I have two questions from my exam. I'm sorry for not remembering what was F.
1) $F$ is 3-dimensional vector field. $\delta$ is a surface which is bounded by $z=x^2+y^2-3$ and $z=1$ and oriented in negative $z$
$$ \iint_{\delta}curlF.\hat N.ds $$
Should I do $z=x^2+y^2-3 = 1$ $\Rightarrow$ $z=x^2+y^2=4$ and use polar coordinates?
And I know we can write $\hat N.ds$ = $\pm \frac{\nabla G}{G_3}dxdy$ when $G(x,y,z)=0$ is equation of surface with (1-1) projection onto a domain in the $xy$-plane. How can I decide what is $\hat N.ds$ when we have two or more equations like $x^2+y^2+z^2=a^2$ and $x+y+z=0$
2) It was more confusing me. I have no idea how can I determine boundaries.
I have found $divF=3y$
$$ \iiint_D 3ydV $$
Equations : $z=1-x^2$ parabolic cylinder and $z=0$, $y=0$, $y+z=2$
What should I say for boundaries?
As you can see I have some problems about these theorems. Can someone suggest some tutorials or books etc for catching them pratically?
Thanks
For the first question you can simply the Stokes theorem. Indeed that integral is egual to : $$\int_{\alpha} (F,T) $$ where $\alpha =(2cos t, 2sin t, 1)$ with $t = [0,2\pi]$ where $T$ is the tangent vector while $(F,T) $ is the dot product