BIBLIOGRAPHY. Englewood Cliffs, N.J.:Prentice-Hall, 1969.

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1 35 BIBLIOGRAPHY Abramowitz, M. and Stegun, I.A., Handbook of Mathematical Functions, 10th ed, New York:Dover, 197. Akivis, M.A., Goldberg, V.V., An Introduction to Linear Algebra and Tensors, New York:Dover, 197. Aris, Rutherford, Vectors, Tensors, and the Basic Equations of Fluid Mechanics, Englewood Cliffs, N.J.:Prentice-Hall, 196. Atkin, R.J., Fox, N., An Introduction to the Theory of Elasticity, London:Longman Group Limited, Bishop, R.L., Goldberg, S.I.,Tensor Analysis on Manifolds, New York:Dover, Borisenko, A.I., Tarapov, I.E., Vector and Tensor Analysis with Applications, New York:Dover, Chorlton, F., Vector and Tensor Methods, Chichester,England:Ellis Horwood Ltd, Dodson, C.T.J., Poston, T., Tensor Geometry, London:Pittman Publishing Co., Eisenhart, L.P., Riemannian Geometry, Princeton, N.J.:Univ. Princeton Press, Eringen, A.C., Mechanics of Continua, Huntington, N.Y.:Robert E. Krieger, D.J. Griffiths, Introduction to Electrodynamics, Prentice Hall, Flügge, W., Tensor Analysis and Continuum Mechanics, New York:Springer-Verlag, 197. Fung, Y.C., A First Course in Continuum Mechanics, Englewood Cliffs,N.J.:Prentice-Hall, Goodbody, A.M., Cartesian Tensors, Chichester, England:Ellis Horwood Ltd, 198. Hay, G.E., Vector and Tensor Analysis, New York:Dover, Hughes, W.F., Gaylord, E.W., Basic Equations of Engineering Science, New York:McGraw-Hill, Jeffreys, H., Cartesian Tensors, Cambridge, England:Cambridge Univ. Press, Lass, H., Vector and Tensor Analysis, New York:McGraw-Hill, Levi-Civita, T., The Absolute Differential Calculus, London:Blackie and Son Limited, Lovelock, D., Rund, H.,Tensors, Differential Forms, and Variational Principles, New York:Dover, Malvern, L.E., Introduction to the Mechanics of a Continuous Media, Englewood Cliffs, N.J.:Prentice-Hall, McConnell, A.J., Application of Tensor Analysis, New York:Dover, Newell, H.E., Vector Analysis, New York:McGraw Hill, Schouten, J.A., Tensor Analysis for Physicists,New York:Dover, Scipio, L.A., Principles of Continua with Applications, New York:John Wiley and Sons, Sokolnikoff, I.S., Tensor Analysis, New York:John Wiley and Sons, Spiegel, M.R., Vector Analysis, New York:Schaum Outline Series, Synge, J.L., Schild, A., Tensor Calculus, Toronto:Univ. Toronto Press, Bibliography

2 353 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 Prefix Multiplication factor Symbol tera 10 1 T giga 10 9 G mega 10 6 M kilo 10 3 K hecto 10 h deka 10 da deci 10 d centi 10 c milli 10 3 m micro 10 6 µ nano 10 9 n pico 10 p Basic Units. Basic units of measurement Unit Name Symbol Length meter m Mass kilogram kg Time second s Electric current ampere A Temperature degree Kelvin K Luminous intensity candela cd Supplementary units Unit Name Symbol Plane angle radian rad Solid angle steradian sr

3 354 DERIVED UNITS Name Units Symbol Area square meter m Volume cubic meter m 3 Frequency hertz Hz (s Density kilogram per cubic meter kg/m 3 Velocity meter per second m/s Angular velocity radian per second rad/s Acceleration meter per second squared m/s Angular acceleration radian per second squared rad/s Force newton N (kg m/s Pressure newton per square meter N/m Kinematic viscosity square meter per second m /s Dynamic viscosity newton second per square meter N s/m Work, energy, quantity of heat joule J (N m Power watt W (J/s Electric charge coulomb C (A s Voltage, Potential difference volt V (W/A Electromotive force volt V (W/A Electric force field volt per meter V/m Electric resistance ohm Ω (V/A Electric capacitance farad F (A s/v Magnetic flux weber Wb (V s Inductance henry H (V s/a Magnetic flux density tesla T (Wb/m Magnetic field strength ampere per meter A/m Magnetomotive force ampere A Physical constants. lim n 4 arctan 1 π ( 1+ n 1 n e Euler s constant γ γ lim ( n log n n speed of light in vacuum.99795(10 8 ms electron charge (10 9 C Avogadro s constant 6.05(10 3 mol Plank s constant 6.656(10 34 Js Universal gas constant JK mol JKg K Boltzmann constant (10 3 JK Stefan Boltzmann constant (10 8 Wm K 4 Gravitational constant 6.67(10 1 Nm kg

4 355 APPENDIX B CHRISTOFFEL SYMBOLS OF SECOND KIND 1. Cylindrical coordinates (r, θ, z (x 1,x,x 3 x r cos θ y r sin θ z z r 0 0 θ π <z< h 1 1 h r h 3 1 x + y r, y/x tanθ z Constant Cylinders Planes Planes. 1 r r. Spherical coordinates (ρ, θ, φ (x 1,x,x 3 x ρ sin θ cos φ y ρ sin θ sin φ z ρ cos θ ρ 0 0 θ π 0 φ π h 1 1 h ρ h 3 ρ sin θ x + y + z ρ x + y tan θz y x tan φ Spheres Planes. Cones 1 ρ 1 ρ sin θ sin θ cos θ ρ ρ 3 3 cotθ 3 3

5 Parabolic cylindrical coordinates (,η,z (x 1,x,x 3 x η y 1 ( η z z << <z< η 0 h 1 + η h + η h 3 1 x (y Parabolic cylinders x η (y + η Parabolic cylinders z Constant Planes. 1 + η η + η η + η η η + η + η 4. Parabolic coordinates (,η,φ (x 1,x,x 3 x η cos φ y η sin φ z 1 ( η 0 η 0 0 <φ<π h 1 + η h + η h 3 η x + y (z Paraboloids x + y η (z + η Paraboloids y x tan φ Planes. 1 + η η + η 1 + η η + η η + η 1 η + η 1 1 η η η η

6 Elliptic cylindrical coordinates (,η,z (x 1,x,x 3 x cosh cos η y sinh sin η z z 0 0 η π <z< h 1 sinh +sin η h sinh +sin η h 3 1 x cosh + y sinh 1 x cos η y sin η 1 1 sinh cosh sinh +sin η 1 sinh cosh sinh +sin η 1 1 sin η cos η 1 1 sinh +sin η 6. Elliptic coordinates (,η,φ (x 1,x,x 3 x (1 η ( 1 cos φ y (1 η ( 1 sin φ z η z Constant 1 < 1 η 1 0 φ<π Elliptic cylinders Hyperbolic cylinders Planes. sin η cos η sinh +sin η sin η cos η sinh +sin η sinh cosh 1 1 sinh +sin η h 1 η 1 h η 1 η h 3 (1 η ( 1 x 1 + y 1 + z 1 Prolate ellipsoid z η x 1 η y 1 1 η Two-sheeted hyperboloid y x tan φ Planes η η 1 η η η 1 ( + (1 η ( η 1 ( + ( 1 η η η ( 1 η (+ ( η ( + η ( 1 η η 1 η 1 η 1 η η 3 1 η

7 Bipolar coordinates (u, v, z (x 1,x,x 3 a x cosh v cos u, 0 u<π a y cosh v cos u, <v< z z <z< h 1 h h a (cosh v cos u h (x a coth v + y a sinh v x +(y a cot u a sin u z Constant cos u +coshv Cylinders Cylinders Planes. cos u +coshv 8. Conical coordinates (u, v, w (x 1,x,x 3 x uvw ab, b >v >a >w, u 0 h 1 1 y u (v a (w a h a a b z v (v b (w b b b a x v + x w + x + y + z u y v a + z v b 0, y w a + z w b 0, u (v w (v a (b v h 3 u (v w (w a (w b Spheres Cones Cones. v b v v a + v + v v w 3 w v w w a + w w b + w 1 u ( v w (b v ( a + v 1 u ( v w ( a + w ( b + w v ( b v ( a + v (v w ( a + w ( b + w 3 w ( a + w ( b + w (b v ( a + v (v w 1 1 u w 3 v w u 3 v 3 v w

8 Prolate spheroidal coordinates (u, v, φ (x 1,x,x 3 x a sinh u sin v cos φ, u 0 y a sinh u sin v sin φ, 0 v π h 1 h h a (sinh u +sin v z a cosh u cos v, 0 φ<π h 3 a sinh u sin v x (a sinh u + y a sinh u + z a cosh u 1, Prolate ellipsoids x (a cos v y (a sin v z 1, (a cos v Two-sheeted hyperpoloid y x tan φ, Planes. 1 cosh u sinh u sin v + sinh cos v sin vsinh u u sin v + sinh u cos v sin v 1 cos v sin v sin v + sinh u 1 sin v + sinh u 1 cosh u sinh u cosh u sinh u sin v + sinh u 1 sin v + sinh u 1 sin v cosh u sinh u sin v + sinh u cos v sin v sin v + sinh u cosh u sinh u cos v sin v 10. Oblate spheroidal coordinates (,η,φ (x 1,x,x 3 x a cosh cos η cos φ, 0 h 1 h y a cosh cos η sin φ, π η π h a (sinh +sin η z a sinh sin η, 0 φ π h 3 a cosh cos η x (a cosh + y (a cosh + z 1, (a sinh Oblate ellipsoids x (a cos η + y (a cos η z 1, (a sin η One-sheet hyperboloids y x tan φ, Planes. 1 cosh sinh sin η + sinh cos η sin ηcosh sin η + sinh cos η sin η 1 cos η sin η sin η + sinh 1 sin η + sinh 1 cosh sinh cosh sinh sin η + sinh 1 sin η + sinh 1 cos η cosh sinh 3 sin η + sinh sinh 31 cosh cos η sin η 3 sin η + sinh sin η 3 cos η

9 360. Toroidal coordinates (u, v, φ (x 1,x,x 3 a cos φ x cosh v cos u, a sin φ y cosh v cos u, a z cosh v cos u, 0 u<π <v< 0 φ<π h 1 h h a (cosh v cos u h 3 a sinh v (cosh v cos u ( x + y + z a cos u a sin u, ( x + y a cosh v + z a sinh v, y x tan φ, Spheres Tores planes cos u +coshv cos u +coshv cos u +coshv (cos u cosh v 1 cos u cosh v 1 cos u cosh v

10 Confocal ellipsoidal coordinates (u, v, w (x 1,x,x 3 x (a u(a v(a w (a b (a c, u < c <b <a y (b u(b v(b w (b a (b c, c <v<b <a z (c u(c v(c w (c a (c b, c <b <v<a h (u v(u w 1 4(a u(b u(c u h (v u(v w 4(a v(b v(c v h (w u(w v 3 4(a w(b w(c w 1 1 (a u + 1 (b u + 1 (c u + 1 (u v + 1 (u w 1 (a v + 1 (b v + 1 (c v + 1 ( u + v + 1 (v w 3 1 (a w + 1 (b w + 1 (c w + 1 ( u + w + 1 ( v + w ( 1 a u ( b u ( c u (v w 1 (a v(b v(c v(u v(u w ( 1 a u ( b u ( c u 1 ( v + w 1 (u v(a w(b w(c w(u w ( a v ( b v ( c v 13 (u w (a u(b u(c u( u + v(v w ( a v ( b v ( c v 1 ( u + w ( u + v(a w(b w(c w(v w 3 (u v ( a w ( b w ( c w 3 3 (a u(b u(c u( u + w( v + w 3 ( u + v ( a w ( b w ( c w 31 3 (a v(b v(c v( u + w( v + w 3 (u v (u w ( u + v (v w ( u + w ( v + w

11 36 APPENDIX C VECTOR IDENTITIES The following identities assume that A, B, C, D are differentiable vector functions of position while f,f 1,f are differentiable scalar functions of position. 1. A ( B C B ( C A C ( A B. A ( B C B( A C C( A B 3. ( A B ( C D( A C( B D ( A D( B C 4. A ( B C+ B ( C A+ C ( A B 0 5. ( A B ( C D B( A C D A( B C D C( A B C D( A B C 6. ( A B ( B C ( C A( A B C 7. (f 1 + f f 1 + f 8. ( A + B A + B 9. ( A + B A + B 10. (fa( f A + f A. (f 1 f f 1 f + f f 1 1. (fa f A + f( A 13. ( A B B ( A A ( B ( 14. ( A A A A ( A 15. ( A B ( B A +( A B + B ( A+ A ( B 16. ( A B( B A B( A ( A B + A( B 17. ( f f 18. ( f ( A0 0. ( A ( A A