Toroidal coordinates: Difference between revisions

Jump to navigation Jump to search
Line 153: Line 153:
*<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 \mathbf{B}\cdot\nabla f \rangle = 0~,\qquad \forall~ \mathrm{single~valued~} f, ~ \mathrm{if}~ \nabla\cdot\mathbf{B} = 0 ~\mathrm{and}~ \nabla V\cdot\mathbf{B} = 0 </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>.


204

edits

Navigation menu