What is the relation between $\mathbf F$ and G,H in Green's Theorem?

Question

Green's theorem is given by:

What is the relation between $\mathbf F$ and G,H?

Answer 1

$\mathbf F$ is a vector function $\mathbf F=(G,H)$ where $G$ and $H$ are the components of the vector $\mathbf F$ with respect to the basis $\hat x, \hat y$.