[tex]\mathbf J=\begin{bmatrix}\dfrac{\partial x}{\partial u}&\dfrac{\partial x}{\partial v}\\\\\dfrac{\partial y}{\partial u}&\dfrac{\partial y}{\partial v}\end{bmatrix}=\begin{bmatrix}2u+v&u\\v^2&2uv\end{bmatrix}[/tex]
The Jacobian has determinant
[tex]\det\mathbf J=\begin{vmatrix}2u+v&u\\v^2&2uv\end{vmatrix}=(2u+v)2uv-uv^2=4u^2v+uv^2=uv(4u+v)[/tex]
