I was thinking about coordinates change and, using polar coordinates on a 2D plane, i found an incoherence in my equations
$$x=r cos \theta \\ y=r sin \theta$$
from wich $$dx=cos\theta dr-rsin \theta d\theta \\ dy=sin \theta dr+rcos \theta d \theta$$
then, multiplying $dx$ and $dy$ $$dxdy=cos\theta sin \theta dr^2 + rcos \theta^2 dr d\theta - rsin \theta^2 d\theta dr- r^2 sin\theta cos\theta d\theta^2$$
while, using the standard procedure with the determinant of the Jacobian we obtain a different result: $$dxdy=rd\theta dr$$ I'm sure $dxdy=rd\theta dr$ is the correct result but i don't understand what is wrong with the procedure above
