Watching a video about harmonic functions I've seen the "Gauss-Greens-Theorem" written as:
$$\int_{B(a,b)}\text{div}\,F\,\mathrm{dx\,dy} = \int_{\partial B(a,b)}F_x\,\mathrm{dy}-F_y\,\mathrm{dx}$$
with $F$ a vector field: $F = \left(\begin{array}{c}F_x\\ F_y\end{array}\right)$ and $B$ a ball around it at position $(a,b)$
What's confusing me is the actual state of that Theorem, because it seems to blend two claims:
$$\begin{align} &\text{Greens Theorem:} \int_A \partial_x\,F_y -\partial_y\,F_x \,\mathrm{dx\,dy} = \int_C F_x\,\mathrm{dx}+F_y\,\mathrm{dy} \\\\ &\text{Gauss Theorem:} \int_A \text{div}\,F\,\mathrm{dV} = \int_{\partial_A} \langle F, N\rangle\,\mathrm{dS} \end{align}$$
I don't see how those come together.
The way you've phrased the Gauss/divergence theorem looks like it is in $3$D, but the situation here is one dimension less. Anyway, note that if we define $G=(-F_y,F_x)$, then we can apply Greens theorem as you have stated, but this time apply it to $G$ to obtain \begin{align} \int_{\partial B(a,b)}F_x\,dy-F_y\,dx&=\int_{\partial B(a,b)}G_y\,dy+G_x\,dx\\ &=\int_{B(a,b)}(\partial_xG_y-\partial_yG_x)\,dx\,dy\\ &=\int_{B(a,b)}\partial_xF_x+\partial_yF_y\,dx\,dy\\ &=\int_{B(a,b)}\text{div}(F)\,dx\,dy. \end{align}