I have a question that I got half through but can't finish it. If anyone could help I would appreciate it.
Question: let C1 be the straight line from (-1,0,0) to (1,0,0) and C2 the semi circle $x^2+y^2=1$, z=0 $y\le0$. Let S be the smooth surface joining C1 to C2 having an upward normal and let $$F=(\alpha x^2-z)i + (xy+y^3+z)j + \beta y^2(z+1)k$$ Find the values for $\alpha$ and $\beta$ for which $I=\int \int_S F\bullet dS$ is independent of the choice of S and find the value of I for these values of $\alpha$ and $\beta$.
I have found $\alpha$ by parametrization both the line and the curve separately and computing $\int_C F\bullet r'(t)dt$ which after I added the two ended up equalling -1/2.
My trouble is finding $\beta$... I have tried using the formulas for Stoke's theorem but non of them yield values for $\beta$ at the end of the computation. Again if anyone can help that would be awesome.