Calc3.6.DivergenceProof

This page is a proof of a theorem stated in Multivariable Calculus, Lecture 6.

Theorem

Suppose that in cartesian coordinates, we have ${\vec {p}}=(x_{0},y_{0},z_{0})$ and ${\vec {F}}({\vec {p}})=f_{1}(x_{0},y_{0},z_{0}){\vec {i}}+f_{2}(x_{0},y_{0},z_{0}){\vec {j}}+f_{3}(x_{0},y_{0},z_{0}){\vec {k}}$, such that each of the fs have continuous partial derivatives. Let S be a sphere of radius r centered at ${\vec {p}}$. Then

$\lim _{r\to 0}\left({\frac {1}{{\frac {4}{3}}\pi r^{3}}}\iint \limits _{S}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset \!\supset {\vec {F}}\cdot {\vec {n}}_{S}dS\right)={\frac {\partial f_{1}}{\partial x}}+{\frac {\partial f_{2}}{\partial y}}+{\frac {\partial f_{3}}{\partial z}}$