📢Tensor Formulation of Vector Calculus Operations

The solution demonstrates how tensor notation translates complex vector calculus operations into component-based index contractions. Crucially, the curl is generalized to arbitrary coordinates by replacing the Cartesian Levi-Civita symbol with the contravariant Levi-Civita tensor density. The resulting formula is clean because the symmetry of the Christoffel symbols ensures they cancel out when contracted with the antisymmetric. Finally, the complex vector identity 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 and the tensor to manage all index raising and lowering.

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