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Calc3.6.DivergenceProof (view source)
Revision as of 22:06, February 9, 2010
, 22:06, February 9, 2010Created page with ':''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...'
:''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==
==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==