🎬how a completely anti-symmetric tensor is constructed from a tensor density

The covariant and contravariant forms of a completely anti-symmetric tensor have an inverse relationship determined by the geometry of the coordinate system. The demo visually proves this by showing that as the off-diagonal component of the metric tensor changes, the covariant component decreases while the contravariant component increases. This confirms that multiplying by $\sqrt{ g }$ (for the covariant form) and by $\sqrt{ g }^{-1}$ (for the contravariant form) correctly scales the tensor to match the underlying geometry.

The Metric Tensor Covariant Derivatives and Tensor Densities | Condensed Notesviadean on Notion