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Toroidal coordinates: Difference between revisions
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→Contravariant From
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It is worthwhile to note that the Clebsch form of <math> \mathbf{B} </math> corresponds to a [[:Wikipedia: Magnetic potential|magnetic vector potential]] | It is worthwhile to note that the Clebsch form of <math> \mathbf{B} </math> corresponds to a [[:Wikipedia: Magnetic potential|magnetic vector potential]] | ||
<math> \mathbf{A} = \nu\nabla\psi </math> (or <math> \mathbf{A} = \psi\nabla\nu </math> as they differ only by the Gauge transformation <math> \mathbf{A} \to \mathbf{A} - \nabla (\psi\nu)</math>). | <math> \mathbf{A} = \nu\nabla\psi </math> (or <math> \mathbf{A} = \psi\nabla\nu </math> as they differ only by the Gauge transformation <math> \mathbf{A} \to \mathbf{A} - \nabla (\psi\nu)</math>). | ||
The general form of the stream function is | |||
:<math> | |||
\nu(\psi,\theta,\phi) | |||
= \frac{1}{2\pi}(\dot{\Psi}_{tor}\theta | |||
- \dot{\Psi}_{pol}\phi) | |||
+ \tilde{\nu}(\psi,\theta,\phi) | |||
</math> | |||
where <math>\tilde{\nu}</math> is a differentiable function periodic in the two angles. This general form can be derived by using the fact that <math> \mathbf{B}</math> is a physical function (and hence singe-valued). The specific form for the coefficients in front of the secular terms (i.e. the non-periodic terms) can be obtained from the FSA properties [[Useful properties of the FSA]]. | |||
== Magnetic == | == Magnetic == | ||