
$$
\xymatrix{
	{\colvectwo{r}{\theta}}\ar[rr]^{F} & {} & {\colvectwo{x}{y}}
		= F(r,\theta)
		= \colvectwo{r\cos\theta}{r\sin\theta} \\
%
	F^*(dx)=d(r\cos\theta) = \cos\theta \,dr - r\sin\theta \,d\theta
		\ar@{<-|}[rr]^{F^*}
		& {} & dx\\
%
	F^*(dy)=d(r\sin\theta) = \sin\theta\,dr + r\cos\theta \,d\theta
		\ar@{<-|}[rr]^{F^*}
		& {} & dy\\
%
	\fracDr\ar@{|->}[rr]^{F_*} & {} &
		F_*\left(\fracDr\right)
		= \frac{\Dx}{\Dr}\fracDx + \frac{\Dy}{\Dr}\fracDy
		%= {\colvectwo{\Dx/\Dr}{\Dy/\Dr}}
		= {\colvectwo{\cos\theta}{\sin\theta}} \\
%
	\fracDtheta\ar@{|->}[rr]^{F_*} & {} &
		F_*\left(\fracDtheta\right)
		= \frac{\Dx}{\D\theta}\fracDx + \frac{\Dy}{\D\theta}\fracDy
		%= {\colvectwo{\Dx/\D\theta}{\Dy/\D\theta}}
		= {\colvectwo{-r\sin\theta}{r\cos\theta}}.
}
$$
