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Flux coordinates: Difference between revisions
→Useful properties of FSA
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*<math> \int_{\mathcal{V}}\nabla\cdot\Gamma\; d\mathcal{V} = \langle\Gamma\cdot\nabla V\rangle = V'\langle\Gamma\cdot\nabla \psi\rangle</math> | *<math> \int_{\mathcal{V}}\nabla\cdot\Gamma\; d\mathcal{V} = \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(\mathbf{x}), ~ \mathrm{if}~ \nabla\cdot\mathbf{B} = 0 ~\mathrm{and}~ \nabla \psi\cdot\mathbf{B} = 0 </math> | *<math> \langle \mathbf{B}\cdot\nabla f \rangle = \langle \nabla\cdot(\mathbf{B} f) \rangle = 0~,\qquad \forall~ \mathrm{single~valued~} f(\mathbf{x}), ~ \mathrm{if}~ \nabla\cdot\mathbf{B} = 0 ~\mathrm{and}~ \nabla \psi\cdot\mathbf{B} = 0 </math> | ||
*<math> \langle \nabla \psi\cdot\nabla\times \mathbf{A} \rangle = 0~. | *<math> \langle \nabla \psi\cdot\nabla\times \mathbf{A} \rangle = -\langle \nabla\cdot( \nabla\psi\times\mathbf{A}) \rangle = 0~. | ||
</math> | </math> | ||
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</math> | </math> | ||
In the above <math>V' = \frac{dV}{d\psi}</math>. | In the above <math>V' = \frac{dV}{d\psi}</math>. Some [[:Wikipedia: Vector calculus identities|vector identities]] are useful to derive the above identities. | ||
=== Magnetic field representation in flux coordinates === | === Magnetic field representation in flux coordinates === | ||