1) Compute, using green's theorem,
$\int_\Gamma (xy+x+y)dx+(xy+x+y)dy$
$\Gamma : \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ .
2) $f:R^3\to R^3,~f(x,y,z)=(z,x,-3y^2z)$ and $A=\partial\Omega$ the edge of the cylinder sector.
$\Omega =${$(x,y,z)\in R^3 ;x\geq 0, y\geq 0, 0\leq z\leq 5, x^2+y^2\leq 16$}.
Compute the flow of $f$ through $A$ directly and by using Gauss' theorem.
About 1: I haven't fully understood the green theorem yet. Can't I just set up a double integral with boundaries defined by $\Gamma$? Although I probably won't get any points for it since I need to use green theorem.
About 2: I'm kinda lost here. Firstly, I don't know what they mean by the cylinder sector. Are they talking about a disc cut off a cylinder or a smaller cylinder inside the original cylinder? Secondly, I don't know how to approach this kind of questions yet. Anyone got some tips or guidelines for these? I would appreciate it.
Hint:
Regarding 1: Green's theorem allows to calculate the surface integral (here some ellipse) with a line integral over the boundary. That looks like the curve integral is given. You can choose the parametrization.
Regarding 2: This cylinder with extruded sector might help:
You are supposed to calculate the flux $\Phi = \int\limits_E f \cdot dA$ through the edge surface $E$.