A seemingly simple exercise on the divergence theorem

[Feynman's fan]

Here is the problem statement:

Compute the surface integral
\int_{S^2}f \cdot n \ dS \ \ where f(x,y,z)=(y^3, z^3, x^3)^T

I thought it's a straightforward exercise on the divergence theorem, yet it looks like \operatorname{div} f = 0. So the surface integral is zero?

Am I missing some sort of a trick here? The exercise isn't supposed to be that easy.

Any hints are very appreciated!

 

[BruceW]

true, $\operatorname{div} f = 0$ but what is the surface? Is it a closed surface or not closed and do they give some specific surface to integrate over?

 

[LCKurtz]

Here is the problem statement:



I thought it's a straightforward exercise on the divergence theorem, yet it looks like \operatorname{div} f = 0. So the surface integral is zero?

Am I missing some sort of a trick here? The exercise isn't supposed to be that easy.

Any hints are very appreciated!

If $S^2$ means the unit sphere, then you are correct. Unless there is something missing in the translation of $(y^3,z^3,x^3)^T$.

 

[BruceW]

ah, right, $S^2$ means sphere. I forgot about that. hmm, yeah it looks like zero is the right answer then. well, don't look a gift horse in the mouth :)