# Curl and Vector Cross-Product Identity in General Coordinates

The solution demonstrates how tensor notation translates complex vector calculus operations into component-based index contractions. Crucially, the curl ( $$\nabla \times v$$ ) is generalized to arbitrary coordinates by replacing the Cartesian Levi-Civita symbol with the contravariant Levi-Civita tensor density $$\left(\eta^{a b c}\right)$$, resulting in $$(\nabla \times v)^c=\eta^{a b c} \partial\_a v\_b$$. This formula is clean because the symmetry of the Christoffel symbols ensures they cancel out when contracted with the antisymmetric $$\eta^{a b c}$$. Finally, the complex vector identity $$v \times(\nabla \times w)+w \times(\nabla \times v)$$ is expressed in covariant components by nesting the tensor form of the curl inside the tensor form of the cross product, requiring multiple applications of the metric ( g ) and the $$\eta$$ tensor to manage all index raising and lowering.

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