Heinbockel - Tensor Calculus - Part

352 BIBLIOGRAPHY • Abramowitz, M. and Stegun, I.A., Handbook of Mathematical Functions, 10th ed, New York:Dover, 1972. • Akivis, M.A., Goldberg, V.V., An Introduction to Linear Algebra and Tensors, New York:Dover, 1972. • Aris, Rutherford, Vectors, Tensors, and the Basic Equations of Fluid Mechanics, Englewood Cliffs, N.J.:Prentice-Hall, 1962. • Atkin, R.J., Fox, N., An Introduction to the Theory of Elasticity, London:Longman Group Limited, 1980. • Bishop, R.L., Goldberg, S.I.,Tensor Analysis on Manifolds, New York:Dover, 1968. • Borisenko, A.I., Tarapov, I.E., Vector and Tensor Analysis with Applications, New York:Dover, 1968. • Chorlton, F., Vector and Tensor Methods, Chichester,England:Ellis Horwood Ltd, 1976. • Dodson, C.T.J., Poston, T., Tensor Geometry, London:Pittman Publishing Co., 1979. • Eisenhart, L.P., Riemannian Geometry, Princeton, N.J.:Univ. Princeton Press, 1960. • Eringen, A.C., Mechanics of Continua, Huntington, N.Y.:Robert E. Krieger, 1980. • D.J. Griffiths, Introduction to Electrodynamics, Prentice Hall, 1981. • Fl¨ ugge, W., Tensor Analysis and Continuum Mechanics, New York:Springer-Verlag, 1972. • Fung, Y.C., A First Course in Continuum Mechanics, Englewood Cliffs,N.J.:Prentice-Hall, 1969. • Goodbody, A.M., Cartesian Tensors, Chichester, England:Ellis Horwood Ltd, 1982. • Hay, G.E., Vector and Tensor Analysis, New York:Dover, 1953. • Hughes, W.F., Gaylord, E.W., Basic Equations of Engineering Science, New York:McGraw-Hill, 1964. • Jeffreys, H., Cartesian Tensors, Cambridge, England:Cambridge Univ. Press, 1974. • Lass, H., Vector and Tensor Analysis, New York:McGraw-Hill, 1950. • Levi-Civita, T., The Absolute Differential Calculus, London:Blackie and Son Limited, 1954. • Lovelock, D., Rund, H. ,Tensors, Differential Forms, and Variational Principles, New York:Dover, 1989. • Malvern, L.E., Introduction to the Mechanics of a Continuous Media, Englewood Cliffs, N.J.:Prentice-Hall, 1969. • McConnell, A.J., Application of Tensor Analysis, New York:Dover, 1947. • Newell, H.E., Vector Analysis, New York:McGraw Hill, 1955. • Schouten, J.A., Tensor Analysis for Physicists,New York:Dover, 1989. • Scipio, L.A., Principles of Continua with Applications, New York:John Wiley and Sons, 1967. • Sokolnikoff, I.S., Tensor Analysis, New York:John Wiley and Sons, 1958. • Spiegel, M.R., Vector Analysis, New York:Schaum Outline Series, 1959. • Synge, J.L., Schild, A., Tensor Calculus, Toronto:Univ. Toronto Press, 1956.

Bibliography

353 APPENDIX A UNITS OF MEASUREMENT The following units, abbreviations and prefixes are from the Syst`eme International d’Unit`es

(designated SI in all Languages.)

Prefixes. Abreviations Multiplication factor 1012 109 106 103 102 10 10−1 10−2 10−3 10−6 10−9 10−12

Symbol T G M K h da d c m µ n p

Basic units of measurement Name Length meter Mass kilogram Time second Electric current ampere Temperature degree Kelvin Luminous intensity candela

Symbol m kg s A ◦ K cd

Prefix tera giga mega kilo hecto deka deci centi milli micro nano pico

Basic Units.

Unit

Unit Plane angle Solid angle

Supplementary units Name radian steradian

Symbol rad sr

354 Name Area Volume Frequency Density Velocity Angular velocity Acceleration Angular acceleration Force Pressure Kinematic viscosity Dynamic viscosity Work, energy, quantity of heat Power Electric charge Voltage, Potential difference Electromotive force Electric force field Electric resistance Electric capacitance Magnetic flux Inductance Magnetic flux density Magnetic field strength Magnetomotive force

DERIVED UNITS Units square meter cubic meter hertz kilogram per cubic meter meter per second radian per second meter per second squared radian per second squared newton newton per square meter square meter per second newton second per square meter joule watt coulomb volt volt volt per meter ohm farad weber henry tesla ampere per meter ampere

Symbol m2 m3 −1 Hz (s ) kg/m3 m/s rad/s m/s2 rad/s2 N (kg · m/s2 ) N/m2 m2 /s N · s/m2 J (N · m) W (J/s) C (A · s) V (W/A) V (W/A) V/m Ω (V/A) F (A · s/V) Wb (V · s) H (V · s/A) T (Wb/m2 ) A/m A

Physical constants. 4 arctan 1 = π = 3.14159 26535 89793 23846 2643 . . . n  1 = e = 2.71828 18284 59045 23536 0287 . . . lim 1 + n→∞ n Euler’s constant γ = 0.57721 56649 01532 86060 6512 . . .   1 1 1 γ = lim 1 + + + · · · + − log n n→∞ 2 3 n speed of light in vacuum = 2.997925(10)8 m s−1 electron charge = 1.60210(10)−19 C Avogadro’s constant = 6.02252(10)23 mol−1 Plank’s constant = 6.6256(10)−34 J s Universal gas constant = 8.3143 J K −1 mol−1 = 8314.3 J Kg −1 K −1 Boltzmann constant = 1.38054(10)−23 J K −1 Stefan–Boltzmann constant = 5.6697(10)−8 W m−2 K −4 Gravitational constant = 6.67(10)−11 N m2 kg −2

355 APPENDIX B CHRISTOFFEL SYMBOLS OF SECOND KIND 1. Cylindrical coordinates (r, θ, z) = (x1 , x2 , x3 ) x = r cos θ

r≥0

h1 = 1

y = r sin θ

0 ≤ θ ≤ 2π

h2 = r

z=z

−∞
h3 = 1

The coordinate curves are formed by the intersection of the coordinate surfaces x2 + y 2 = r2 ,

Cylinders

y/x = tan θ

Planes

z = Constant 

1 22



 = −r

2 12

Planes. 

 =

2 21

 =

1 r

2. Spherical coordinates (ρ, θ, φ) = (x1 , x2 , x3 ) x = ρ sin θ cos φ

ρ≥0

h1 = 1

y = ρ sin θ sin φ

0≤θ≤π

h2 = ρ

z = ρ cos θ

0 ≤ φ ≤ 2π

h3 = ρ sin θ

The coordinate curves are formed by the intersection of the coordinate surfaces x2 + y 2 + z 2 = ρ2

Spheres

x2 + y 2 = tan2 θ z y = x tan φ 

 1 = −ρ 22   1 = −ρ sin2 θ 33   2 = − sin θ cos θ 33



Cones Planes.

   2 2 1 = = 12 21 ρ     3 3 1 = = ρ 13 31     3 3 = = cot θ 32 23

356 3. Parabolic cylindrical coordinates (ξ, η, z) = (x1 , x2 , x3 ) x = ξη 1 y = (ξ 2 − η 2 ) 2 z=z

−∞
p ξ 2 + η2 p h2 = ξ 2 + η 2

η≥0

h3 = 1

−∞<ξ <∞

h1 =

The coordinate curves are formed by the intersection of the coordinate surfaces x2 = −2ξ 2 (y −

ξ2 ) 2

Parabolic cylinders

η2 ) 2 z = Constant

x2 = 2η 2 (y +

Parabolic cylinders Planes. 

 1 −ξ = 2 ξ + η2 22     1 1 η = = 2 ξ + η2 12 21     2 2 ξ = = 2 ξ + η2 21 12



 1 ξ = 2 ξ + η2 11   2 η = 2 ξ + η2 22   2 −η = 2 ξ + η2 11

4. Parabolic coordinates (ξ, η, φ) = (x1 , x2 , x3 ) x = ξη cos φ

ξ≥0

y = ξη sin φ 1 z = (ξ 2 − η 2 ) 2

η≥0

p ξ 2 + η2 p h2 = ξ 2 + η 2

0 < φ < 2π

h3 = ξη

h1 =

The coordinate curves are formed by the intersection of the coordinate surfaces x2 + y 2 = −2ξ 2 (z − x2 + y 2 = 2η 2 (z + y = x tan φ 

 1 = 11   2 = 22   1 = 22   2 = 11   2 = 33

ξ 2 ξ + η2 η 2 ξ + η2 −ξ 2 ξ + η2 −η 2 ξ + η2 −ηξ 2 ξ 2 + η2

ξ2 ) 2

η2 ) 2

Paraboloids Paraboloids Planes. 

 1 = 33     1 1 = = 21 21     2 2 = = 21 12     3 3 = = 32 23     3 3 = = 13 31

−ξη 2 ξ 2 + η2 η 2 ξ + η2 ξ 2 ξ + η2 1 η 1 ξ

357 5. Elliptic cylindrical coordinates (ξ, η, z) = (x1 , x2 , x3 ) x = cosh ξ cos η

ξ≥0

y = sinh ξ sin η

0 ≤ η ≤ 2π

q sinh2 ξ + sin2 η q h2 = sinh2 ξ + sin2 η

z=z

−∞
h3 = 1

h1 =

The coordinate curves are formed by the intersection of the coordinate surfaces y2 x2 + =1 cosh2 ξ sinh2 ξ y2 x2 − =1 cos2 η sin2 η

Elliptic cylinders Hyperbolic cylinders

z = Constant 

 1 = 11   1 = 22     1 1 = = 12 21

Planes. 

 2 sin η cos η = 22 sinh2 ξ + sin2 η   2 − sin η cos η = 11 sinh2 ξ + sin2 η     2 2 sinh ξ cosh ξ = = 12 21 sinh2 ξ + sin2 η

sinh ξ cosh ξ sinh2 ξ + sin2 η − sinh ξ cosh ξ sinh2 ξ + sin2 η sin η cos η sinh2 ξ + sin2 η

6. Elliptic coordinates (ξ, η, φ) = (x1 , x2 , x3 ) s p

(1 − η 2 )(ξ 2 − 1) cos φ p y = (1 − η 2 )(ξ 2 − 1) sin φ

1≤ξ<∞

z = ξη

0 ≤ φ < 2π

x=

h1 = s

−1≤η ≤1

h2 = h3 =

ξ 2 − η2 ξ2 − 1 ξ 2 − η2 1 − η2

p (1 − η 2 )(ξ 2 − 1)

The coordinate curves are formed by the intersection of the coordinate surfaces y2 z2 x2 + + =1 ξ2 − 1 ξ2 − 1 ξ2 x2 y2 z2 − − =1 η2 1 − η2 1 − η2 y = x tan φ 

 ξ 1 ξ + 2 =− 2 −1 + ξ ξ − η2 11   2 η η − 2 = 2 1−η ξ − η2 22
  ξ −1 + ξ 2 1 =− (1 − η 2 ) (ξ 2 − η 2 ) 22

  ξ −1 + ξ 2 1 − η 2 1 =− ξ 2 − η2 33
  η 1 − η2 2 = (−1 + ξ 2 ) (ξ 2 − η 2 ) 11

Prolate ellipsoid Two-sheeted hyperboloid Planes

 −1 + ξ 2 η 1 − η 2 2 = 33 ξ 2 − η2   1 η =− 2 ξ − η2 12   2 ξ = 2 ξ − η2 21   3 ξ = −1 + ξ 2 31   3 η =− 1 − η2 32 

358 7. Bipolar coordinates (u, v, z) = (x1 , x2 , x3 ) a sinh v , 0 ≤ u < 2π cosh v − cos u a sin u , −∞ < v < ∞ y= cosh v − cos u z=z −∞
h21 = h22

x=

h22 =

a2 (cosh v − cos u)2

h23 = 1

The coordinate curves are formed by the intersection of the coordinate surfaces a2 sinh2 v a2 x2 + (y − a cot u)2 = sin2 u z = Constant

(x − a coth v)2 + y 2 =

Cylinders Cylinders Planes.



 2 sinh v = 11 − cos u + cosh v   1 sinh v = cos u − cosh v 12   2 sin u = cos u − cosh v 21



 1 sin u = cos u − cosh v 11   2 sinh v = cos u − cosh v 22   1 sin u = − cos u + cosh v 22

8. Conical coordinates (u, v, w) = (x1 , x2 , x3 ) uvw , b 2 > v 2 > a2 > w 2 , ab r u (v 2 − a2 )(w2 − a2 ) y= a a2 − b 2 r v (v 2 − b2 )(w2 − b2 ) z= b b 2 − a2

x=

u≥0

h21 = 1 u2 (v 2 − w2 ) − a2 )(b2 − v 2 ) u2 (v 2 − w2 ) h23 = 2 (w − a2 )(w2 − b2 ) h22 =

(v 2

The coordinate curves are formed by the intersection of the coordinate surfaces x2 + y 2 + z 2 = u2 2

2

Spheres

2

y z x + 2 + 2 = 0, v2 v − a2 v − b2 y2 z2 x2 + 2 + 2 = 0, 2 2 w w −a w − b2 

 v v 2 v − + 2 = 2 b − v2 −a2 + v 2 v − w2 22   3 w w w − − =− 2 v − w2 −a2 + w2 −b2 + w2 33
  u v 2 − w2 1 =− 2 (b − v 2 ) (−a2 + v 2 ) 22
  u v 2 − w2 1 =− (−a2 + w2 ) (−b2 + w2 ) 33

  v b2 − v 2 −a2 + v 2 2 =− 2 (v − w2 ) (−a2 + w2 ) (−b2 + w2 ) 33

Cones Cones.

 w −a2 + w2 −b2 + w2 3 = 2 22 (b − v 2 ) (−a2 + v 2 ) (v 2 − w2 )   2 1 = u 21   2 w =− 2 v − w2 23   3 1 = u 31   3 v = 2 v − w2 32 

359 9. Prolate spheroidal coordinates (u, v, φ) = (x1 , x2 , x3 ) x = a sinh u sin v cos φ,

u≥0

h21 = h22

y = a sinh u sin v sin φ,

0≤v≤π

h22 = a2 (sinh2 u + sin2 v)

z = a cosh u cos v,

h23 = a2 sinh2 u sin2 v

0 ≤ φ < 2π

The coordinate curves are formed by the intersection of the coordinate surfaces y2 z2 x2 + + = 1, 2 2 (a sinh u) a sinh u) a cosh u)2 y2 z2 x2 − − = 1, 2 2 (a cos v) (a sin v) (a cos v)2

Prolate ellipsoids Two-sheeted hyperpoloid

y = x tan φ, 

 1 cosh u sinh u = 11 sin2 v + sinh2 u   2 cos v sin v = 22 sin2 v + sinh2 u   1 cosh u sinh u =− 2 22 sin v + sinh2 u   1 sin2 v cosh u sinh u =− 33 sin2 v + sinh2 u   2 cos v sin v =− 2 11 sin v + sinh2 u

Planes.



 2 cos v sin vsinh2 u =− 33 sin2 v + sinh2 u   1 cos v sin v = 12 sin2 v + sinh2 u   2 cosh u sinh u = 21 sin2 v + sinh2 u   3 cosh u = sinh u 31   3 cos v = sin v 32

10. Oblate spheroidal coordinates (ξ, η, φ) = (x1 , x2 , x3 ) x = a cosh ξ cos η cos φ, y = a cosh ξ cos η sin φ, z = a sinh ξ sin η,

ξ≥0 π π − ≤η≤ 2 2 0 ≤ φ ≤ 2π

h21 = h22 h22 = a2 (sinh2 ξ + sin2 η) h23 = a2 cosh2 ξ cos2 η

The coordinate curves are formed by the intersection of the coordinate surfaces y2 z2 x2 + + = 1, (a cosh ξ)2 (a cosh ξ)2 (a sinh ξ)2 y2 z2 x2 + − = 1, (a cos η)2 (a cos η)2 (a sin η)2 y = x tan φ, 

 1 cosh ξ sinh ξ = 11 sin2 η + sinh2 ξ   2 cos η sin η = 22 sin2 η + sinh2 ξ   1 cosh ξ sinh ξ =− 2 22 sin η + sinh2 ξ   1 cos2 η cosh ξ sinh ξ =− 33 sin2 η + sinh2 ξ   2 cos η sin η =− 2 11 sin η + sinh2 ξ



Oblate ellipsoids One-sheet hyperboloids Planes.

 2 cos η sin ηcosh2 ξ = 33 sin2 η + sinh2 ξ   1 cos η sin η = 12 sin2 η + sinh2 ξ   2 cosh ξ sinh ξ = 21 sin2 η + sinh2 ξ   3 sinh ξ = cosh ξ 31   3 sin η =− cos η 32

360 11. Toroidal coordinates (u, v, φ) = (x1 , x2 , x3 ) a sinh v cos φ , cosh v − cos u a sinh v sin φ , y= cosh v − cos u a sin u , z= cosh v − cos u

x=

0 ≤ u < 2π −∞ < v < ∞ 0 ≤ φ < 2π

h21 = h22 h22 =

a2 (cosh v − cos u)2

h23 =

a2 sinh2 v (cosh v − cos u)2

The coordinate curves are formed by the intersection of the coordinate surfaces  a cos u 2 a2 , = x2 + y 2 + z − sin u sin2 u 2 p cosh v a2 , x2 + y 2 − a + z2 = sinh v sinh2 v y = x tan φ, 

 1 = 11   2 = 22   1 = 22   1 = 33   2 = 11

sin u cos u − cosh v sinh v cos u − cosh v sin u − cos u + cosh v sin usinh v 2 − cos u + cosh v sinh v − cos u + cosh v



Spheres Tores planes

 2 sinh v (cos u cosh v − 1) =− 33 cos u − cosh v   1 sinh v = cos u − cosh v 12   2 sin u = cos u − cosh v 21   3 sin u = cos u − cosh v 31   3 cos u cosh v − 1 = cos u sinh v − cosh v sinh v 32

361 12. Confocal ellipsoidal coordinates (u, v, w) = (x1 , x2 , x3 ) (a2 − u)(a2 − v)(a2 − w) , (a2 − b2 )(a2 − c2 ) (b2 − u)(b2 − v)(b2 − w) , y2 = (b2 − a2 )(b2 − c2 ) (c2 − u)(c2 − v)(c2 − w) , z2 = (c2 − a2 )(c2 − b2 )

x2 =

u < c2 < b 2 < a 2 c2 < v < b 2 < a 2 c2 < b 2 < v < a 2

(u − v)(u − w) 4(a2 − u)(b2 − u)(c2 − u) (v − u)(v − w) h22 = 4(a2 − v)(b2 − v)(c2 − v) (w − u)(w − v) h23 = 4(a2 − w)(b2 − w)(c2 − w) h21 =



 1 1 1 1 1 1 + + + + = 2 2 2 2 (a − u) 2 (b − u) 2 (c − u) 2 (u − v) 2 (u − w) 11   1 1 1 1 2 1 + + + + = 2 2 2 2 (a − v) 2 (b − v) 2 (c − v) 2 (−u + v) 2 (v − w) 22   1 1 1 1 3 1 + + + + = 2 2 2 2 (a − w) 2 (b − w) 2 (c − w) 2 (−u + w) 2 (−v + w) 33


    a2 − u b2 − u c2 − u (v − w) 1 1 −1 = = 2 (a2 − v) (b2 − v) (c2 − v) (u − v) (u − w) 22 12 2 (u − v)


    a2 − u b2 − u c2 − u (−v + w) 1 1 −1 = = 2 (u − v) (a2 − w) (b2 − w) (c2 − w) (u − w) 33 2 (u − w) 13


    a2 − v b2 − v c2 − v (u − w) 2 2 −1 = = 2 2 2 2 (a − u) (b − u) (c − u) (−u + v) (v − w) 2 (−u + v) 11 21
2
2
    2 2 −1 − v b − v c − v (−u + w) a 2 = = 2 2 2 2 (v − w) 23 2 (−u + v) (a − w) (b − w) (c − w) (v − w) 33  
2
2
  2 −1 3 (u − v) a − w b − w c − w 3 = = 2 (−u + w) 31 2 (a2 − u) (b2 − u) (c2 − u) (−u + w) (−v + w) 11  
2
2
  2 3 −1 (−u + v) a − w b − w c − w 3 = = 2 (−v + w) 32 2 (a2 − v) (b2 − v) (c2 − v) (−u + w) (−v + w) 22

362 APPENDIX C VECTOR IDENTITIES ~ B, ~ C, ~ D ~ are differentiable vector functions of position while The following identities assume that A, f, f1 , f2 are differentiable scalar functions of position.

1.

~ · (B ~ × C) ~ =B ~ · (C ~ × A) ~ =C ~ · (A ~ × B) ~ A

2.

~ × (B ~ × C) ~ = B( ~ A ~ · C) ~ − C( ~ A ~ · B) ~ A

3.

~ × B) ~ · (C ~ × D) ~ = (A ~ · C)( ~ B ~ · D) ~ − (A ~ · D)( ~ B ~ · C) ~ (A

4.

~ × (B ~ × C) ~ +B ~ × (C ~ × A) ~ +C ~ × (A ~ × B) ~ = ~0 A

5.

~ × B) ~ × (C ~ × D) ~ = B( ~ A ~·C ~ × D) ~ − A( ~ B ~ ·C ~ × D) ~ (A ~ A ~·B ~ × C) ~ − D( ~ A ~·B ~ × C) ~ = C(

6.

~ × B) ~ · (B ~ × C) ~ × (C ~ × A) ~ = (A ~·B ~ × C) ~ 2 (A

7.

∇(f1 + f2 ) = ∇f1 + ∇f2

8.

~ + B) ~ = ∇·A ~ +∇·B ~ ∇ · (A

9.

~ + B) ~ =∇×A ~+∇×B ~ ∇ × (A

10.

~ = (∇f ) · A ~ + f∇ · A ~ ∇(f A)

11.

∇(f1 f2 ) = f1 ∇f2 + f2 ∇f1

12.

~ =)∇f ) × A ~ + f (∇ × A) ~ ∇ × (f A)

13. 14.

~ × B) ~ =B ~ · (∇ × A) ~ −A ~ · (∇ × B) ~ ∇ · (A ! ~2 ~ × (∇ × A) ~ ~ · ∇)A ~ = ∇ |A| −A (A 2

15.

~ · B) ~ = (B ~ · ∇)A ~ + (A ~ · ∇)B ~ +B ~ × (∇ × A) ~ +A ~ × (∇ × B) ~ ∇(A

16.

~ × B) ~ = (B ~ · ∇)A ~ − B(∇ ~ ~ − (A ~ · ∇)B ~ + A(∇ ~ · B) ~ ∇ × (A · A)

17.

∇ · (∇f ) = ∇2 f

18.

∇ × (∇f ) = ~0

19.

~ =0 ∇ · (∇ × A)

20.

~ ~ = ∇(∇ · A) ~ − ∇2 A ∇ × (∇ × A)