Consider the standard transformation equations between Cartesian and polar coordinates:
\begin{align*} x&=r \cos \theta\\ y&=r \sin \theta \end{align*}
and the inverse: $r=\sqrt{x^2+y^2}, \theta=\arctan\frac{y}{x}$.
Now consider the following product of derivatives: ${\displaystyle f=\frac{\partial r(x,y)}{\partial y}\frac{\partial y(r,\theta)}{\partial r}}.$ By the chain rule ${\displaystyle f=\frac{\partial r(x,y)}{\partial y}\frac{\partial y(r,\theta)}{\partial r} = \frac{\partial r}{\partial r} = 1}.$ However, if we calculate each multiplicand in isolation, then transform the mixed-coordinate result into a single coordinate system, we get:
\begin{align*} \frac{\partial r(x,y)}{\partial y}& =\frac{y}{\sqrt{x^2+y^2}}=\sin\theta\\ \frac{\partial y(r,\theta)}{\partial r}& = \sin\theta \end{align*}
and therefore, ${\displaystyle f=\frac{\partial r(x,y)}{\partial y}\frac{\partial y(r,\theta)}{\partial r} = \sin^2\theta}$
But we've shown by the chain rule that $f=1$!
:giantfireball:
I must be abusing the chain rule in some way (in the original context in which I stumbled on this, the correct result is $\sin^2\theta$), but I can't see what I did wrong. What's going on?