<s>[INST] How do you show that $d\theta = \frac{x dy - y dx }{x^2 + y^2}$? If $(r, \theta)$ are polar coordinates on $\mathbb{R}^2\setminus \{ (0,0)\}$, then how do I show/prove that 
\begin{equation*}
d\theta =\dfrac{x dy - y dx}{x^2 + y^2}?
\end{equation*}
 [/INST] On $R^2\setminus\{0\}$ we have $\theta=\Im(\ln(x+iy))$ (here $\Im$ stands for "the imaginary part of"). Therefore $$d\theta = \Im\left(\frac1{x+iy}\right)dx+\Im\left(\frac{i}{x+iy}\right)dy=\Im\left(\frac{x-iy}{x^2+y^2}\right)dx+\Im\left(\frac{i(x-iy)}{x^2+y^2}\right)dy=$$$$\frac{-y}{x^2+y^2}dx+\frac{x}{x^2+y^2}dy=\frac{xdy-ydx}{x^2+y^2}.$$
</s>