🎬how two dynamic inputs determine the covariant components of the resulting vector

The visualization effectively translates the complex, high-order tensor identity $Z =v \times(\nabla \times w)+w \times(\nabla \times v)$ into a dynamic geometric result. It demonstrates how the covariant components ( $Z_x$ and $Z_y$ ) of the resultant vector Z are instantaneously determined by the relative orientation and interaction of the two input vector fields ( v and w ). The rotation of the inputs causes the magnitude and sign of the covariant components to fluctuate dynamically, proving that this complex identity represents an active, geometry-dependent coupling between the two vector fields and their rotational tendencies.

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