I've done many examples questions on calculating the divergence of a vector field, but it's always been the case that I get the same divergence no matter which point I use in the calculation (I use the geometric definition of divergence, which involves imagining a sphere/cube becoming smaller and smaller around a point, hence why I need to consider a specific point).
Does this hold in general? That is, is the divergence of a vector field the same no matter which point you use in your calculation (assuming you're using the geometric method to calculate)?
If the point does in fact matter, then it wouldn't make sense to say "find the divergence of $\vec F$" because the answer would depend on the point. Is that correct?
On the contrary, it's quite unusual for the divergence of a vector field to be independent of the point! You can see this from the formula $\nabla \cdot \mathbf{F} = \partial F_1/\partial x_1 + \partial F_2/\partial x_2 + \partial F_3/\partial x_3$; if you pick some functions $F_i$ arbitrarily (try some polynomials for example), the expression that you get will usually depend on $(x_1,x_2,x_3)$. So the divergence of a vector field is a function, and it makes perfect sense to ask about finding this function.