Magneto-Fluid Dynamics Fundamentals and Case Studies of Natural Phen
This book concerns the generation of electric currents and of electric space charges inside conducting media that move in magnetic fields. The authors postulate nothing but the Maxwell equations. They discuss at length the disk dynamo, which serves as a m
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w (2) (3)
VxA =
-f-f.^-%y-%^ V
dA^ dy
dAx dx
dAy dy
dAy^^^^^^A^_^A^.^ dz dz
(4) (5)
A =
dA, dz
+
dAy dx
dx
dA^ dy
^^f^^'f^^'f^^'f + dy^ + dz^ dx^
V^A = (V^^x) X + (VMj,) y + (V^yl,) z = V ( V • A) - V x (V x A)
Cylindrical coordinates (p, 0, z)
(7) V.A = 1A(,^,) + 1 ^ + pdp
(8)
VxA:
la^ p d
1 aA^ [sin0 Si^
dAr de
1
a
^2 Qr [^ dr ) ^ r^ sin 0 90 (15)
a^^
V^A - V ( V • A) - V X (V X A)
d{rA^) 0 dr J
Identities *Starred equations are valid only in Cartesian coordinates. (1) (2)
(A X B ) • C = A • ( B X C)
Ax{BxC)
= B{A • C) - C{A • B)
(3) '^{f9) = fV9 + g'^f (4) V(a/6) = (1/6) Va - {a/b^) Vb (5) *
V(A •B) = {B- V)A + {A • V)B + B x (V x A) + A x (V x B) (6) V . ( / A ) = ( V / ) - A + / ( V - A ) (7)
V • {Ax B) = B • {V X A) - A- {V X B)
(8) ( V . V ) / = V V (9) V x ( V / ) = 0 (10) V • (V X A) = 0 (11) V X (/A) = (V/) X A + / ( V X A) (12) *
Vx{AxB)
= {B- V)A - (A • V ) B + (V • B)A - (V • A)B
(13) V X (V X A) = V(V • A) - V^A The above equation defines the Laplacian of a vector in other than Cartesian coordinates. (14) ( A . V ) B = ( ^ ^^, ^ + + A ,^ ^ + AA ,—^ ) x dx ^ dy '^ dz + ( A . ^ +A , ^ + A.^)y dy ^ dz dB^ ,. dB, dB^ . dB, (, dB, dz
Theorems The divergence theorem:
AdA= f AdA= f' JA
V-Adv
Jv
where A is the area of the closed surface that bounds the volume v. Stokes's theorem: i Adl=
f {V
xA)-dA,
where A is the open surface bounded by the closed cvirve C.
ASTRONOMY AND ASTliOPH YS iCS [J BR A R Y LIBRARY Series Editors:
G. Bomer, Garching, Germany A. Burkert, Munchen, Germany W. B. Burton, Charlottesville, VA, USA and Leiden, The Netherlands M. A. Dopita, Canberra, Australia A. Eckart, Koln, Germany T. Encrenaz, Meudon, France B. Leibundgut, Garching, Germany J. Lequeux, Paris, France A. Maeder, Sauvemy, Switzerland V. Trimble, College Park, MD, and Irvine, CA, USA
ASTR()N()M'\' ,\N!) ASll^'^CiPlIYSICS LIBH-Jr^ G. Bomer • A. Burkert • W. B. Burton • M. A. Dopita A. Eckart • T. Encrenaz • B. Leibundgut • J. Lequeux A. Maeder • V. Trimble The Stars By E. L. Schatzman and R Praderie Modem Astrometry 2nd Edition By J. Kovalevsky The Physics and Dynamics of Planetary Nebulae By G. A. Gurzadyan Galaxies and Cosmology By F. Combes, P. Boisse, A. Mazure and A. Blanchard Observational Astrophysics 2nd Edition By P. Lena, F. Lebrun and F. Mignard Physics of Planetary Rings Celestial Mechanics of Continuous Media By A. M. Fridman and N. N. Gorkavyi Tools of Radio Astronomy 4th Edition, Corr. 2nd printing By K. Rohlfs and T. L. Wilson Tools of Radio Astronomy Problems and Solutions 1st Edition, Corr. 2nd printing By T. L. Wilson and S. Hiittemeister Astrophysical Formulae 3rd Edition (2 volumes) Volume I: Radiation, Gas Processes and High Energy Astrophysics Volume II: Space, Time, Matter and Cosmology By K. R. Lang Galaxy Formation By M. S. Longair Astrophysical Concepts 4th Edition By M. Harwit forthcoming Astrometry of Fundamental Catalogues The Evolution from Optical to Radio Reference Frames By H. G. Walter and O. J. Sovers Compact Stars. Nuclear Physics, Parti
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