 Divergence theorem

Let $V$ be a bounded region in Euclidean space with piecewise smooth boundary and outward unit normal vector $n$. For a continuously differentiable vector field $F$ on a neighborhood of its closure,The total outward flux equals the accumulated divergence inside. Contributions across shared internal faces cancel when a region is divided into small pieces.

For $F=(x,y,z)$ on a ball of radius $R$, the divergence is $3$, giving $3(4\pi R^3/3)=4\pi R^3$. On the boundary, $F\cdot n=R$, so the surface integral is also $R(4\pi R^2)=\boxed{4\pi R^3}$. This agreement turns the derivative into a geometric flux measurement. The required regularity matters: a singular field at the center cannot be included without checking its contribution separately.