🧄Verification of the Product Rule for Jacobian Determinants and Tensor Density Transformation

The verification confirms that Jacobian determinants follow a crucial product rule for successive coordinate transformations ( $y \rightarrow y^{\prime} \rightarrow y^{\prime \prime}$ ), where the total Jacobian, $J^{\prime \prime}$, is the product of the individual Jacobians, $J J^{\prime}=J^{\prime \prime}$. 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, $J^{\prime}=1 / J$, 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 w; 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 ( $J^{\prime \prime}$ ) 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