Unofficial Errata for Introduction to Tensor Analysis and the Calculus of Moving Surfaces (hardcover), Pavel Grinfeld
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1 Unofficial Errata for Introduction to Tensor Analysis and the Calculus of Moving Surfaces (hardcover), Pavel Grinfeld Alex J. Yuffa June 5, 2017 Other typos found by George Katanics: PDF Part I: Chapters Page 19, line 3: to R (α) and... should read to R(α) and Page 42, last line: (added on 10/21/15) g(x) 1 = ln x... should read g(x) = 1 ln x Page 57, equation below (5.9): U W =... should read U V = Page 58, Exercise 71, should read the contravariant metric tensor Z ij [Submitted by Daniel Volinski, danielvolinski@yahoo.es] (added on 05/02/17) 5. Page 64, last line in Fig. 5.3 caption: Z 1 = i and Z 1. and Z 2 = j and Z 2 should read Z 1 = i and Z 2. and Z 2 = j and Z 1 [Submitted by Paul McCartney, pauljmccartney@gmail.com] (added on 09/24/15) 6. Page 66, Exercise 76: in polar coordinates should read in cylindrical coordinates [Submitted by Paul McCartney, pauljmccartney@gmail.com] (added on 09/24/15) 7. Page 69, 1st line: (added on 04/17/15) δ k j. Since δ k j... should read δ k i. Since δ k i Page 73, 3rd and 4th line in Sec. 5.15: the volume element Z. I think you meant to write: the length of a curve L = b a Z ij dz i dt dz j dt dt. 1
2 Because the volume element has not yet been introduced, see Sec Page 89, line below equation (6.62): is a object... should be is an object Page 89, equation (6.64) should read S i = S j Z ji 11. Page 90, equation (6.65) should read S i = S j Z ji 12. Page 90, equation (6.69) should read T i j = T i j 13. Page 98, equation (7.32) should read (added on 03/24/17) (u, v) = M ij u i v j 14. Page 102, Exercise 112 should read r = x A 1 b [Submitted by Ihor Yalovetskyi, i.yalovecky@gmail.com] (added on 05/02/17) 15. Page 114, (8.52) should read (added on 04/14/14) i Z j = Z j Z i Γk ijz k 16. Page 114, (8.53) should read (added on 04/14/14) Z j Z i = Γk ijz k 17. Page 118, 5th line after equation (8.66): in Sec should be in Sec Page 121, equation (8.79) should read T i = S ij U kl j V kl 2
3 19. Page 123, 2nd line: we will is it... should read we will use it Page 125, equation (8.106) should read Γ k i j T k = ( Γ k ijj i i J j j J k k + J k i j k Jk ) T l J l k = Γk ijj i i J j j T k + J k i j T k [Submitted by Andrew Szymczak, drew.szymczak@gmail.com] (added 06/05/17) 21. Page 125, equation (8.107) should read j T i = T i Z j J i i J j j + J i i j T i Γ k ijj i i J j j T k J k i j T k [Submitted by Andrew Szymczak, drew.szymczak@gmail.com] (added 06/05/17) 22. Page 127, equation (8.124) should read Γ i k m m Tj = Γi kmtj m Ji i Jk k J j j +... [Submitted by Andrew Szymczak, drew.szymczak@gmail.com] (added 06/05/17) 23. Page 127, equation (8.125) should read Γ m j i k Tm = ( ) T sj i i i Jm s [Submitted by Andrew Szymczak, drew.szymczak@gmail.com] (added 06/05/17) 24. Page 127, equation (8.126) should read Γ m j i k Tm = Γm jktmj i j j J k i k Ji +... [Submitted by Andrew Szymczak, drew.szymczak@gmail.com] (added 06/05/17) 25. Page 129, line above (8.132) should read (added 5/2/17) Riemann Christoffel tensor vanishes 26. Page 129, 2nd line after (8.132) should read (added on 04/17/15) the Riemann-Christoffel symbol... Thank you George Katanics 27. Page 132, last paragraph, line 2: properties of the covariant derivative... should read properties of the derivative Page 137, missing the word indices in the italicized definition, i.e., the definition should read when the upper and lower indices are Page 139, line between (9.31) and (9.32) should read (added on 04/17/15) δ ijk rsta r l a s ma t n = Aδ ijk lmn 3
4 30. Page 141, equation (9.48) should read D 1 1 = 1 2! ( ) δ 123 rst a r 1a s 2a t 3 + δrst 132 a r 1a s 3a t Page 144, equation (9.59) should read Z = r 2 sin θ 32. Page 146, 2 lines above equation (9.72): is a called relative tensor... should read is called a relative tensor Page 147, 5th line below equation (9.78): invariant or weight 2. should read invariant of weight Page 151, 5th line after (9.97) should read (added on 04/17/15) constructing the cross product... instead of constructing the gradient Page 157, (9.139) should read (added on 04/17/15) Part II: Chapters (n (k 1))δ i 1 i k 1 j 1 j k 1 = δ i 1 i k 1 i k j 1 j k 1 i k 1. Page 162, equation (10.2c) should read z(θ, φ) = Ccos θ [Submitted by James Pedid, jamespedid@gmail.com] (added on 05/02/17) 2. Page 164, equation (10.11a) should be labeled as (10.11) 3. Page 165, 2nd line below (10.17): the ambient and the surface covariant bases... should read the ambient and the surface covariant metric tensors Page 166, (10.22) should read Z iα = Z i βs αβ 5. Page 168, (10.34) should read Z i S α = Zi Z i Z i S α S α S α 6. Page 169, 3rd line below (10.46): P or either 0 or 1 should read P are either 0 or 1 7. Page 170, Exercise 218: Denote the tensor N i N j by T i j should read Denote the tensor Z i αz α j by T i j 4
5 8. Suggestion: On page 173, move the word Further, and (10.73) below equation (10.77). Because (10.73) is not needed in the derivation of (10.74) but is needed in the derivation of (10.78). 9. Page 174, 2nd line in Section 10.8: with respect the... should read with respect to the Page 175, (10.82) should read Γ α βγ = 1 2 Sαω ( Sωβ S γ + S ωγ S β S ) βγ S ω 11. Page 175, (10.85) should read Γ α βγ = Z α i Z i β S γ + Γi jkz α i Z j β Zk γ 12. Page 177, 2nd line from the bottom: that is. deformations should read that is, deformations 13. Page 180, (10.109) should read z(θ, φ) = r sin φ 14. Page 181, (10.113) should read Γ Φ ΘΘ = (R + r cos φ) sin φ r 15. Page 181, second part of (10.118) should read N i = cos θ 1+r (z) 2 sin θ 1+r (z) 2 r (z) 1+r (z) Page 182, (10.122) should read Γ Z ΘΘ = r(z)r (z) 1 + r (z) 2 5
6 17. Page 183, in (10.131) delete extra comma after N i 18. Page 183, 1st equation in (10.132) should read [ ] 1 Zα i = y ; (x) 19. Page 183, 1st equation in (10.134) should read S = 1 + y (x) Page 183, 2nd line from the bottom should read (added on 11/03/14) to polar coordinates (r, θ)... instead of to Cartesian coordinates (r, θ) Page 189, (11.14) should read S δ α S β = S δ S β S α Γδ αβ 22. Page 190, Example 237 should read Exercise Page 190, (11.21) should read γ T = T S = (T i Z i ) γ S γ = T i S γ Z i + T i Z i S γ 24. Page 190, (11.22) should read γ T = T i S Z γ i + T i Z i Z j Z j S γ 25. Page 190, (11.23) should read γ T = T i S γ Z i + T i Z j γγ k ijz k 26. Page 190, (11.24) should read ( ) T i γ T = S + γ Zk γ Γ i kmt m Z i 27. Page 191, (11.25) should read ( ) T γ T i i Z i = S + γ Zk γ Γ i kmt m Z i 6
7 28. Page 191, (11.26) should read γ T i = T i S γ + Zk γ Γ i kmt m 29. Page 191, (11.27) should read γ T i j = T i j S γ + Zk γ Γ i kmt m j Z k γ Γ m kjt i m 30. Page 191, (11.28) should read γ T iα jβ = T iα jβ S γ + Zk γ Γ i kmt mα jβ Zγ k Γ m kjtmβ iα + Γ α γωtjβ iω Γ ω γβtjω iα 31. Page 192, (11.32) should read γ T i j = T i j S γ + Zk γ Γ i kmt m j Z k γ Γ m kjt i m 32. Suggestion: Move Exercise 253 on page 202 to the end of Section Page 203, (12.22) should read ( α β T γ δ β α T γ δ ) Sδ = R γ ωαβ T ω R δ ωαβs ω T γ δ 34. Page 203, (12.23) should read ( α β T γ δ β α T γ δ ) Sδ = R γ ωαβ T ω δ S δ R ω δαβt γ ω S δ 35. On page 207, (12.50) is only correct if one chooses S 1 = φ and S 2 = r (with the normal away from the z-axis). If one chooses S 1 = r and S 2 = φ, then the 1, 1 element in all of the matrices in (12.50) becomes the 2, 2 element. 36. Page 208, (12.59) should read B α α = r (z)r(z) r (z) 2 1 r(z)(1 + r (z) 2 ) 3/2 7
8 37. Page 212, delete period on the line right above (12.85). 38. Page 212, (12.85) should read C αβ = B ω ωb αβ KS αβ 39. Page 187, line between (11.5) and (11.6): The Laplacian... should read The surface Laplacian Page 192, it would be better to say The object Z k / S γ is... Z k γ... on the line below (11.33). 41. Page 199, 2nd paragraph, 3rd line: in the perspective in the Riemann Christoffel... should read in the perspective is the Riemann Christoffel Page 217, 1st line: Z i and that the embedded should read Z i and the embedded Page 217, (13.15) should read s = b a U du Page 221, line above (13.34): a product its length κ... should read a product of its length κ Page 221, (13.40) should read B α α = sin t (1 + cos 2 t) 3/ Page 222, 1st line in the Fig caption: The first two plots... should read The top two plots... Also, add a period at the end of the caption. 47. Page 224, (13.53) should read dq ds = κp P + T ( κt + τq). 48. Page 228, line between (10.3) and (13.87): (added on 10/21/15) the embedded surface... should read the embedded curve Page 229, line below (13.18): Christoffel symbol Γ α βγ symbol Γ α βγ should read Christoffel 8
9 50. Page 229, (13.93) should read Γ Ω ΦΨ = Γ α βγs Ω α S β Φ Sγ Ψ + Sα Φ U Ψ SΩ α. 51. Page 229, line between (13.100) and (13.101): (added on 10/21/15) We next the curvature... should read We define the curvature Page 231, line below (13.113): By equation (10.31)... should read By equation (10.55) Page 237, 3rd sentence in the caption of Fig. 14.1: The firgure includes... should read The figure includes... Also, add a period at the end of the caption. 54. Page 240, (14.16) should read (added on 10/22/15) B3 B2 B1 A 3 A 2 A 1 F (Z) dz 1 dz 2 dz 3 = B3 B2 B1 A 3 A 2 A 1 F (Z ) J dz 1 dz 2 dz Page 241, (14.17) should read (added on 10/22/15) = B3 B2 B1 A 3 A 2 A 1 F (Z(Z )) Z J dz 1 dz 2 dz 3 Note that by Z we really mean Z = Z = Z (Z ), i.e., the determinate of Z ij is expressed in the prime coordinates. For example, when switching from polar coordinates to Cartesian coordinates we must write Z as Z = x 2 + y 2 and not as Z = r, then Z J = x 2 + y 2 1 x. 2 +y Page 241, line between (14.17) and (14.18): (added on 10/21/15) Sect. 9.9 that... invariant of weight 1 should read Sect. 9.9 (Exercise 194) that... invariant of weight Page 241, (14.18) should read (added on 10/22/15) Z = J Z 58. Page 242, line between (14.24) and (14.25): (updated on 10/22/15) Sincev should read Since 59. Page 244, (14.43) should read (Note the capital Z instead of z.) T i = ε ijk N j n α Z k α 60. Page 245, (14.52) should read F i T i du = m F i ε ijk N j δm k ds m F i ε ijk N j N k N m ds. U S S 9
10 Bibliography [23 ] F. Harley should read H. Flanders [Submitted by Don Benson, dbenson9@shaw.ca] (added on 05/03/17) Index 1. Shift tensor (added on 01/09/17) 10
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