The task is to rewrite the following differential form in polar coordinates:
$w = \sqrt{x^2 + y^2} \, dx \land dy $
I did it by a direct substitution:
$\begin{cases}x = r \cos \varphi \\ y = r \sin \varphi \end{cases}, \;\;\;\;\;\;\;\;\; \begin{cases} dx = \cos \varphi \, dr - r \sin \varphi \, d \varphi \\d y = \sin \varphi \, dr + r \cos \varphi \, d \varphi \end{cases} $
$\begin{multline} w = r (cos \varphi \, dr - r \sin \varphi \, d \varphi) \land (\sin \varphi \, dr + r \cos \varphi \, d \varphi) = \\ r^2 (\cos^2 \varphi \, dr \land d \varphi - \sin^2 \varphi \, d \varphi \land dr) = r^2 \, dr \land d \varphi \end{multline}$
Is there any other algebraical way to do it? That is I don't mean any drawing (geometry), just tossing symbols around, like pulling some terms under $d$.