= Rational square diffeomorphism
{title2=$(u,v)\mapsto(1-u,(1-v)/(1-uv))$}
The displayed map is a <diffeomorphism> of the open unit square onto itself. For fixed $u$, the second coordinate decreases strictly from one to zero as $v$ increases from zero to one. Its inverse is $u=1-x$, $v=(1-y)/[1-(1-x)y]$, and its positive <Jacobian determinant> is $[1-(1-x)y]^2/x$. Composing it with $(t,w)\mapsto((1-t)/(1-wt),1-w)$ gives determinant $[1-(1-x)y][1-(1-x^2)y]^2/[x(1-y)]$. The reciprocal determinant therefore integrates to one over the target square by the <change of variables formula>, despite boundary singularities.
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