Section 3.12 (pg. 299) of Lawrence Perko's Differential Equations and Dynamical Systems (3rd edition) on index theory states "It is one of the most interesting facts of the index theory that the index of the surface $S$, $I_{\mathbf{f}}(S)$, is independent of of the vector field $\mathbf{f}$... This result is the famous Poincaré Index Theorem."
It then gives the formula for the index which is defined as
$ I_{\mathbf{f}}(C) = \frac{1}{2\pi} \oint_C \frac{PdQ - QdP}{P^2 + Q^2}$
for some vector field $\mathbf{f} = \begin{pmatrix} P(x,y) \\ Q(x,y)\end{pmatrix}$
Example 1 (starting on the same page) says:
Let $C$ be the circle of radius one, centered at the origin, and let us compute the index of $C$ relative to the vector fields
$\mathbf{f}(x) = \begin{pmatrix} x \\ y\end{pmatrix} \quad \mathbf{g}(x) = \begin{pmatrix} -x \\ -y\end{pmatrix} \quad \mathbf{h}(x) = \begin{pmatrix} -y \\x \end{pmatrix} \quad \mathbf{k}(x) = \begin{pmatrix} x \\-y \end{pmatrix}$
the results being
$I_{\mathbf{f}}(C) = 1, \quad I_{\mathbf{g}}(C) = 1, \quad I_{\mathbf{h}}(C) = 1, \quad I_{\mathbf{k}}(C) = -1$
Doesn't this violate what was just stated in the Poincaré index theorem? That is, it looks to me that the index of the same surface $C$ is not independent of the vector field since $I_{\mathbf{k}}(C)$ is not equal to the others. Is there some qualification to the theorem that I'm missing or something I haven't understood? I am working on solving for the index of $C$ relative to a more complicated vector field and also seing that it is not equal to 1.