So I have a question about the Jacobian. Suppose we consider the transformation from (x,y) to (r,$\theta$). We have :$$ \begin{align} x = r\cos(\theta) \\ y = r\sin(\theta) \end{align} $$
Now, if we consider dx and dy, we have We have :$$ \begin{align} dx = dr\cos(\theta) - r\sin(\theta)d\theta \tag{1}\\ dy = dr\sin(\theta) + r\cos(\theta)d\theta \tag{2} \end{align} $$
Since in a polar integral we equate the areas as : $$\begin{align}dxdy=rdrd\theta\end{align} \tag{3}$$shouldn't multiplying the (1) and (2) yield the same answer as (3)? When I try to multiply I do not get a matching answer whatsoever. I don't understand how to concile this with the determinant of a Jacobian matrix.
