🎬The metric determinant is a scalar density with a weight of two and its square root is a scalar dens

Tensors are independent of the coordinate system, but tensor densities are not. A true tensor (like the Transformed True Tensor Value in the demo) represents a physical quantity that is a property of the object itself. Its components may change when you transform the coordinate system, but its underlying value remains constant. A tensor density (like the permutation symbol) is a quantity that's tied to the coordinate system. Its value changes with the Jacobian, which measures how much the volume (in 3D) or area (in 2D) of the coordinate grid is scaled during a transformation.

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