How do we know that the divergence of a vector field exists?

Question

How do we prove that the limit of $${\displaystyle \left.\operatorname {div} \mathbf {F} \right|_{\mathbf {x_{0}} }=\lim _{V\rightarrow 0}{1 \over |V|}} \unicode{x222F}_{\displaystyle \scriptstyle S(V)} {\displaystyle \mathbf {F} \cdot \mathbf {\hat {n}} \,dS}$$ exists? $$$$ I understand how to intuitively derive the formula for divergence in Cartesian coordinates given that the limit exists (since we can then choose an easy shape for the volume), but I don't know how to prove that the limit exists in the first place.

Answer 1

Of course you need some hypotheses here. Usually you would need something like $F$ differentiable, and $V $ "decent".

One way to go would be to prove the Divergence Theorem, so if you let $F'=P_x+Q_y+R_z$, you get that $$ {1 \over |V|}{\iint}_{\displaystyle \scriptstyle S(V)} {\displaystyle \mathbf {F} \cdot \mathbf {\hat {n}} \,dS}=\frac1{|V|}\,\iiint_V\,F'\,dV. $$ Now you can use the Mean Value Theorem, or some differentiation result like Lebesgue's differentiation to get that the right-hand-side goes to $F'(x_0)$ as $|V|\to0$.