I've been studying double integrals from the point of view of differential forms and pullbacks. In my book there's an example of how to convert a triangular region of integration between points (0, 0), (0, 1) and (1, 0) with line $x + y =1$ into a rectangular integration region in variables $u, v$. If $\alpha$ is the map from the rectangular region to the triangular region, its pullback $\alpha^{*}$ works as $\alpha^{*}x = uv$ and $\alpha^{*}y = u(1 - v)$.
This change in terms of $u$ and $v$ does not seem intuitive at all for me. My question is: is there a reasoning that you can use to convert any region of integration to a rectangular region?