Could someone direct me to a proof showing the equivalence between the following two definitions of the divergence of vector field $F$ at $x$? (1) $\lim_{|V| \to 0} \cfrac{\alpha({S})}{|V|}$, where $\alpha$ is the surface integral of $F$ over surface $S$ with volume $V$ containing $x$ (2) $\cfrac{\partial F_1}{\partial x_1} + \dots + \cfrac{\partial F_n}{\partial x_n}$; the first makes intuitive sense to me, the second not at all.
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I don't have a full proof, but if I remember correctly, the idea is to choose as the volume $V$ a hypercube centered at $x$ whose surfaces are parallel to the coordinate surfaces. – Vercassivelaunos Aug 12 '20 at 17:07
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1The details are pretty messy, but I wrote up a detailed answer a while back for equivalence between definitions of curl. A very similar approach can be used for the divergence (though I just don't feel like writing out the details :). – peek-a-boo Aug 12 '20 at 18:59