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.

$\complement\cdots$Counselor

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🏗️Computational Analysis

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$\gg$The Metric Tensor Covariant Derivatives and Tensor Densities

$\ggg$Mathematical Structures Underlying Physical Laws

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