Calculating Flux: x^2 + y^2 + z^2 = 4a^2

[Uncensored]

Homework Statement

Find the flux of x^2 + y^2 + z^2 = 4a^2 through F = (x+yz)i + (y-zx)j + (z-e^x * siny)k
directly without using divergence theorem.

Homework Equations

The Attempt at a Solution

Using spherical coordinates I achieve N = 4asin(phi)cos(theta) i + 4asin(phi)sin(theta) j + 4asin(phi)cos(phi) k

But when subbing in spherical to F, then F dot N, I can only simplify to a very messy integral of: -8a^3sin(phi) + 4a^2cos(phi)sin(phi) * e^(2sin(phi)cos(theta) * sin(2asin(phi)sin(theta)

Any help would be appreciated :)

 

[gabbagabbahey]

Using spherical coordinates I achieve N = 4asin(phi)cos(theta) i + 4asin(phi)sin(theta) j + 4asin(phi)cos(phi) k

Is the radius of that sphere 4a or 2a?...Is that the unit normal to the surface?

But when subbing in spherical to F, then F dot N, I can only simplify to a very messy integral of: -8a^3sin(phi) + 4a^2cos(phi)sin(phi) * e^(2sin(phi)cos(theta) * sin(2asin(phi)sin(theta)

Show the steps of your simplification; I get 2a-e^{2a\sin\theta\cos\phi}\sin\left(2a\sin\theta\sin\phi\right) for \textbf{F}\cdot\textbf{n}.

 

[Uncensored]

and yes radius is 2a

 

[LCKurtz]

Your integrand should be

8a^3\sin\phi - 4a^2\sin\phi\cos\phi e^{2a\sin\phi\cos\theta}\sin(2a\sin\phi\sin\theta)

 

[Uncensored]

okay I forgot the a on my e^(...) and I simply used -N so the switch of signs means nothing. Where do I go from here? Or is there another way to approach this problem? Stokes Theorem?

 

[LCKurtz]

Of course, the divergence theorem is the easy and obvious way but you ruled that out.

However, you almost have it. Notice that your second term is an odd function of theta. That means if you integrate theta from -pi to pi (which is just as good as 0 to 2pi) you will get zero. So all that is left is to do the double integral on the first term.

\int_0^{2\pi}\int_0^\pi -8a^2 \sin\phi\ d\phi d\theta

which will give you the same answer as the trivial application of the divergence theorem.