View source for Calc3.6.DivergenceProof

:''This page is a proof of a theorem stated in [[Calc3.6|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 ''f''s 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>

==Proof==
[[Category:Mathematics]]