🎬the totally antisymmetric tensor in a flat two-dimensional space

The totally antisymmetric tensor $\eta^{ ab }$ in flat two-dimensional space geometrically implements the Hodge Dual operation. This is visually confirmed by demonstrating that the transformed vector field (Red, $W = \eta V$ ) is consistently maintained at a $9 0 ^{ \circ }$ clockwise rotation relative to the input vector field (Blue, V ), regardless of the input field's rotation or location. This illustrates how $\eta$ is used to establish duality relationships between vector spaces and their associated differential forms.

Vanishing Divergence of the Levi-Civita Tensor | Proof and Derivationviadean on Notion