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Toroidal coordinates: Difference between revisions
→Covariant Form
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= \oint(\beta d\psi + d\chi) | = \oint(\beta d\psi + d\chi) | ||
</math> | </math> | ||
and choosing an integration circuit contained within a flux surface. | and choosing an integration circuit contained within a flux surface <math>(d\psi = 0)</math>. Then we get | ||
:<math> | :<math> | ||
\int_S \mu_0\mathbf{j}\cdot d\mathbf{S} | \int_S \mu_0\mathbf{j}\cdot d\mathbf{S} | ||
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I_{pol}^d = \int_S \mu_0\mathbf{j}\cdot d\mathbf{S}\; ; ~\mathrm{with}~ \partial S ~\mathrm{such~that}~ (\Delta\theta = 0, \Delta\phi = 2\pi)~. | I_{pol}^d = \int_S \mu_0\mathbf{j}\cdot d\mathbf{S}\; ; ~\mathrm{with}~ \partial S ~\mathrm{such~that}~ (\Delta\theta = 0, \Delta\phi = 2\pi)~. | ||
</math> | </math> | ||
here the superscript <math>d</math> is meant to indicate the flux is computed through a disc limited by the integration line, as opposed to the ribbon limited by the integration line on one side and the magnetic axis on the other that was used for the definition of poloidal magnetic flux <math>\Psi_{ | here the superscript <math>d</math> is meant to indicate the flux is computed through a disc limited by the integration line, as opposed to the ribbon limited by the integration line on one side and the magnetic axis on the other that was used for the definition of poloidal magnetic flux <math>\Psi_{pol}</math> above these lines. | ||
Similarly | Similarly | ||
:<math> | :<math> | ||