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 <math>\vec{p}=(x_0,y_0,z_0)</math> and <math>\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}</math>, such that each of the fs have continuous partial derivatives. Let S be a sphere of radius r centered at <math>\vec{p}</math>. Then

<math>\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}</math>