Question

Let \[ \Phi(u,v) = (u-v,u+v,uv) \] and let D be the unit disk in the uv plane. Find the area of \[\ \Phi(D)\]
I got the right result by finding the \[\ ||\Phi_u\times \Phi_v || \] and then converting to polar coordinates. and integrating \[\ \int\int ||\Phi_r\times \Phi_\theta || rdrd\theta \]
My question is: Would I get the same result if I first parametrized u and v as \[\ u = rcos\theta; v=rsin\theta\]So that \[ \Phi(r,\theta) = (rcos\theta-rsin\theta,rcos\theta+rsin\theta,r^2cos\theta sin\theta) \]and then computed \[\ ||\Phi_r\times \Phi_\theta || \], and would I have to add Jacobian?