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Flux coordinates: Difference between revisions
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→Coordinates and basis vectors
| Line 17: | Line 17: | ||
\to | \to | ||
\mathbf{e}_i\cdot\mathbf{e}^j | \mathbf{e}_i\cdot\mathbf{e}^j | ||
= \delta_{i}^{j} | = \delta_{i}^{j}~, | ||
</math> | |||
and therefore relates to the contravariant vectors as | |||
:<math> | |||
\mathbf{e}_i | |||
= \frac{\mathbf{e}^j\times\mathbf{e}^k}{\mathbf{e}^i\cdot\mathbf{e}^j\times\mathbf{e}^k} | = \frac{\mathbf{e}^j\times\mathbf{e}^k}{\mathbf{e}^i\cdot\mathbf{e}^j\times\mathbf{e}^k} | ||
= \sqrt{g}\;\mathbf{e}^j\times\mathbf{e}^k ~, | = \sqrt{g}\;\mathbf{e}^j\times\mathbf{e}^k ~, | ||