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.
The Curl in General Coordinates : The curl of a vector $v$, traditionally an operation defined using the Cartesian Levi-Civita symbol $\left(\varepsilon^{a b c}\right)$, must be written using the contravariant Levi-Civita tensor density ( $\eta^{a b c}=\varepsilon^{a b c} / \sqrt{g}$ ) in general coordinates. The resulting contravariant component is:
$$ (\nabla \times v)^c=\eta^{a b c} \partial_a v_b $$
This formula is valid because the terms involving Christoffel symbols ( $\Gamma_{a b}^d$ ) in the covariant derivative cancel out when contracted with the antisymmetric $\eta^{a b c}$.
Cross Product via Tensors $\times$ : The cross product of two vectors, $A \times B$, is expressed using the $\eta^{a b c}$ tensor and the covariant components of the vectors:
$$ (A \times B)^c=\eta^{c a b} A_a B_b $$
The covariant component $(A \times B)_d$ is obtained by lowering the index using the metric:
$$ g_{d c}(A \times B)^c . $$
Complex Identity in Tensor Notation $\theta$ : To express the complex vector identity $v \times (\nabla \times w)+w \times(\nabla \times v)$ in covariant components, we need two applications of the metric ( $g$ ) and the $\eta$ tensor, effectively multiplying the expressions. The final expression is lengthy because it involves substituting the tensor form of the curl into the tensor form of the cross product:
$$ [v \times(\nabla \times w)]d=g{d c} g_{b e} \eta^{c a b} \eta^{e m n} v_a \partial_m w_n $$
The final step is simply summing this expression with the term where $v$ and $w$ are swapped. The key is that the entire vector operation is translated into a series of index contractions.
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