I was wondering about a simple detail. As you probably already know, there are two definitions curl and divergence can be defined in the following way (in $\mathbb{R^{2}}$ but the question is also valid for higher dimensions) :
$$ \text{rot(f)(u)} = \lim_{\epsilon \to \,0}\frac{1}{\pi r^{2}}\int_{\partial B_{\epsilon}(u)}f\; \cdot \; \text{d}s $$
and:
$$ \text{div}(f)(u) = \lim_{\epsilon \to \,0} \frac{1}{4\epsilon^{2}}\int_{u + B_{ \infty}( \epsilon )} f \; \cdot \text{d}n$$
So here is my question. Is there a reason beyond semantics, why the divergence is defined over a square, as opposed to a disk ? Intuitively, you should be able to define div with a disk...
What do i mean by semantics ? I have seen divergence being introduced first in the calculus class on a square, before generalizing it to functions with a smooth boundary. So therefore it makes sense to introduce it over squares to begin with.