Volume Jacobian for downslope stretched coordinates ($V=P{\cot }^2\theta \iint hd\xi d\eta$)

With constant porosity $P$ and $X=x\tan \theta$, $Y=y\tan \theta$, a translating frame $\xi =X-T$, $\eta =Y$ has area Jacobian determinant $dxdy={\cot }^2\theta d\xi d\eta$. Actual fluid volume is $P\iint hdxdy$, proving the displayed normalization. Replacing the cotangent factor by a tangent factor is incompatible with these coordinates except at a slope of $\pi /4$.