Let $\varphi$ a differential transformation such that $\varphi (x,y)=(f(x,y),g(x,y))$ and $D\subset U$ such that $\varphi$ restricted to $\partial D=\gamma$ be distinct zero and we define $i(\varphi ,D)=\frac{1}{2\pi }\int _{\gamma }\theta _{0}$, where $\theta _{0}=\frac{fdg-gdf}{f^{2}+g^{2}}$ a 1-form with $f^{2}+g^{2}\neq 0$. We say $\varphi$ have a simple positive zero if $\varphi(p)=0$ and $det(D_{p}\varphi)>0$, conversely we say $\varphi$ have a simple negative zero if $\varphi(p)=0$ and $det(D_{p}\varphi)<0$.
If $\varphi$ have a simple zero then prove that $i(\varphi ,D)=\pm 1$
I have a hint; think in the expansion $\varphi (p)=D_{p}\varphi p+O(\left \| p \right \|^{2})$ and in the correct homotopy. I´ve tried to use the expresion, but I really don´t see how to use it. Of course I´ve already tried to attack the problem with out the hint, but I don´t see how is useful to know that $\int _{\gamma }f^{2}+g^{2}\neq 0$
I´ve been thinking in this problem for a few days, does anybody can help me? Beacause I´m really stuck with this. Thanks!