I've googled a lot but in most places, they usually only show how to compute it which I by now can do. But I wanna understand why the cross product between nabla and the vector give us the curl. So far I've managed to understand that when we are talking about the curl we are interested in the vector that points upwards or downwards (perpendicular) to the actual curl (rotation) like the green vector down here:
And then since the cross product between two vectors give us another vector orthogonal to both I finally understood why we do $\nabla\times \vec{F}$. Am I correct so far? Otherwise feel free to correct me (I actually want you to). But then my first question is why does this "green" vector in the picture define the curl? For example in Stoke's theorem we use curl to calculate higher dimension of "Green", but in Green's theorem we don't take any perpendicular vectors (obviously because z is non existent). This brings me to my second question which is how is the curl related to Green's thorem? Until now I had assumed that the green's theorem was just like line integral where we are only interested in the the field vector's component's that are parallel to the curve's tangents.
But in general I am confused and I'd really appreciate it if someone could just clear these things up for me?
For vector fields $V,W$ and a function $f$, we have \begin{align*} \nabla\times (fV)&=(\nabla f)\times V+f(\nabla\times V)\\ \nabla\times (V+W)&=\nabla\times V + \nabla\times W \end{align*}
If one also knows that the constant coordinate vector fields $e_1,e_2,e_3$ have zero curl, one can derive the curl formula as follows.
For any vector field $V$, there are functions $a,b,c$ such that $V=ae_1+be_2+ce_3$. Then
\begin{align*}\require{cancel} \nabla \times V&=\nabla \times (ae_1)+\nabla \times (be_2)+\nabla \times (ce_3)\\ &=(\nabla a)\times e_1+\cancel{a(\nabla\times e_1)}+(\nabla b)\times e_2+\cancel{b(\nabla\times e_2)}+(\nabla c)\times e_3+\cancel{c(\nabla\times e_3)}\\ &=(\nabla a)\times e_1+(\nabla b)\times e_2+(\nabla c)\times e_3 \end{align*}
And this last expression is formally equivalent to the rule of taking the cross product of the vector field with the "vector" $\nabla=(\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z})^t$.