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Flux coordinates: Difference between revisions
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→Jacobian
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</math> | </math> | ||
It can be seen that <ref name='Dhaeseleer'></ref> <math>g \equiv \det(g_{ij}) = J^2 \Rightarrow J = \sqrt{g}</math> | It can be seen that <ref name='Dhaeseleer'></ref> <math>g \equiv \det(g_{ij}) = J^2 \Rightarrow J = \sqrt{g}</math> | ||
=== Gradient, Divergence and Curl in curvilinear coordinates === | |||
The gradient of a funcion f is naturally given in the contravariant base vectors: | |||
:<math> | |||
\nabla f = \frac{\partial f}{\partial u^i}\nabla u^i = \frac{\partial f}{\partial u^i}\mathbf{e}^i~. | |||
</math> | |||
The divergence of a vector \mathbf{A} is best expressed in terms of its contravariant components | |||
:<math> | |||
\nabla\cdot\mathbf{A} = \frac{1}{\sqrt{g}}\frac{\partial}{\partial u^i}(\sqrt{g}A^i)~, | |||
</math> | |||
while the curl is | |||
:<math> | |||
\nabla\times\mathbf{A} = \frac{\epsilon^{ijk}}{\sqrt{g}}\frac{\partial}{\partial u^i}(\sqrt{g}A_j)\mathbf{e}_k | |||
</math> | |||
given in temr of the covariant base vectors, where <math>\epsilon^{ijk}</math> is the [[::Wikipedia:Levi-Civita symbol| Levi-Civita]] symbol. | |||
== Flux coordinates == | == Flux coordinates == | ||