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.
• Flügge, 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
Système International d’Unitès
(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
−∞<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
−∞<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
−∞<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
−∞<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)