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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Confirming divergence theorm example

**Physics Forums | Science Articles, Homework Help, Discussion**