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.
The Product Rule for Jacobian Determinants
The central mathematical result is the product rule for Jacobian determinants under successive transformations:
$$ J J^{\prime}=J^{\prime \prime} $$
This means the Jacobian determinant for a composite coordinate transformation ( $y \rightarrow y^{\prime} \rightarrow y^{\prime \prime}$ ) is the product of the individual Jacobian determinants. This directly parallels the chain rule for differentiation applied to the transformation matrices: $\operatorname{det}( C )=\operatorname{det}( A ) \operatorname{det}( B )$, where $C =AB$.
Jacobian of the Inverse Transformation
When the successive transformations return to the original coordinates $\left(y^{\prime \prime}=y\right)$, the product rule simplifies to:
$$ J J^{\prime}=1 \quad \Longrightarrow \quad J^{\prime}=\frac{1}{J} $$
This shows that the Jacobian determinant of an inverse transformation is the reciprocal of the Jacobian determinant of the original transformation.
Consistency of Tensor Density Transformations
The transformation law for a tensor density of weight $w$ is path-independent. The final components of the tensor density ( $T^{\prime \prime}$ ) obtained by performing two successive transformations ( $y \rightarrow y^{\prime} \rightarrow y^{\prime \prime}$ ) are identical to the components obtained by a single, direct transformation ( $y \rightarrow y^{\prime \prime}$ ). This consistency is ensured by two factors:
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