For (iii): along a periodic orbit $\Gamma$, the vector field $f$ is everywhere tangent to the curve and nonzero. As one traverses $\Gamma$ counterclockwise once, the angle $\psi$ of $f$ rotates by exactly $2\pi$ in the positive sense. To see why the sign is positive: the vector $f$ is tangent to $\Gamma$ and points in the direction of motion. As the curve is traversed, this tangent direction rotates by a full $2\pi$ counterclockwise. Under the transformation $f \mapsto -f$, the angle $\psi$ changes by $\pi$ at every point, but the definition of $I_\Gamma$ via the formula $d\psi = (f_1 \, df_2 - f_2 \, df_1)/(f_1^2 + f_2^2)$ is invariant under this sign change (numerator and denominator both change sign by the same factor), so $I_\Gamma = +1$ regardless of orientation. For (i) and (ii): a detailed computation using the Jacobian of $f$ at $x_0$ is required. For a node or focus, the linearisation has a net positive rotation of $\psi$ around any small enclosing curve, giving index $+1$. For a saddle, the alternating stable and unstable directions produce a net rotation of $-2\pi$, giving index $-1$.