🧄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.

Curl and Vector Cross-Product Identity in General Coordinates | Proof and Derivationviadean on Notion