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Toroidal coordinates: Difference between revisions
→Useful properties of FSA
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*<math> \langle\nabla\cdot\Gamma\rangle = \frac{d}{dV}\langle\Gamma\cdot\nabla V\rangle = \frac{1}{V'}\frac{d}{d\psi}V'\langle\Gamma\cdot\nabla \psi\rangle</math> | *<math> \langle\nabla\cdot\Gamma\rangle = \frac{d}{dV}\langle\Gamma\cdot\nabla V\rangle = \frac{1}{V'}\frac{d}{d\psi}V'\langle\Gamma\cdot\nabla \psi\rangle</math> | ||
*<math> \int_{V}\nabla\cdot\Gamma\; dV = \langle\Gamma\cdot\nabla V\rangle = V'\langle\Gamma\cdot\nabla \psi\rangle</math> | *<math> \int_{V}\nabla\cdot\Gamma\; dV = \langle\Gamma\cdot\nabla V\rangle = V'\langle\Gamma\cdot\nabla \psi\rangle</math> | ||
*<math> \langle \sqrt{g}^{-1}\rangle = \frac{4\pi^2}{V'} | *<math> \langle \sqrt{g}^{-1}\rangle = \frac{4\pi^2}{V'} | ||
</math> | </math> | ||
*<math> \langle \mathbf{B}\cdot\nabla f \rangle = 0~,\qquad \forall~ \mathrm{single~valued~} f(\mathbf{x}), ~ \mathrm{if}~ \nabla\cdot\mathbf{B} = 0 ~\mathrm{and}~ \nabla V\cdot\mathbf{B} = 0 </math> | |||
*<math> \langle \mathbf{B}\cdot\nabla \theta\rangle =2\pi\frac{d\Psi_{pol}}{dV} \qquad (\mathrm{Note:}~ \theta(\mathbf{x})~\mathrm{is~not~single~valued}) | |||
</math> | |||
*<math> \langle \mathbf{B}\cdot\nabla \phi\rangle =2\pi\frac{d\Psi_{tor}}{dV} \qquad (\mathrm{Note:}~ \phi(\mathbf{x})~\mathrm{is~not~single~valued}) | |||
</math> | |||
where <math>V' = \frac{dV}{d\psi}</math>. | where <math>V' = \frac{dV}{d\psi}</math>. | ||