Answer
$J(u,v) = -1 - 16uv $
Work Step by Step
Let $u = x + 2v^2$, $v = 2u^2 - y$.
Finding the Jacobian, we have: \[ J(u,v) = \left| \frac{\partial(u,v)}{\partial(x,y)} \right| = \left| \frac{\partial(x+2y^2)}{\partial u} \frac{\partial(x+2y^2)}{\partial v} - \frac{\partial(2u^2-y)}{\partial u} \frac{\partial(2u^2-y)}{\partial v} \right| = \left| (1)(-1) - (4u)(4v) \right| = -1 - 16uv \] Result : \[ J(u,v) = -1 - 16uv \]
