Divergence Theorem Homework: Find Divergence

[Tyst]

Homework Statement

Here is a link to the problem:
http://www.brainmass.com/homework-help/physics/electromagnetic-theory/68800

The Attempt at a Solution

To find the divergence

1/r^2*d(r)*(r^2*r^2*cos(theta))
+[1/r*sin(theta)]*d(theta)*(sin(theta)*r^2*cos(phi))
-[1/r*sin(theta)]*d(phi)*(r^2*cos(theta)*sin(phi))

Which gives

1/r^2*4*r^3*cos(theta)
+[1/r*sin(theta)]*(cos(theta)*r^2*cos(phi))
-[1/r*sin(theta)]*(r^2*cos(theta)*cos(phi))

Is this correct?

Looks correct to this point

Following this i get

=4*r*cos(theta)

What do you mean by "following this"? How did you get that and for what?

which gives me the right answer when i continue on with the question, however i am unsure about my second step... shouldn't i have to differentiate the 1/r*sin(theta) in the second term? And the 1/r^2 in the first? Or does the d(variable) only apply to the expressions written after it? As i guess you can tell, I'm confused and this is probably a really stupid question... Thanks in advance for you help ladies and gents.

It would help if you used parenthes:
div v= (1/r^2)D_r(r²v_r)+ (1/r sin theta)D_theta(sin theta v_theta)+ (1/r sin theta) D_phi(v_phi)

Yes, the derivative only applies to the expression immediately following. Usually it is in the derivative symbol or in parentheses to indicate that.

 

[Defennder]

I'm a little confused as to the notational convention for $$\phi \ \mbox{and} \ \theta$$. Which one in this question is the azimuthal angle to the xy plane?

 

[Tyst]

Phi is the azimuthal angle

 

[Tyst]

4rcos(theta) came from simplifying the expressions above it, it is the divergence.

And thank you - you answered my question!