Volume Jacobian for downslope stretched coordinates
= Volume Jacobian for downslope stretched coordinates
{title2=$V=P\cot^2\theta\iint h\,d\xi\,d\eta$}
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> $dx\,dy=\cot^2\theta\,d\xi\,d\eta$. Actual fluid volume is $P\iint h\,dx\,dy$, proving the displayed normalization. Replacing the cotangent factor by a tangent factor is incompatible with these coordinates except at a slope of $\pi/4$.