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Flux coordinates: Difference between revisions
→Gradient, Divergence and Curl in curvilinear coordinates
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=== Gradient, Divergence and Curl in curvilinear coordinates === | === Gradient, Divergence and Curl in curvilinear coordinates === | ||
The gradient of a | The gradient of a function f is naturally given in the contravariant base vectors: | ||
:<math> | :<math> | ||
\nabla f = \frac{\partial f}{\partial u^i}\nabla u^i = \frac{\partial f}{\partial u^i}\mathbf{e}^i~. | \nabla f = \frac{\partial f}{\partial u^i}\nabla u^i = \frac{\partial f}{\partial u^i}\mathbf{e}^i~. | ||
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\nabla\times\mathbf{A} = \frac{\varepsilon_{ijk}}{\sqrt{g}}\frac{\partial}{\partial u^i}(\sqrt{g}A_j)\mathbf{e}_k | \nabla\times\mathbf{A} = \frac{\varepsilon_{ijk}}{\sqrt{g}}\frac{\partial}{\partial u^i}(\sqrt{g}A_j)\mathbf{e}_k | ||
</math> | </math> | ||
given in | given in terms of the covariant base vectors, where <math>\varepsilon_{ijk}</math> is the [[::Wikipedia:Levi-Civita symbol| Levi-Civita]] symbol. | ||
== Flux coordinates == | == Flux coordinates == | ||