The following is the problem I'm working on.
If $\overrightarrow {F} = <x,y^2z,-xy^2>$, calculate the circulation of $\overrightarrow {F}$ over the surface $z=x^2+y^2$ bounded by $C$ using a line integral. $C$ is given by $$C_1 : \overrightarrow {r}=<t,t,0> (0\le t \le 2\sqrt{2})$$ $$C_2 : \overrightarrow {r} = <4\cos(t),4\sin(t),0> ({\pi \over 4} \le t \le \pi)$$ $$C_3 : \overrightarrow {r} = <t-4,0,0> (0 \le t \le 4)$$
I tried to solve it using $\int_C \overrightarrow {F}・d\overrightarrow {r}$, but the integral seems to be too complex and I don't think I'm doing this right.
I don't think neither Green's theorem or the Fundamental theorem of line integrals are applicable either because I don't think the curve is closed or $\overrightarrow {F}$ not seeming to be conservative.
Can someone explain this problem ?
I don't need much of the calculation. It's more of a conceptual idea.
It would also be great if someone could explain what it physically represents.
Best Answer
The curve $C$, as given, is a curve in the plane (it traces out the boundary of a 'wedge of pie'); the first thing to do is lift it to the surface using the equation $z = x^2 + y^2$.
The curve is given as three smooth pieces, so you will need to calculate three separate line integrals; these become ordinary integrals over $t$ via $$ d\vec{r} = \left(\frac{dx}{dt},\frac{dy}{dt},\frac{dz}{dt}\right)dt $$
(Edit: I'm guessing by your question that you know the following, but I'll add it for completeness. By Stokes' theorem, the line integral will be equal to the surface integral $$ \int\!\!\!\!\int_S (\nabla\times\vec{F})\cdot\vec{n}~ dA $$ where $S$ is the part of the surface bounded by the curve, $\vec{n}$ is a unit normal vector field, and $dA$ is the area element.)