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Flux coordinates: Difference between revisions
→Magnetic coordinates
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</math> | </math> | ||
where <math>G</math> is periodic in the angles, preserves the straightness of the field lines. The spatial function <math>G(\psi, \theta_f, \phi_f)</math>, is called the ''generating function''. It can be obtained from a [[magnetic differential equation]] if we know the Jacobians of the two | where <math>G</math> is periodic in the angles, preserves the straightness of the field lines. The spatial function <math>G(\psi, \theta_f, \phi_f)</math>, is called the ''generating function''. It can be obtained from a [[magnetic differential equation]] if we know the Jacobians of the two magnetic coordinate systems <math> \sqrt{g_f}</math> and <math> \sqrt{g_F}</math>. In fact taking <math>\mathbf{B}\cdot\nabla</math> on any of the transformation of the angles and using the known expressions for the contravariant components of <math>\mathbf{B}</math> in flux coordinates we get | ||
:<math> | :<math> | ||
2\pi\mathbf{B}\cdot\nabla G = \frac{1}{\sqrt{g_F}} - \frac{1}{\sqrt{g_f}}~. | 2\pi\mathbf{B}\cdot\nabla G = \frac{1}{\sqrt{g_F}} - \frac{1}{\sqrt{g_f}}~. | ||