Then I made a test in Mathematica. I transformed a unit square (left figure) into a quadrilateral (right figure):
Then transformation equation is$ \begin{cases} x(u, v) = u(1 + v)\\ y(u, v) = v(1 + 3u) \end{cases}, $ and the Jacobian is $1+3u+v$. But it seems not make sense. The area of origin unit square is $1$, and the area of transformed quadrilateral is $3$.
How can I derive the area of parallelogram using Jacobian?
