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


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}


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