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Flux coordinates: Difference between revisions
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→Gradient, Divergence and Curl in curvilinear coordinates
m (→Covariant Form) |
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while the curl is | while the curl is | ||
:<math> | :<math> | ||
\nabla\times\mathbf{A} = \frac{\varepsilon_{ijk}}{\sqrt{g}}\frac{\partial}{\partial u^i} | \nabla\times\mathbf{A} = \frac{\varepsilon_{ijk}}{\sqrt{g}}\frac{\partial A_j}{\partial u^i}\mathbf{e}_k \Rightarrow | ||
\left(\nabla\times\mathbf{A}\right)^k = \frac{\varepsilon_{ijk}}{\sqrt{g}}\frac{\partial}{\partial u^i} | \left(\nabla\times\mathbf{A}\right)^k = \frac{\varepsilon_{ijk}}{\sqrt{g}}\frac{\partial A_j}{\partial u^i} | ||
</math> | </math> | ||
given in terms of the covariant base vectors, where <math>\varepsilon_{ijk}</math> is the [[::Wikipedia:Levi-Civita symbol| Levi-Civita]] symbol. | given in terms of the covariant base vectors, where <math>\varepsilon_{ijk}</math> is the [[::Wikipedia:Levi-Civita symbol| Levi-Civita]] symbol. | ||