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.
Let $S$ be a smooth surface joining $C_1$ to $C_2$ having an upward normal and $S^*$ be any other smooth surface spanned on the contour $C={C_1}\cup{C_2}.$ Integral over surface $S$ $$ I=\iint\limits_{S^{+}} \vec{F}d\vec{S}=\iint\limits_{S^{+}} ({\vec{F}}\cdot{\vec{n})\, dS} $$ will be independent upon the choice of $S$ if $$\iint\limits_{S^{+}} \vec{F}\,d\vec{S}=\iint\limits_{{S^{*}}^{+}} \vec{F}\,d\vec{S}.$$ This is equivalent to $$\iint\limits_{S^{+}\cup{\,{S^{*}}^{-}}} \vec{F}\,d\vec{S}=0.$$ By the divergence theorem, $$\iint\limits_{S^{+}\cup{\,{S^{*}}^{-}}} \vec{F}\,d\vec{S} = \iiint\limits_{V} {(\vec{\nabla} \cdot \vec{F}})\,dV =0,$$ therefore ${\vec{\nabla}} \cdot {\vec{F}}=0$ in $V.$ For given ${\vec{F}}=(\alpha x^2-z)\vec{i}+ (xy+y^3+z)\vec{j} + \beta y^2(z+1)\vec{k}$ $${\vec{\nabla}} \cdot {\vec{F}}=x(2\alpha-1)+y^2(\beta+3)=0 \;\;\; \Rightarrow \;\;\; \begin{cases} 2\alpha-1=0 \\ \beta+3=0 \end{cases} \;\;\; \Rightarrow \;\;\; \begin{cases} \alpha=\dfrac{1}{2}, \\ \beta=-3. \end{cases}$$ Finally, if we choose ${S^{*}}^{+},$ bounded by $C={C_1}\cup{C_2}$ on the $XOY$-plane and oriented by $\vec{n}_{*}=0\vec{i}+0\vec{j}+1\vec{k} $, then on ${S^{*}}^{-}$ $$\vec{F}=\left.(\alpha x^2-z)\vec{i}+ (xy+y^3+z)\vec{j} + \beta y^2(z+1)\vec{k}\right|_{z=0}=\alpha x^2\vec{i}+ (xy+y^3)\vec{j} + \beta y^2\vec{k} $$ and $${\vec{F}}\cdot{\vec{n}_{*}}=\beta y^2=-3y^2. $$ Then $$\iint\limits_{S^{+}} \vec{F}\,d\vec{S}=\iint\limits_{{S^{*}}^{+}} \vec{F}\,d\vec{S}=\iint\limits_{\operatorname{int}C} (-3y^2) \,dx\,dy.$$