Simple proof of integration in polar coordinates?

In every instance I saw of integration in polar coordinates the Jacobian component is made use of, not that I have a trouble with the Jacobian, yet I asked yourself if there is a less complex means to show this which will certainly additionally offer me some even more instinct concerning the Jacobian.

If I attempt to merely write the differentials:

\begin{align} 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\\ \end{align}

In a double indispensable you incorporate $dxdy$, so if I attempt to connect in the outcomes I'll get something which is not $r d\theta dr$ \begin{align} dxdy & = \left(dr \cos \theta - r \sin \theta\ d\theta \right) \left( dr \sin \theta + r \cos \theta\ d\theta\right)\\ & = dr^2 \cos \theta \sin \theta - r^2 d\theta^2 \cos \theta\ \sin\ \theta + r\ dr\ d\theta\ (\cos^2 \theta\ - \sin^2\theta ) \end{align}

I do not assume I can go anywhere from below, I'm not exactly sure if it is simply an estimation blunder or the whole reasoning misbehaves.

Just how do I get this right?


2022-07-25 20:47:02
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