# 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.

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