How to Compute F on a Given Surface with Downward Pointing Normal?

[lembeh]

Homework Statement

Let S be the surface given by the graph z = 4 - x² - y² above the xy-plane (that it is, where z $$\geq$$ 0) with downward pointing normal, and let

F (x,y,z) = xcosz i - ycosz j + (x² + y² ) k

Compute $$\oint\oints$$$$\oint$$_s F dS. (F has a downward pointing normal)

(Hint: Its easy to see that div F = 0 on all R³. This implies that there exists a vector field G such that F = Curl G, although it doesn't tell you what G is)

Homework Equations

z = 4 - x² - y² above the xy-plane (that it is, where z $$\geq$$ 0) with downward pointing normal

F (x,y,z) = xcosz i - ycosz j + (x² + y² ) k

Compute $$\oint\oints$$$$\oint$$_s F dS. (F has a downward pointing normal)

The Attempt at a Solution

Im getting throw off a bit by the hint. I know its something to do with the surface not being defined around the origin but that's about it.

Homework Statement

See above

Homework Equations

How do I solve this?!

 

[HallsofIvy]

Looks to me like the hint is suggesting you use Stokes' theorem:
$$\int \vec{G}\cdot\d\vec{r}= \int\int \nabla G\cdot d\vec{S}$$

 

[lembeh]

Right, but how do I compute this? My daughter hasnt gone past Green's theorem yet in class...I saw this problem on her homework but she couldn't solve it. I can help her and know some Multivariable calculus (but not vector calculus). I want to help her get through this. I would really appreciate it if someone spelt out the solution for me. So I could learn this and help her out with this. I hope that's not an unreasonable request :)