# 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.

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