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Let $ F: U \subset \mathbb R ^2 \to \mathbb R^2$ be a map with $ F(x,y) = (f(x,y) , g(x,y))$ ,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 $D$.

Definition: $\displaystyle{ n(F;D) =\frac{1}{2 \pi} \int_c \theta _0}$ where $\displaystyle{ \theta_0 = \frac{g df - fdg}{f^2 + g^2}}$

One can easily check that the index $n(F;D)$ is the winding number of the curve $ F \circ c$ about the origin and so index is an integer.

edit: I didn't write all hypothesis. I am really sorry for that. I hope now is clear.

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    Maybe I'm missing something stupid, but if $f(x,y) = g(x,y) = 1$ are constant functions, then they have no zeros on $\partial D$. Then the index is clearly $0$ since $df = dg = 0$.2012-06-20
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    @JasonDeVito: I have edit the question. I missed to write the hypothesis. I am sorry...2012-06-20
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    Doesn't my counter example still apply?2012-06-20
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    Now I see! I'll think about it.2012-06-20

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