Is there such a thing as a difference vector field (a discrete differential vector field)?

Question

Let's just work in $\mathbb{R^2}$. We know that we may write an arbitrary vector field $v$ on $\mathbb{R}^2$ as $$v = v_1\frac{\partial}{\partial x} + v_2\frac{\partial}{\partial y}.$$ My question is, does this same sort of thing exist for finite differences? It seems to me that we could easily define a difference vector field $w$ by $$w = w_1\frac{\delta}{\delta x} + w_2\frac{\delta}{\delta y}$$ where $$w[f(x,y)] = w_1\frac{f(x+h,y)-f(x,y)}{h} + w_2\frac{f(x,y+h)-f(x,y)}{h}.$$ Has this been studied anywhere? If it has, if anyone could provide me with a reference, I would greatly appreciate it. Thank you!