Consider a map $\gamma: [0,T] \rightarrow [0,1] \times [0,1]$ such that is continous and surjective. Compute the winding number of $\gamma$ in $a$ ($ind(\gamma,a)$), where $a \notin Im(\gamma)$.
I've justed started studying winding numbers where comes to me that problem. I do know how to compute it using integrals and topological ways, however, I do not know how to do it since $\gamma$ is no explicited given.
On
Assuming the intent is for $a \in \mathbb{R}^2 \setminus ([0,1]^2)$, then we can draw a line through $a$ such that the image of $\gamma$ lies entirely in one of the two open half-planes.
This implies the angle on $\gamma$ about $a$ lies entirely in some $(\theta, \theta+\pi)$ for some $\theta$ (which we can set to $\theta = 0$ or $\theta = -\pi/2$ if we really want). This allows us to conclude the winding number for the closed curve $\gamma$ is $0$.
Hint: The winding number should be zero. If $\gamma$ is surjective, then $a\notin\rm{im}\gamma\implies a$ is outside of $\gamma$.
There's a theorem that when $a$ is in the unbounded part of $\Bbb C\setminus\rm{im}\gamma$ the index is zero.
(Intuitively, we have a space filling curve, and there's no way for it to wrap around a point without hitting it.)