Answer
$4u^2v+uv^2$
Work Step by Step
Since, $Jacobian =\begin{vmatrix} \dfrac{\partial x}{\partial a}&\dfrac{\partial x}{\partial b}\\\dfrac{\partial y}{\partial a}&\dfrac{\partial y}{\partial b}\end{vmatrix}$
Here, we have $\dfrac{\partial x}{\partial a}=2u+v$ and $\dfrac{\partial x}{\partial b}=u$
Also, $\dfrac{\partial y}{\partial a}=v^2$ and $\dfrac{\partial y}{\partial b}=2uv$
Now, $Jacobian =\begin{vmatrix} \dfrac{\partial x}{\partial a}&\dfrac{\partial x}{\partial b}\\\dfrac{\partial y}{\partial a}&\dfrac{\partial y}{\partial b}\end{vmatrix}=\begin{vmatrix} 2u+v&u\\v^2&2uv\end{vmatrix}$
or, $=(2u+v) \times 2uv -u \times v^2$
or, $=4u^2v+2uv^2-uv^2$
or, $=4u^2v+uv^2$
