Toroidal coordinates: Difference between revisions

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<ref>Morse and Feshbach, ''Methods of theoretical physics'', McGraw-Hill, New York, 1953 ISBN 007043316X</ref>
<ref>Morse and Feshbach, ''Methods of theoretical physics'', McGraw-Hill, New York, 1953 ISBN 007043316X</ref>
<ref>[[:Wikipedia:Toroidal coordinates]]</ref>
<ref>[[:Wikipedia:Toroidal coordinates]]</ref>
<ref>F. Alladio, F. Chrisanti, ''Analysis of MHD equilibria by toroidal multipolar expansions'', Nucl. Fusion '''26''' (1986) 1143</ref>
<ref>[http://dx.doi.org/10.1016/0010-4655(94)90112-0 B.Ph. van Milligen and A. Lopez Fraguas, ''Expansion of vacuum magnetic fields in toroidal harmonics'', Computer Physics Communications '''81''', Issues 1-2 (1994) 74-90]</ref>


:<math>R = R_p \frac{\sinh \zeta}{\cosh \zeta - \cos \eta}</math>
:<math>R = R_p \frac{\sinh \zeta}{\cosh \zeta - \cos \eta}</math>
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The coordinate ''&eta;'' is a poloidal angle and runs from 0 to 2&pi;.  
The coordinate ''&eta;'' is a poloidal angle and runs from 0 to 2&pi;.  
This system is orthogonal.
This system is orthogonal.
The Laplace equation separates in this system of coordinates, thus allowing an expansion of the vacuum magnetic field in toroidal harmonics.
<ref>F. Alladio, F. Chrisanti, ''Analysis of MHD equilibria by toroidal multipolar expansions'', Nucl. Fusion '''26''' (1986) 1143</ref>
<ref>[http://dx.doi.org/10.1016/0010-4655(94)90112-0 B.Ph. van Milligen and A. Lopez Fraguas, ''Expansion of vacuum magnetic fields in toroidal harmonics'', Computer Physics Communications '''81''', Issues 1-2 (1994) 74-90]</ref>


== Magnetic ==
== Magnetic ==

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