# Vector calculus question.

1. Nov 21, 2008

### lembeh

1. The problem statement, all variables and given/known data

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 doesnt tell you what G is)

2. Relevant 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)

3. 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 thats about it.

1. The problem statement, all variables and given/known data

See above

2. Relevant equations

How do I solve this?!
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Nov 21, 2008

### HallsofIvy

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

3. Nov 21, 2008

### 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 couldnt 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 thats not an unreasonable request :)