I have been working on a Stokes' Theorem problem. (Thank you @Prasiortle for your help in [this problem.][1]) This is the question and my work so far. The reason why I'm feeling hesitant is that although my integral is very neat, the final answer seems to be messier - which is not characteristic of these textbook problems. I don't see my error - but I don't really know what I'm doing yet. Did I mess up over here? I am taking an independent study course, so I don't have a teacher that I can ask these questions to - I appreciate all advice!
Thanks!
*Edit: I had had this down in pictures, but I will change it to mathJax. I saw that I had forgotten to put the norm of my normal vector in to the dS, so I added that. This makes my answer more manageable than it was originally, but I'm still not sure (especially because it is a large number). Here is my work:
Use Stokes’ Theorem to evaluate $\int_{C}\ F\bullet\ dr$ where $C$ is the boundary of the portion of the paraboloid $x=y^2+z^2$ with $x\le4,$ n to the back, $F =<yz,y-4,2xy>$
Parametrize $C: x=r^2,y=r\cos{\left(\theta\right)},\mathrm{\mathrm{\ }and\ }z=r\sin{\left(\theta\right)}$ and so: $\langle r^2,r\ cos(\theta),r\ sin(\theta)\rangle\\ C_r=\left\langle 2r,cos{\left(\theta\right)},sin{\left(\theta\right)}\right\rangle\\ C_\theta=\left\langle0,-rsin{\left(\theta\right)},rcos{\left(\theta\right)}\right\rangle\\ \left|\begin{matrix}i&j&k\\2r&cos\left(\theta\right)&sin\left(\theta\right)\\0&-rsin\left(\theta\right)&rcos\left(\theta\right)\\\end{matrix}\right|=\frac{\left\langle r,-2r^2cos{\left(\theta\right)},-2r^2sin{\left(\theta\right)}\right\rangle}{r\sqrt{1+4r^2}}\\ \nabla\times F=\left\langle2x,-y,-z\right\rangle=\left\langle2r^2,-rcos{\left(\theta\right)},-rsin{\left(\theta\right)}\right\rangle\\ \left(\nabla\times F\right)\bullet{n}=\frac{4r^2}{\sqrt{4r^2+1}}\\ \int_{0}^{2\pi}\int_{0}^{4}{\frac{4r^2}{\sqrt{4r^2+1}}\left(r\sqrt{1+4r^2}\right)drd\theta}=512\pi$