Hello, I am having trouble confirming that the flux integral is equal to the divergence over a volume. I am making a silly mistake & its just one of those days that I can't eyeball it. Here is the problem.(adsbygoogle = window.adsbygoogle || []).push({});

I want to compute the flux integral for

[itex]

\vec{ F}=x\hat i+y\hat j-z\hat k

[/itex]

out of theclosedcone

[itex]

x^2+y^2=z^2 \,\,\,\,\,\,\,0\leq z\leq1

[/itex]

Let us first do this the easy way using the divergence theorm.

[itex]

\int\int \vec F \cdot d \textbf{S}=\int \int \int \nabla \cdot \vec F dV

[/itex]

[itex]

\nabla \cdot \vec F=1

[/itex]

[itex]

\int \int \int \nabla \cdot \vec F dV=\int_0^{2\pi} d\theta \int_0^1 r dr \int_r^1 dz=\frac{\pi}{3}

[/itex]

This agrees with the formula for the volume of a cone of radius one and height one. Now, let us apply the flux integral directly.

First lets look at the cone.

[itex]

\int\int \vec F \cdot d \textbf{S}=\int\int \vec F \cdot d \textbf{S}_1+\int\int \vec F \cdot d \textbf{S}_2

[/itex]

where S_{1}is the cone and S_{2}is the lid of the cone, namely the unit disk z=1.

[itex]

f=x^2+y^2-z^2=0

[/itex]

[itex]

\nabla f=(2x,2y,-2z)

[/itex]

[itex]

\hat n=\frac{\nabla f}{|\nabla f|}=\frac{(2x,2y,-2z)}{2 \sqrt{x^2+y^2+z^2}}=\frac{(2x,2y,-2z)}{2 \sqrt{2}z}

[/itex]

[itex]

d\textbf{S}_1=\hat n \sqrt{\frac{\partial f}{\partial x}^2+\frac{\partial f}{\partial y}^2+\frac{\partial f}{\partial z}^2} dx dy=\hat n 2 \sqrt{x^2+y^2+z^2} dx dy= \hat n 2 \sqrt{2}z dx dy =(2x,2y,-2z)dx dy

[/itex]

[itex]

\int\int \vec F \cdot d \textbf{S}_1=\int\int \vec F \cdot (2x,2y,-2z)dx dy=\int\int 2x^2+2y^2+2z^2 dx dy =\int\int 4x^2+4y^2 dx dy=\int_0^{2\pi}d\theta \int_0^1 r dr 4 r^2=2\pi

[/itex]

Over the disk [itex]\hat n=\hat k[/itex] and z=1

[itex]

\int\int \vec F \cdot d \textbf{S}_2=-\int_0^{2\pi}d\theta\int_0^1 r dr =-\pi

[/itex]

Hence,

[itex]

2\pi-\pi=\pi

[/itex]

So this is the wrong answer. I don't see where I went wrong. Can someone please help? Thank you.

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# Confirming divergence theorm example

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