multi integral , stokes I need to help my brother with this question for his diagnose test tomorrow. I don't want to teach him wrong. How would you smarties solve this one? I think it's Stokes theorem but it's been a while since I used it.
how to solve for a field given "F=x,y,z" from this form:
dubbel integral of : curl(F) N dS 
 A: The rotational of $\bf{F}$ is given by:
$$\mbox{rot}{\bf{F}} = (x-y,-y,1) $$
Also, the unitary normal on the surface of the sphere in cartesian coordinates is, obviously:
$$\hat{n} =\frac{x \hat{x}+y\hat{y}+z\hat{z}}{\sqrt{x^{2}+y^{2}+z^{2}}}.$$ 
Thus, we have:
$$\mbox{rot}{\bf{F}}\cdot \hat{n} = \frac{x^{2}-xy-y^{2}+z}{\sqrt{x^{2}+y^{2}+z^{2}}}$$
Now you can use spherical coordinates:
$$ x = 2\cos\theta\sin\phi \quad y = 2\sin\theta\sin\phi \quad z = 2\cos\phi$$
with $0 \le \theta \le 2\pi$ and $0 \le \phi \le \frac{\pi}{2}$. Thus, you have to solve:
$$\int_{0}^{\frac{\pi}{2}}\int_{0}^{2\pi}\bigg{[}2\cos^{2}\theta\sin^{2}\phi-2\cos\theta\sin\theta\sin^{2}\phi-2\sin^{2}\theta\sin^{2}\phi+\cos\phi\bigg{]}2\sin\phi d\theta d\phi$$
A: You can use Stokes' theorem to say that
$$\iint_{S_1}\operatorname{rot} \mathbf{F} \cdot \mathbf{N}\:dS = \iint_{S_2}\operatorname{rot} \mathbf{F} \cdot \mathbf{N}\:dS$$
where $S_1$ and $S_2$ are any two surfaces that share the same boundary. We can use this to choose a different, easier surface to integrate over. Notice that the half sphere shares a boundary with the disk $x^2+y^2=4$ in the $z=0$ plane. So we have that
$$I = \iint_{x^2+y^2=4 \:\cap\:z=0} \operatorname{rot} \mathbf{F} \cdot \mathbf{N}\:dS$$
Now since we are on the plane $z=0$, we have that $\mathbf{N} = (0,0,1)$ pointing up to make sure it has the correct orientation, giving that
$$I = \iint_{x^2+y^2\leq 4} 1 \: dA = 4\pi $$
the area of the disk.
