I know that Jacobian is necessary to change the area element in integration, for example, \begin{equation} \Delta A= dx dy= \left|\frac{\partial x,\partial y}{\partial r ,\partial \theta}\right| drd\theta \end{equation} But why is the following thing is an illegal move,($x=r\cos\theta, y=r\sin\theta$), $$dx=dr\cos\theta-r\sin\theta d\theta, dy=dr\sin\theta+r\cos\theta d\theta$$ And so, $$\Delta A= dx dy= \left(dr\cos\theta-r\sin\theta d\theta\right)\left(dr\sin\theta+r\cos\theta d\theta\right)$$
Why the last equation is wrong ? If it is not wrong then, how is it connected to the first equation?
