∂uj Appendix E: Euclidean and Riemannian Manifolds 353

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:  ÀÁ a11 a12 10 aij ¼ ¼ a21 a22 01

The angle between two base vectors results from Eq. (E.40):

aij cos θij ¼ pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi aðÞii Á aðÞjj a12 0 ) cos θ12 ¼ pffiffiffiffiffiffi pffiffiffiffiffiffi ¼ pffiffiffi pffiffiffi ¼ 0 a11 Á a22 1 Á 1

Thus,  π À1 a12 À1 θ12 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

∂r ∂r N ¼  ¼ r  r P ∂ui ∂uj , i , j ðE:42Þ  ai  aj for i, j ¼ 1, 2 ¼ α ak where α is the scalar factor; ak is the contravariant basis of the curvilinear coordinate of uk. Multiplying Eq. (E.42) by the covariant basis ak, the scalar factor α results in 354 Appendix E: Euclidean and Riemannian Manifolds ÀÁ ÀÁ α ak Á a ¼ αδk ¼ α ¼ a  a Á a kÀÁk ÂÃi j k ðE:43Þ ) α ¼ ai  aj Á ak  ai; aj; ak

The scalar factor α equals the scalar triple product that is given in [2]: ÀÁ ÀÁ α  ½Š¼a1; a2; a3 ai  aj Á ak ¼ ðÞÁak  ai aj ¼ aj  ak Á 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 q31ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÀÁ32 33 21 22 23 11 12 13 ¼ det aij  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.   ¼ ai aj ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiai aj ð : Þ nP ÀÁ E 45 ai  aj 2 aðÞii Á aðÞjj À aij

Note that 2 jjai ¼ ai Á ai ¼ aðÞii , no summation over (ii) The Lagrange identity results from the cross product of two vectors a and b.

jj¼a  b jjÁa jjb sin ðÞ)a; b jja  b 2 ¼ jja 2 Á jjb 2 sin 2ðÞa; b ÀÁ ¼ jja 2 Á jjb 2 Á 1 À cos 2ðÞa; b ¼ ðÞjja Á jjb 2 À ðÞjja Á jjb Á cos ðÞa; b 2 ¼ ðÞjja Á jjb 2 À ðÞa Á b 2

Thus, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jja  b ¼ jja 2 Á jjb 2 À ðÞa Á b 2 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 [2]. Appendix E: Euclidean and Riemannian Manifolds 355

u2 a2 a du2 2 du1 P(u1,u2) dS a1 du1 a1 r(u1,u2) du2

0 u1

Fig. E.11 Surface area in the curvilinear coordinates

ZZ ∂r ∂r S ¼ Â duiduj ∂ i ∂ j ZZ u u

i j ¼ ai  aj du du ZZ ð : Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ÀÁ E 47 2 2 2 i j ¼ jjai Á aj À ai Á aj du du ZZ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÀÁ 2 i j ¼ aðÞii Á aðÞjj À aij du du

Therefore, ZZ 1 2 S ¼ jja1 Â a2 du du for i ¼ 1; j ¼ 2 ZZ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðE:48Þ 2 1 2 ¼ a11 Á a22 À ðÞa12 du du

In Eq. (E.47), the vector length squared can be calculated as 2 jjai ¼ ai Á ai ¼ aðÞii , no summation over (ii)

E.3 Kronecker Delta

The Kronecker delta is very useful in tensor analysis and is defined as 356 Appendix E: Euclidean and Riemannian Manifolds  ∂uj 0fori 6¼ j δ j  ¼ ðE:49Þ i ∂ui 1fori 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 [2, 5]. We summarize a few properties of the Kronecker delta: Property 1 The chain rule of differentiation of the Kronecker delta using the contraction rule (cf. Appendix A)

∂uj ∂uj ∂uk δ j ¼ ¼ ¼ δ jδ k ðE:50Þ i ∂ui ∂uk ∂ui k i

Property 2 Kronecker delta in Einstein summation convention

δ i jk ¼ δ i 1k þÁÁÁþδ i ik þÁÁÁþδ i Nk j a 1a i a Na ¼ 0 þÁÁÁþ1 Á aik þÁÁÁþ0 ðE:51Þ ¼ aik

Property 3 Product of Kronecker deltas

δ iδ j ¼ δ i δ1 þÁÁÁþδ iδ i þÁÁÁþδ i δ N j k 1 k i k N k ¼ þÁÁÁþ Á δ i þÁÁÁþ ð : Þ 0 1 k 0 E 52 ¼ δ i k

Note that δðÞi  δ1 ¼ δ2 ¼ÁÁÁ¼δ N ¼ ðÞi 1 2 N 1 (no summation over the free index i); However, δ i  δ1 þ δ2 þÁÁÁþδ N ¼ i 1 2 N N (summation 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 [1–3]. The Levi-Civita permutation symbols can simply be defined as Appendix E: Euclidean and Riemannian Manifolds 357

j = 1,2,3

⎛ 0 1 0⎞ ⎜ ⎟ i = 1,2,3 ⎜−1 0 0⎟ ⎜ ⎟ ⎝ 0 0 0⎠ ε = ⎛0 0 −1⎞ ijk ⎜ ⎟ ⎜0 0 0 ⎟ ⎜ ⎟ ⎝1 0 0 ⎠ ⎛0 0 0⎞ ⎜ ⎟ ⎜0 0 1⎟ ⎜ ⎟ ⎝0 −1 0⎠

Fig. E.12 27 Levi-Civita permutation symbols

8 < þ1ifðÞi, j, k is an even permutation; ε ¼ À1ifðÞi, j, k is an odd permutation; ijk : ð : Þ 0ifi ¼ j, or i ¼ k; or j ¼ k E 53 , ε ¼ 1ðÞÁi À j ðÞÁj À k ðÞk À i for i, j, k ¼ 1, 2, 3 ijk 2

Here, we abstain from giving an exact definition of even and odd permutation because this would go beyond the scope of this book. The reader is referred to the literature [8, 9]. According to Eq. (E.53), the Levi-Civita permutation symbols can be expressed as  εijk ¼ εjki ¼ εkij ðÞeven permutation ; εijk ¼ ðE:54Þ Àεikj ¼Àεkji ¼ÀεjikðÞodd permutation

The 27 Levi-Civita permutation symbols for a three-dimension coordinate system are graphically displayed in Fig. E.12.

References

1. Klingbeil, E.: Tensorrechnung fur€ Ingenieure (in German). B.I.-Wissenschafts-verlag, Mann- heim (1966) 2. Nayak, P.K.: Textbook of Tensor Calculus and Differential Geometry. PHI Learning, New Delhi (2012) 3. Kay, D.C.: Tensor Calculus. Schaum’s Outline Series. McGraw-Hill, New York, NY (2011) 4. Griffiths, D.J.: Introduction to Quantum Mechanics, 2nd edn. Pearson Prentice Hall, Upper Saddle River, NJ (2005) 5. Oeijord, N.K.: The Very Basics of Tensors. IUniverse, New York, NY (2005) 6. Springer, C.E.: Tensor and Vector Analysis. Dover Publications, Mineola, NY (2012) 7. Lang, S.: Fundamentals of Differential Geometry. Springer, Berlin (1999) 358 Appendix E: Euclidean and Riemannian Manifolds

8. Lee, J.: Introduction to Smooth Manifolds. Springer, Berlin (2000) 9. Fecko, M.: Differential Geometry and Lie Groups for Physicists. Cambridge University Press, Cambridge (2011) 10. Riemann, B.: U¨ ber die Hypothesen, welche der Geometrie zu Grunde liegen. Springer Spektrum, Berlin (2013) Appendix F: Probability Function for the Quantum Interference

In case of the interference pattern, the probability function of the electrons for finding the electron at position z in a long observing time is the squared amplitude of the sum of the wave functions ψA(z) and ψB(z). The complex wave functions ψA(z) and ψB(z) are displayed in a complex plane, as shown in Fig. F.1.

Im

ψ B

ψ AB

θ ϕB ψ α A ϕ A 0 Re

Fig. F.1 The sum function ψ AB of two complex wave functions

© Springer-Verlag Berlin Heidelberg 2017 359 H. Nguyen-Scha¨fer, J.-P. Schmidt, Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers, Mathematical Engineering, DOI 10.1007/978-3-662-48497-5 360 Appendix F: Probability Function for the Quantum Interference

Using the law of cosines for a triangle, the probability function PAB(z) at position z is calculated as ðÞkkψ 2 ¼ kkψ þ ψ 2 PAB z AB A B ¼ kkψ 2 þ kkψ 2 À 2kkψ Á kkψ cos θ A B A B ðF:1Þ ¼ kkψ 2 þ kkψ 2 þ kkψ Á kkψ α A B 2 A B cos ¼ ðÞþ ðÞþ kkψ Á kkψ α PA z PB z 2 A B cos where PA(z) and PB(z) are the probability functions at position z of the wave functions ψA(z) and ψB(z). The wave functions are expressed in the complex form as

φ ψ ¼ kkÁψ ðÞ¼φ þ φ kkψ i A ; A A cos A i sin A A e φ ðF:2Þ ψ ¼ kkψ Á ðÞ¼φ þ φ kkψ i B B B cos B i sin B B e

The complex conjugate of the wave function of ψB results as ψ * ¼ kkψ Á ðÞcos φ À i sin φ B B B B ð : Þ À φ F 3 ¼ kkÁψ ½Š¼ðÞþÀφ ðÞÀφ kkψ i B B cos B i sin B B e

The product of two complex wave functions is calculated as

ðÞφ Àφ ψ ψ * ¼ kkψ Á kkψ ei A B A B A B ðF:4Þ ¼ kkψ Á kkψ Á ½ŠðÞþφ À φ ðÞφ À φ A B cos A B i sin A B

The difference angle between the phase angles of the wave functions results as φ À φ ¼ α ) φ À φ ¼Àα ð : Þ B A A B F 5

Substituting Eqs. (F.4) and (F.5), one calculates the real part of Eq. (F.4) ÀÁ ψ ψ * ¼ kkψ Á kkψ ðÞφ À φ Re A B A B cos A B ¼ kkψ Á kkψ cos ðÞÀα A B ðF:6Þ ¼ kkψ Á kkψ cos α ÀÁA B ¼ ψ * ψ Re A B

Substituting Eq. (F.6) into Eq. (F.1), one obtains the probability function PAB(z)at position z ÀÁ P ðÞ¼z kkψ 2 þ kkψ 2 þ 2Re ψ ψ * AB A B ÀÁA B ¼ P ðÞþz P ðÞþz 2Re ψ ψ * ðF:7Þ A B ÀÁA B ¼ ðÞþ ðÞþ ψ * ψ PA z PB z 2Re A B

The third term on the RHS of Eq. (F.7) represents the quantum interference in the probability function. Appendix G: Lorentz and Minkowski Transformations in Spacetime

The Lorentz and Minkowski transformations deal with the special relativity theory (SRT) in a four-dimensional spacetime. Let S be an inertial coordinate system; S0 be a moving coordinate system with a velocity v relative to S (s. Fig. G.1). In the case that the velocity v is parallel to the coordinate x, the Lorentz transformation between two coordinate systems is written as

0 x À vt x ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffi  γ Á ðÞx À vt 2 À v 1 2 0 c y ¼ y 0 ð : Þ z ¼ z G 1 vx t À  0 2 vx t ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffic  γ Á t À v2 c2 1 À c2

z z′

r = ct y y′ r′ = ct′

v < c x x′ S S′

Fig. G.1 Inertial and moving coordinate systems

© Springer-Verlag Berlin Heidelberg 2017 361 H. Nguyen-Scha¨fer, J.-P. Schmidt, Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers, Mathematical Engineering, DOI 10.1007/978-3-662-48497-5 362 Appendix G: Lorentz and Minkowski Transformations in Spacetime

The transformed coordinates of the moving system S0 are written in the transfor- mation matrix L as 2 3 2 3 0 12 3 0 γ Àγ x x 00 v x 6 0 7 6 7 B 0100C6 7 6 y 7 ¼ 6 y 7 ¼ B C6 y 7 ð : Þ 4 0 5 L4 5 @ 0010A4 5 G 2 z z γv z 0 À 00 γ t t c2 t where c is the constant light speed, t and t0 are the time coordinates in the systems S and S0, respectively; γ is called the Lorentz factor, which is defined as

1 γ  qffiffiffiffiffiffiffiffiffiffiffiffi  1 À v2 1 c2

The Lorentz backtransformation results from Eq. (G.2)as 2 3 2 3 0 12 3 0 γ γ 0 x x 00 v x 6 7 6 0 7 B 0100C6 0 7 6 y 7 ¼ À16 y 7 ¼ B C6 y 7 ð : Þ 4 5 L 4 0 5 @ 0010A4 0 5 G 3 z z γv z 0 00 γ 0 t t c2 t

Thus, the coordinates of the inertial system S are written as

0 0  x þ vt 0 0 x ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ γ Á x þ vt 2 À v 1 2 0 c y ¼ y ¼ 0 ðG:4Þ z z 0 0 vx  t þ 0 2 0 vx t ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffic ¼ γ Á t þ v2 c2 1 À c2

Note that the larger the moving speed v, the shorter the length of an object in the moving coordinate system S0 becomes. This effect is called the length contraction. rffiffiffiffiffiffiffiffiffiffiffiffiffi 2 0 l v l ¼ ¼ l Á 1 À  l ðG:5Þ γ c2

On the contrary, the larger the moving speed v, the longer the time interval between two events encounters compared to the standing coordinate system S. This effect is defined as the time dilation. This result indicates that the clock runs more slowly in the moving coordinate system. If the moving speed v reaches the light speed c, the Appendix G: Lorentz and Minkowski Transformations in Spacetime 363 clock would stop running since the time interval between two events will go to infinity (t0 !1) in the moving coordinate system.

0 t t ¼ γt ¼ qffiffiffiffiffiffiffiffiffiffiffiffi  t ðG:6Þ À v2 1 c2

However, if the velocity v is in an arbitrary direction in the coordinate system S, the propagating distances of a light signal transmitting with a constant light speed c at the times t and t0 result in the coordinate systems S and S0 as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ qxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 þ y2 þ z2 ¼ ct; 0 0 ðG:7Þ r ¼ x0 2 þ y0 2 þ z0 2 ¼ ct

Having squared both sides of Eq. (G.7), one obtains

x2 þ y2 þ z2 À c2t2 ¼ 0; 0 2 0 2 0 2 0 2 ðG:8Þ x þ y þ z À c2t ¼ 0:

The generalized Lorentz transformation between two arbitrary coordinate systems results from Eq. (G.8) for any proportional parameter σ as

0 2 0 2 0 2 0 2 ÀÁ x þ y þ z À c2t ¼ σ Á x2 þ y2 þ z2 À c2t2 ¼ 0 ðG:9Þ

Equation (G.9) is also valid for σ ¼ 1; therefore, the generalized Lorentz transfor- mation becomes

0 2 0 2 0 2 0 2 x þ y þ z À c2t ¼ x2 þ y2 þ z2 À c2t2 ðG:10Þ

The Minkowski transformation in the field-free spacetime with four independent dimensions of x1, x2, x3, and x4 is written as 0 1 2 3 0 12 3 x1 ¼ x x1 1000 x B C 6 7 B C6 7 B x2 ¼ y C 6 x2 7 B 0100C6 y 7 @ A , 4 5 ¼ @ A4 5 ðG:11Þ x3 ¼ pz ffiffiffiffiffiffiffi x3 0010 z x4 ¼ À1ct ¼ jct x4 000jc t

Therefore,

2 þ 2 þ 2 þ 2 ¼ 2 þ 2 þ 2 À 2 2 ð : Þ x1 x2 x3 x4 x y z c t G 12

Thus, the Minkowski transformation satisfies the generalized Lorentz transforma- tion of Eq. (G.10) using in the special relativity theory, cf. Fig. 5.11. Both trans- formations are used for spacetimes without gravitational field. 364 Appendix G: Lorentz and Minkowski Transformations in Spacetime

The general relativity theory (GRT) is the field theory of gravitation, in which the spacetime is curved with nonzero Riemannian curvature at any point on its surface. On the contrary, the SRT is only valid in a flat spacetime in which gravity does not exist. Hence, the GRT is a geometric theory of gravity that determines the structure of spacetime by means of curvatures at each point on the spacetime surface. The principle of equivalence indicates that the effect of an accelerating coordi- nate system has the same effect of gravity, in which the spacetime is affected by a gravitational field. As a result, the geometric structure of an accelerated spacetime must be curved. If a matter is under a gravitational or electromagnetic field in a spacetime, it is moved in the spacetime. In turn, the matter curves the spacetime by means of its gravitational or electromagnetic force acting upon the spacetime. Therefore, the four-dimensional Minkowski transformation is a pseudo-Euclidean flat spacetime is only obtained in the absence of gravity. In the GRT, a metric tensor gμv is necessary for the curved spacetime under a gravitational field, in which the geodesic distance ds (i.e. the shortest distance between two arbitrary points) is written in a four-dimensional spacetime as

2 ¼ ; 8μ 8 ¼ ð : Þ ds gμvdxμdxv , v 1, 2, 3, 4 G 13

Using the Schwarzschild metric tensor for a gravitational field, the geodesic in a curved spacetime is expressed in the rectangular coordinates as

ds2 ¼ g dx2 þ g dx2 þ g dx2 þ g dx2 11 1 22 2 33 3 44 4 ðG:14Þ ¼ 2 þ 2 þ 2 þ 2 g11dx g22dy g33dz g44dt

The Schwarzschild metric tensor in Eq. (G.14) is written as 0 1 g 000 B 11 C ¼ B 0 g22 00C ð : Þ gμv @ A G 15 00g33 0 000g44

The principle components of the Schwarzschild metric tensor are defined as

1 g ¼ g ¼ g  ; 11 22 33 2MG 1 À c2r ðG:16Þ 2MG g Àc2 1 À 44 c2r where G is the gravitational constant (G ¼ 6.673 Â 10À11 m3 kgÀ1 sÀ2); M is the mass of the object; r is the radial coordinate of the spherical spacetime. Appendix H: The Law of Large Numbers in Statistical Mechanics

Let X1, X2, ..., XN be N independent variables that have the probability densities ρ1(X1), ρ2(X2),..., ρN(XN), respectively. The probability densities satisfy the condition

þ1ð ρ ðÞ ¼ ¼ ... ð : Þ i Xi dXi 1 fori 1, 2, , N H 1 À1

The joint probability density of N independent variables gives the relation

ρðÞ; ...; ... ¼ ρ ðÞ...ρ ðÞ :::: ð : Þ X1 XN dX1 dXN 1 X1 N XN dX1 dXN H 2

Integrating the joint probability density over the entire space, one obtains using Eq. (H.1)

þ1ð þ1ð þ1ð þ1ð ... ρðÞX ; ...; X dX ...dX ¼ ρ ðÞX dX ... ρ ðÞX dX 1 N 1 N 1 1 1 N N N ðH:3Þ À1 À1 À1 À1 ¼ 1

Let ΔX1,...,ΔXN be the differences of the independent variables X1, ...,XN to their average value, respectively. They can be expressed as

ΔXi ¼ Xi À hiXi fori ¼ 1, 2, ..., N ðH:4Þ where the expectation (average value) of the variable Xi is defined as

© Springer-Verlag Berlin Heidelberg 2017 365 H. Nguyen-Scha¨fer, J.-P. Schmidt, Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers, Mathematical Engineering, DOI 10.1007/978-3-662-48497-5 366 Appendix H: The Law of Large Numbers in Statistical Mechanics

þ1ð hi ρ ðÞ ð : Þ Xi Xi i Xi dXi H 5 À1

Similarly, the variance of variable Xi is written as DEDE 2 2 VarðÞ Xi ðÞΔXi ¼ ðÞXi À hiXi þ1 ð ðH:6Þ ¼ ðÞÀ hi2ρ ðÞ Xi Xi i Xi dXi À1

The mean value Xm of the N independent variables is defined as

1 XN Xm ¼ Xj ðH:7Þ N j¼1

Using Eqs. (H.5) and (H.7), the expectation of the mean value Xm is calculated as

þ1ð

hiXm  XmρðÞX1; ...; XN dX1 ...dXN À1 þ1ð ¼ ρ ðÞ...ρ ðÞ ... ð : Þ Xm 1 X1 N XN dX1 dXN H 8 À1 þ1ð XN XN ¼ 1 ρ ðÞ  1 hi Xi i Xi dXi Xi N ¼ N ¼ i 1 À1 i 1

The variance of the mean value Xm is written as DEDE 2 2 VarðÞ Xm ðÞΔXm ¼ ðÞXm À hiXm þ1 ð ðH:9Þ 2 ¼ ðÞXm À hiXm ρðÞX1; ...; XN dX1 ...dXN À1

Substituting Eqs. (H.7) and (H.8) into Eq. (H.9), one obtains the variance of Xm Appendix H: The Law of Large Numbers in Statistical Mechanics 367

þ1ð DEXN ðÞΔ 2 ¼ 1 ðÞÀ hi2ρ ðÞ Xm 2 Xi Xi i Xi dXi N ¼ i 1 À1 ðH:10Þ XN DE ¼ 1 ðÞΔ 2  σ2 2 Xi m N i¼1

Thus, the standard deviation of the mean value Xm results as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u DEuXN DE 2 1 t 2 σm ¼ ðÞΔXm ¼ ðÞΔXi ðH:11Þ N i¼1

In the case of equal probabilities for all N independent variables, the standard deviation of the mean value Xm results as DE DEðÞΔ 2 N Xi VarðÞ X σ2 ¼ ðÞΔ 2 ¼  i m Xm pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N2 N ðH:12Þ VarðÞ Xi σ ) σm ¼ pffiffiffiffi  pffiffiffiffi N N where σ is the standard deviation of variable Xi. Equation (H.12)isthe law of large numbers in statistical mechanics. It denotes that the mean value Xm of N independent variables Xi for i ¼ 1, 2,...,N has a 1/2 standard deviation that is only 1/N of the standard deviation of variable Xi in case of equal probabilities for N independent variables. In statistics, there are two laws of large numbers, the weak and strong laws. The weak law states that the mean value of N independent variables converges in probability towards the expected value when the number of variables is very large.

XN 1 P lim Xm  lim Xi ¼ hiXm ðH:13Þ N!1 N!1 N i¼1

In this case, the probability density function for the weak law is written for any positive real number ε as

lim ρðÞ¼jjXm À hiXm > ε 0 ðH:14Þ N!1

The strong law denotes that the mean value of N independent variables always converges to the expected value when the number of variables is very large. 368 Appendix H: The Law of Large Numbers in Statistical Mechanics

1 XN lim Xm  lim Xi ¼ hiXm ðH:15Þ N!1 N!1 N i¼1

In this case, the probability density function for the strong law results as 

ρ lim Xm ¼ hiXm ¼ 1: ðH:16Þ N!1 Mathematical Symbols in This Book

• First partial derivative of a second-order tensor with respect to uk

∂Tij Tij  ð1Þ ,k ∂uk

Do not confuse Eq. (1) with the symbol used in some books:

∂Tij Tij  þ Γ i Tmj þ Γ j Tim ,k ∂uk km km

This symbol is equivalent to Eq. (2) used in this book. • Covariant derivative of a second-order tensor with respect to uk

ijj  ij þ Γ i mj þ Γ j im ð Þ T k T, k kmT kmT 2

• Second partial derivative of a first-order tensor with respect to uj and uk

∂2T T  i ð3Þ i,jk ∂uj∂uk

Do not confuse Eq. (3) with the symbol used in some books:

∂2T ∂T ∂T T  i À Γ m T À Γ m m À Γ m m i,jk ∂uj∂uk ik, j m ik ∂uj ij ∂uk ∂T þΓ mΓ n T À Γ m i þ Γ mΓ n T ij mk n jk ∂um jk im n

This symbol is equivalent to Eq. (4) used in this book. • Second covariant derivative of a first-order tensor with respect to uj and uk

© Springer-Verlag Berlin Heidelberg 2017 369 H. Nguyen-Scha¨fer, J.-P. Schmidt, Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers, Mathematical Engineering, DOI 10.1007/978-3-662-48497-5 370 Mathematical Symbols in This Book

 À Γ m À Γ m À Γ m Tikj Ti, jk ik, jTm ik Tm, j ij Tm, k ð Þ þΓ mΓ n À Γ m þ Γ mΓ n 4 ij mkTn jk Ti,m jk imTn

• Christoffel symbols of first kind Γijk instead of ½Šij, k used in some books • Christoffel symbols of second kind k Γk instead of used in some books. ij ij Further Reading

1. Aris, R.: Vectors, Tensors, and the Basic Equations of Fluid Mechanics. Dover Publications, New York, NY (1989) 2. Ba¨r, C.: Elementare Differentialgeometrie (in German), Zweiteth edn. De Gruyter, Berlin (2001) 3. Cahill, K.: Physical Mathematics. Cambridge University Press, Cambridge (2013) 4. Chen, N.: Aerothermodynamics of Turbomachinery, Analysis and Design. Wiley, Singapore (2010) 5. De, U.C., Shaikh, A., Sengupta, J.: Tensor Calculus, 2nd edn. Alpha Science, Oxford (2012) 6. Dirac, P.A.M.: The Principles of Quantum Mechanics, 4th edn. Oxford Science, Oxford (1958) 7. Griffiths, D.J.: Introduction to Quantum Mechanics, 2nd edn. Pearson Prentice Hall, Upper Saddle River, NJ (2005) 8. Grinfeld, P.: Introduction to Tensor Analysis and the Calculus of Moving Surfaces. Springer Science + Business Media, New York, NY (2013) 9. Itskov, M.: Tensor Algebra and Tensor Analysis for Engineers – With Applications to Continuum Mechanics, 2nd edn. Springer, Berlin (2010) 10. Kay, D.C.: Tensor Calculus. Schaum’s Outline Series. McGraw-Hill, New York, NY (2011) 11. Klingbeil, E.: Tensorrechnung fur€ Ingenieure (in German). B.I.-Wissenschafts-verlag, Mann- heim (1966) 12. Kuhnel,€ W.: Differentialgeometrie – Kurven, Fla¨chen, Mannigfaltigkeit (in German), 6th edn. Springer-Spektrum, Wiesbaden (2013) 13. Landau, D., Lifshitz, E.M.: The Classical Theory of Fields. Addison-Wesley, Reading, MA (1962) 14. Lang, S.: Fundamentals of Differential Geometry, 2nd edn. Springer, New York, NY (2001) 15. Lawden, D.F.: Introduction to Tensor Calculus, Relativity and Cosmology, 3rd edn. Dover Publications, Mineola, NY (2002) 16. Longair, M.: Quantum Concepts in Physics. Cambridge University Press, Cambridge (2013) 17. Lovelock, D., Rund, H.: Tensors, Differential Forms, and Variational Principles. Dover Publications, New York, NY (1989) 18. McConnell, A.J.: Applications of Tensor Analysis. Dover Publications, New York, NY (1960) 19. Nayak, P.K.: Textbook of Tensor Calculus and Differential Geometry. PHI Learning, New Delhi (2012) 20. Oeijord, N.K.: The Very Basics of Tensors. IUniverse, New York, NY (2005) 21. Penrose, R.: The Road to Reality. Alfred A. Knopf, New York, NY (2005)

© Springer-Verlag Berlin Heidelberg 2017 371 H. Nguyen-Scha¨fer, J.-P. Schmidt, Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers, Mathematical Engineering, DOI 10.1007/978-3-662-48497-5 372 Further Reading

22. Schobeiri, M.: Turbomachinery Flow Physics and Dynamic Performance, 2nd edn. Springer, Berlin (2012) 23. Shankar, R.: Principles of Quantum Mechanics, 2nd edn. Springer, Berlin (1980) 24. Simmonds, J.G.: A Brief on Tensor Analysis, 2nd edn. Springer, New York, NY (1982) 25. Springer, C.E.: Tensor and Vector Analysis. Dover Publications, Mineola, NY (2012) 26. Susskind, L., Lindesay, J.: An Introduction to Black Holes, Information and the String Theory Revolution. World Scientific Publishing, Singapore (2005) 27. Synge, J.L., Schild, A.: Tensor Calculus. Dover Publications, New York, NY (1978) 28. Fecko, M.: Differential Geometry and Lie Groups for Physicists. Cambridge University Press, Cambridge (2011) 29. Lee, J.: Introduction to Smooth Manifolds. Springer, Berlin (2000) 30. Helgason, S.: Differential Geometry, Lie Groups and Symmetric Spaces. AMS, New York, NY (1978) 31. Messiah, A.: Quantum Mechanics – Two Volumes. Dover Publications, New York, NY (2014) 32. Cohen-Tannoudji, C., Diu, B., Laloe, F.: Quantum Mechanics – Two Volumes. Wiley, New York, NY (1977) Index

A Compton wavelength, 291 Absolute probability, 259 Computational fluid dynamics (CFD), 204 Abstract index notation, 97, 239 Congruence, 126 Adjoint, 17 Constitutive equations, 213 Ambient coordinate, 140, 141 Continuity equation, 204 Ampere-Maxwell law, 226 Contravariant basis, 337 Angle between two vectors, 345 Contravariant metric tensor components, 69 Anti-symmetric, 21, 64 Coordinate velocity, 140, 141 Arc length, 340, 341, 350 Coriolis acceleration, 207 Area differential, 105 Cosmological constant, 240 Auto-parallel, 150 Cotangent bundle, 146 Cotangent space, 146 Covariant and contravariant bases, 42 B Covariant basis, 337 Basis tensors, 51 Covariant derivative, 90 Bianchi first identity, 92 Covariant Einstein tensor, 99 Bianchi second identity, 100 Covariant first-order tensor, 88 Black hole, 243 Covariant metric tensor components, 69 Block symmetry, 92 Covariant partial derivative, 88 Bra, 17 Covariant Riemann curvature tensor, 92 Bra and ket, 15 Cross product, 56 Curl (rotation), 188 Curl identity, 191 C Curvature vector, 108 Cartan’s formula, 136 Curved spacetime, 364 Cauchy’s strain tensor, 220, 222 Curved surfaces, 349 Cauchy’s stress tensor, 215 Cyclic property, 92 Cayley-Hamilton theorem, 219 Cylindrical coordinates, 5 CFD. See Computational fluid dynamics (CFD) Characteristic equation, 13, 32, 219 Christoffel symbol, 84 D Codazzi’s equation, 125 Density operator, 273 Commutator, 282 Derivative of the contravariant basis, 84 Complex conjugate, 257 Derivative of the covariant basis, 76 Compton effect, 289 Derivative of the product, 84

© Springer-Verlag Berlin Heidelberg 2017 373 H. Nguyen-Scha¨fer, J.-P. Schmidt, Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers, Mathematical Engineering, DOI 10.1007/978-3-662-48497-5 374 Index

Differential forms, 157 Frenet orthonormal frame, 120 Dirac delta function, 262 Friction stress contravariant tensor, 210 Dirac equation, 310 Divergence, 184, 186 Divergence theorem, 197 G Dual bases, 42 Gauss-Bonnet theorem, 120 Dual space, 146 Gauss-Codazzi equations, 124 Dual vector spaces, 36 Gauss derivative equations, 122 Gauss divergence theorem, 198 Gauss equation, 126 E Gaussian curvature, 113, 117 Eigenfrequency, 13 Gaussian surface parameters, 350 Eigenkets, 30 Gauss’s law for electric fields, 225 Eigenvalue, 11 Gauss’s law for magnetic fields, 225 Eigenvector, 13 Gauss’s Theorema Egregium, 114 Einstein field equations, 239 Gauss theorem, 197 Einstein-Maxwell equations, 240 Geodesic curvature, 109 Einstein tensor, 99 Geodesic equation, 150 Elasticity tensor, 223 Gradient, 182 Electric charge density, 226 Gradient of a contravariant vector, 183 Electric displacement, 224 Gradient of a covariant vector, 183 Electric field strength, 224, 226, 241 Gradient of an invariant, 182–183 Electrodynamics, 224 Gram-Schmidt scheme, 343, 344 Electromagnetic stress-energy tensor, 240 Gravitational constant, 240, 242, 364 Electromagnetic waves, 227 Green’s identities, 199 Elementary k-form, 158 Elliptic point, 111 Energy-momentum tensor, 240 H Energy or rothalpy equation, 210 Entanglement, 249 Heisenberg picture, 300 EPR paradox, 254 Hermitian matrix, 31 Euclidean N-space, 335 Hermitian operator, 29 Euclidean space, 348 Hessian tensor, 113, 116 Euler’s characteristic, 120 Hilbert space, 249 Exterior algebra, 155 Hodge dual, 172 Exterior derivative, 147, 163 Hodge * operator, 171 Exterior product, 158 Homeomorphic, 155 Hooke’s law, 223 Hyperbolic point, 111 F ’ Faraday s law, 225 I First fundamental form, 109 First-kind Christoffel symbol, 79 Identity matrix, 22 First-kind Ricci tensor, 97, 98 Interior derivative, 166 Flat space, 93 Interior product, 136, 166 Flat surface, 348 Intrinsic geometry, 351 Four-current density vector, 229 Intrinsic value, 56 Four-dimensional manifold, 227 Invariant time derivative, 141 Index 375

J Nonlocality, 250 Jacobian, 5, 41 N-order tensor, 35 Jacobi identity, 129 Normal curvature, 109 Normal vector, 353

K Ket orthonormal bases, 18 O Ket transformation, 28 Observables, 270 Ket vector, 16 Orthonormal coordinate, 338 Killing vector, 139 Orthonormalization, 18, 343 Kinetic energy-momentum tensor, 240 Outer product, 22 Klein-Gordon equation, 306 Koszul formula, 152 Kronecker delta, 342, 355 P Parabolic point, 111 Parameterized curve, 103 L Parameterized curves, 352 Lagrange identity, 354 Partial derivatives of the Christoffel symbols, Laplacian of a contravariant vector, 191 92 Laplacian of an invariant, 190 Photon sphere, 245 Levi-Civita connection, 148 Physical tensor component, 74 Levi-Civita permutation, 356 Physical vector components, 203 Levi-Civita permutation symbol, 57 Poincare´ group, 227 Lie derivative, 131 Poincare´ Lemma, 164 Lie dragged, 130 Poincare´ transformation, 227 Light speed, 226 Polar materials, 216 Linear adjoint operator, 21 Positive definite, 21 Lorentz transformation, 227 Principal curvature planes, 113 Principal normal curvatures, 113 Principal strains, 222 M Principal stresses, 218 Magnetic field density, 224 Principle of equivalence, 364 Magnetic field strength, 224 Probability density, 259, 260 Many-worlds interpretation (MWI), 297 Projection operator, 22 Maxwell’s equations, 224 Propagating speed, 227 Mean curvature, 113 Pullback, 168 Metric tensor, 151, 342 Pushforward, 171 Minkowski space, 227 Mixed components, 55 Mixed metric tensor components, 70 Q Moving surface, 140 Qubit, 251 Multi-fold-N-dimensional tensor space, 65 Qubit state, 251 Multilinear functional, 2, 36 MWI. See Many-worlds interpretation (MWI) R Reduced Compton wavelength, 291 N Relative probability, 257 Nabla operator, 181 Ricci curvature, 99 Navier-Stokes equations, 204 Ricci’s lemma, 95 Newton’s constant, 242, 364 Ricci tensor, 97, 239 376 Index

Riemann-Christoffel tensor, 92, 93, 97 Surface curvature tensor, 124–126 Riemann curvature, 117, 118 Surface triangulation, 120 Riemann curvature tensor, 92 Riemannian manifold, 347 Riemann Levi-Civita (RLC) connection, 148 T Tangent bundle, 127, 146 Tangential coordinate velocity, 140, 141 S Tangential surface, 352 Schrodinger€ picture, 300 Tangent space, 127, 146 Schrodinger€ ’s cat, 256 Tangent surface, 127 Schwarzschild metric, 241 Tensor, 35 Schwarzschild space, 242 Tensor product, 51 Schwarzschild’s solution, 241 Time evolution, 298 Second covariant derivative, 90 Transformed ket basis, 25 Second fundamental form, 110 Transpose conjugate, 17 Second-kind Christoffel symbol, 77, 78, 122 Second-kind Ricci tensor, 98, 331 Second-order tensor, 51 U Second partial derivative, 91 Uncertainty principle, 251 Shear modulus, 224 Unit normal vector, 107, 354 Shift tensor, 40 Unit tangent vector, 107 Skew-symmetric, 21 Spacetime, 227 Spacetime dimensions, 224 V Spacetime equations, 224 Volume differential, 202 Spherical coordinates, 8 Star operator, 171 Stokes theorem, 199 W Stress and strain tensors, 212 Wave-particle duality, 286 Surface area, 354 Wedge product, 158 Surface coordinates, 140 Weingarten’s equations, 123