Surface Integrals: Find Value w/Divergence Theorem

[duki]

Homework Statement

Find the value of the surface integrals by using the divergence theorem
$$\vec{F} = (y^2z)\vec{i} + (y^3z)\vec{j} + (y^2z^2)\vec{z}$$

S: $$x^2 + y^2 + z^2$$

Use spherical coordinates.

Homework Equations

The Attempt at a Solution

I've gotten the integral I think. I want to make sure before I go along with evaluating it.

$$\int _0^{2\pi} \int _0^{\pi} \int _0^2 { (7\rho^3 \sin^2{\phi} \sin^2{\theta} \cos{\phi} } \rho^2 d \rho d \phi d \theta$$

My latex is all messed up... maybe a mod can fix it for me?

 

[gabbagabbahey]

Homework Statement

Find the value of the surface integrals by using the divergence theorem
$$\vec{F} = (y^2z)\vec{i} + (y^3z)\vec{j} + (y^2z^2)\vec{z}$$

S: $$x^2 + y^2 + z^2$$

Use spherical coordinates.

Do you mean $x^2+y^2+z^2=4$?

I've gotten the integral I think. I want to make sure before I go along with evaluating it.

$$\int _0^{2\pi} \int _0^{\pi} \int _0^2 { (7\rho^3 \sin^2\phi \sin^2{\theta} \cos{\phi} } \rho^2 d \rho d \phi d \theta$$

My latex is all messed up... maybe a mod can fix it for me?

That doesn't look quite right...what do you get for the divergence of F (in Cartesians and Sphericals)?

 

[duki]

Yes, I meant = 4. Thanks.

I got $$5y^2 2z$$

 

[gabbagabbahey]

Yes, I meant = 4. Thanks.

I got $$5y^2 2z$$

I assume you mean $5y^2 z$?...If so, you're right. What is that in spherical coordinates? What are you using for $dV$ (infinitesimal volume element) in spherical coordinates?

 

[duki]

Ok, I have $$5y^2z$$ in my notes but I thought that was wrong. When I take the partial of $$y^2z^2$$ with respect to z, why does that come out to just z?

 

[gabbagabbahey]

$$\frac{\partial}{\partial z} (y^2 z^2)=y^2 \frac{\partial}{\partial z} (z^2)=2y^2 z$$

 

[duki]

OooOOooOOooooohhh

 

[HallsofIvy]

I fixed the Latex:
1) Use "\" in front of Greek letters: \theta, \phi, \rho.

2) $\rho$ is "rho", not "roe".