I'm having some troubles understanding the proof of a theorem the professor told us, can anybody help me fix some problems? So the theorem states:
Statement:
Let $M$ be a torus embedded in $R^3$, let $X$ be a vector field on $M - \{p_1,...,p_m\}$ where $p_1,...,p_m \in M$, then $$ \sum_{k=0}^m Ind_{p_k}(X) = 0$$
Definitions:
Residue: let $\omega \in \Lambda ^1 (M)$ $$ Res_{p_j} \omega = \int_{\partial B_\epsilon} \omega - \int_{B_\epsilon} d\omega$$ where $\epsilon$ is small and $B_\epsilon$ is actually $\phi^{-1}(B_\epsilon(p_j))$ for $(U,\phi)$ chart on $M$ so that $p_j \in U$.
Complex structure $J$: if $U \subset M$, $J$ is a map $J:TU \rightarrow TU$, such that $\forall q \in U$ it is $J_q: T_q U \rightarrow T_q U$ is linear and $J_q ^2 = -id$. Consequence is that $\forall v \in T_q U$ the pair $(v, Jv$ is a base for $T_q U$.
Index: let $X,Y:M \rightarrow TM$ two vector fields on $M$, let $p \in M$ and suppose $\forall m \in M-\{p\}$ is $X(m) \neq 0$. Suppose also $Y \neq 0$ over all $M$. Suppose $X = uY + vJY$ and let $f:= u + iv$ then $$Ind_p(X) = \frac{1}{2 \pi i}\int_{B_\epsilon} \frac{df}{f}$$.
Proof:
Let $J$ be the complex structure on $M$ induced by the Gauss map. Let $Y$ be a vector field such that $Y(p) \neq 0$ for $p \in M$, then there are $u,v \in C^\infty(M-\{p_1,...,p_m\})$ so that $$X = uY + vJY$$Let $f:= u + iv$,then let $fY:= Re(f)Y + Im(f)JY$, so we have $X = fY$ with $f \in C^\infty(M - \{p_1,...,p_n\})$. But then we have $$Ind_{p_j}(X) \overbrace{=}^{(1)} Res_{p_j}(\frac{df}{f})$$ and for the residue theorem we have $$0 \overbrace{=}^{(2)} \frac{1}{2\pi i}\int_{M}d (\frac{df}{f}) = \sum_{k=0}^m Res_{p_j}(\frac{df}{f}) = \sum_{k=1}^m Ind_{p_j}(X)$$
Problems:
Why (1)? Why $\int_{B_\epsilon(p)} d(\frac{df}{f}) = 0$? As far as I know if $f \in C^\infty(M)$ then $\frac{df}{f}$ would be a closed form, so I would have won, but in this case I don't have smoothness everywhere.
Why (2)? I feel like it has been used Stoke's theorem, but like above I would need $d(\frac{df}{f})$ to be defined on all of $M$, plus it is Stoke's applied to $\frac{df}{f}$ on $M - \{p_1,...,p_n\}$ and it is a little weird.
Edit: also, how is the hypotesis "$M$ is a torus" used? Isn't this theorem more general?