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Toroidal coordinates: Difference between revisions
→Covariant Form
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\chi(\psi, \theta, \phi) = \frac{I_{tor}}{2\pi}\theta + \frac{I_{pol}^d}{2\pi}\phi + \tilde\chi(\psi, \theta, \phi) | \chi(\psi, \theta, \phi) = \frac{I_{tor}}{2\pi}\theta + \frac{I_{pol}^d}{2\pi}\phi + \tilde\chi(\psi, \theta, \phi) | ||
</math> | </math> | ||
The functional dependence on the angular variables is again motivated by the single-valuedness of the magnetic field. The particular form of the coefficients can be obtained noting that | |||
:<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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= \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. We get | ||
:<math> | :<math> | ||
\int_S \mu_0\mathbf{j}\cdot d\mathbf{S} | \int_S \mu_0\mathbf{j}\cdot d\mathbf{S} | ||