Calc3.6.DivergenceProof - Revision history

Aschlafly: Reverted edits by Kevinxu (talk) to last revision by CPalmer

2011-09-17T20:00:06Z

Reverted edits by Kevinxu (talk) to last revision by CPalmer

Kevinxu at 19:52, September 17, 2011

2011-09-17T19:52:06Z

CPalmer: cat

2011-08-09T12:37:32Z

cat

JacobB: Created page with ':''This page is a proof of a theorem stated in Multivariable Calculus, Lecture 6.'' ==Theorem== Suppose that in cartesian coordinates, we have $\vec{p}=(x_0,y_0...'$

2010-02-09T22:06:14Z

Created 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...'

New page

:''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==

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Calc3.6.DivergenceProof - Revision history

Aschlafly: Reverted edits by Kevinxu (talk) to last revision by CPalmer

Kevinxu at 19:52, September 17, 2011

CPalmer: cat

JacobB: Created page with ':''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...'

JacobB: Created page with ':''This page is a proof of a theorem stated in Multivariable Calculus, Lecture 6.'' ==Theorem== Suppose that in cartesian coordinates, we have $\vec{p}=(x_0,y_0...'$