General variable changes and the Jacobian

Material by the IntegralPaso project. Conditions and limits are stated in each guide; external teaching review is pending.

Transform the function, region and area or volume element. Check regularity and injectivity on the chosen domain, apart from sets that do not affect the integral. The engine offers standard coordinates; other changes need a custom setup.

Step-by-step example

1. Introduce a linear change.

   $$x=u+v,\quad y=u-v$$

2. Compute the determinant.

   $$\det\begin{pmatrix}1&1\\1&-1\end{pmatrix}=-2$$

3. Transform the region too.

   $$\iint_Df(x,y)\,dx\,dy=\iint_{D'}2f(u+v,u-v)\,du\,dv$$