📢Path-Independence of Tensor Densities via the Jacobian Product Rule

The verification confirms that Jacobian determinants follow a crucial product rule for successive coordinate transformations, where the total Jacobian, is the product of the individual Jacobians. This rule is a direct consequence of the matrix multiplication property of determinants applied to the chain rule for derivatives. A key corollary is that the Jacobian of an inverse transformation is the reciprocal, when the final coordinates are the initial ones. Ultimately, the product rule guarantees the consistency of the transformation law for a tensor density of weight; whether the transformation is performed in one direct step or multiple successive steps, the resulting tensor components remain the same, as the transformation factors-both the Jacobian power and the partial derivatives-combine via the chain and product rules.

Verification of the Product Rule for Jacobian Determinants and Tensor Density Transformation | Proof and Derivationviadean on Notion