Calculus 3 Flux and Divergence (Gauss) theorem

[tomoyoshki]

Hello so I am stuck at b of this question and hope someone could help

Question B is where I have some trouble

I know that to compute the flux we could use divergence theorem which is simply taking the derivative of vector F. However divF is a really complicated function and I couldn't figure out how to compute it. If it came out to be a constant it would be nice since I could just use that constant times the volume which is 10, but doing div of vector F doesn't seem like the way to go and it is also what the key says. The answer turned out to be -12pi.

So please if anyone could help me out here, I really appreciate it.
Thanks

 

[yoscar04]

Do you know the delta function? I mean more specifically that div([MATH]\vec{r}[/MATH]/r³)=4[MATH]\pi[/MATH][MATH]\delta[/MATH](r)?

 

[yoscar04]

The computation of div H is not that complicated if you write the whole thing orderly. Try first in Cartesian and then in spherical coordinates. Have you taken any course in electrostatics?

 

[tomoyoshki]

Do you know the delta function? I mean more specifically that div([MATH]\vec{r}[/MATH]/r³)=4[MATH]\pi[/MATH][MATH]\delta[/MATH](r)?

Not really, I don't think the book covered it yet.

The computation of div H is not that complicated if you write the whole thing orderly. Try first in Cartesian and then in spherical coordinates. Have you taken any course in electrostatics?

Yes I was able to take the derivative of each vector component respect to x, y or z, but the only thing I am not sure is the limit of integration. The shape in the picture does not explicitly say the range. I assume that theta and phi are 2pi and pi, but how about p then? The question also doesn't give any information regarding the interval for x, y, and z so I am really confused...
Do you have any idea?
No I haven't taken it yet, I am a rising college freshmen and want to take a test to pass calc 3 to earn the credit so I am just learning calc 3 on my own right now....
thanks for the advice though

 

[yoscar04]

Not really, I don't think the book covered it yet.


Yes I was able to take the derivative of each vector component respect to x, y or z, but the only thing I am not sure is the limit of integration. The shape in the picture does not explicitly say the range. I assume that theta and phi are 2pi and pi, but how about p then? The question also doesn't give any information regarding the interval for x, y, and z so I am really confused...
Do you have any idea?
No I haven't taken it yet, I am a rising college freshmen and want to take a test to pass calc 3 to earn the credit so I am just learning calc 3 on my own right now....
thanks for the advice though

Did you study how to compute surface integrals? Here you have spherical symmetry and your function depends only on r, [MATH]\vec H[/MATH]=[MATH]\vec r[/MATH]/r³.

 

[yoscar04]

Try to perform your integral assuming a sphere of radius R. Then try to think why that would solve your problem. Hint: Gauss law: think that the flux coming out of a certain volume does not depend on the particular form you adopt for the closed surface.

 

[yoscar04]

What textbook are you using for this course?

 

[tomoyoshki]

Did you study how to compute surface integrals? Here you have spherical symmetry and your function depends only on r, [MATH]\vec H[/MATH]=[MATH]\vec r[/MATH]/r³.

The textbook I am using is Calculus Early Transcendentals 8th edition. The notation was different that I didn't realize and now with your comment I did get a clue. (I learned p instead r for spherical coordinates). So I first simplified H by switching x^2+y^2+z^2 to p, and then tried to calculate the divergence but end up getting zero. And then I tried to compute the surface integral with u as theta from0-2pi and v as phi from 0-pi, but I got -2pi somehow, and did not even use volume.

The textbook mentioned about divergence theorem or gauss theorem that flux is equal to triple integral of the div of a vector, but only place that mainly mentioned gauss law(electric i was this so I am really confused in someway.

Thank you for being patient with my inability to fully get the question....

 

[yoscar04]

The computation of the surface integral is as follows:
Regarding your computation of the divergence, it is true that div H=0 but only for r[MATH]\neq[/MATH]0. In order to compute the surface integral,
you need to know what is d[MATH]\vec{a}[/MATH]: d[MATH]\vec{a}[/MATH]=r²sin[MATH]\theta[/MATH]d[MATH]\theta[/MATH]d[MATH]\phi[/MATH][MATH]\hat{r}[/MATH].