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This is a question from Do Carmo's Differential Forms and Applications (question 8, chapter 2). Actually, this question was made and answered here. The problem is: The answer redirects the OP to here ($f:\mathbb{S}^1\rightarrow\mathbb{S}^1$ odd $\Rightarrow$ $\mathrm{deg}(f)$ odd (Borsuk-Ulam theorem)), but in this Differential Forms course we didn't had contact with algebraic topology whatsoever, mainly on those firsts chapters. So, there is some way, probably not so hard to prove this statement. FYI, the problem is:

Let $ F: U \subset \mathbb R ^2 \to \mathbb R^2$ be a map with $ F(x,y) = (f(x,y) , g(x,y))$, where it satisfies $F(-q)=-F(q) \quad \forall q \in D \subset U$ where $D$ is a closed disk with center the origin and set $\partial D := c$. Assume that $F$ has no zeros in $\partial D$. (a) Prove that the index $ n(F; D)$ is an odd integer. (b) Prove that $F$ has at least one zero on the disk $D$.

Is good to state that the book defines: $\displaystyle{ n(F;D) =\frac{1}{2 \pi} \int_{F \ \circ \ c} \theta _0}$ where $\displaystyle{ \theta_0 = \frac{fdg - g df}{f^2 + g^2}}$.

I know that $n(F,D)$ is the number of counterclock-wise travels around the point, in this case, the origin. I just can't use any argument. If someone could help with this, it will be very appreciated. Thanks in advance.

big_GolfUniformIndia
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    For starters, the sign on $n(F;D)$ is wrong. – Ted Shifrin Aug 07 '21 at 02:09
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    So what precisely does DoCarmo have in the text? I do not own or know the book. One way or another you’re going to have to argue that $F$ winds a half-integer number of times — by oddness — when you go half around $c$, and then again the same amount when you go the rest if the way. – Ted Shifrin Aug 07 '21 at 23:46
  • @TedShifrin Hello, sorry for the late response. In fact, there is a typo on the $n(F,D)$. I know this, but I can't argue that will be exactly a odd number of times. – big_GolfUniformIndia Aug 08 '21 at 03:08
  • @TedShifrin actually, I did get to prove that the result holds for $F:D -> S^{1}$, but I did not get to argue for $R^{2}$. – big_GolfUniformIndia Aug 08 '21 at 10:21
  • Please revise/edit your post to include precisely what you have done and where you are stuck on the rest. – Ted Shifrin Aug 08 '21 at 21:20

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