Evaluate $$ \int_\gamma F\cdot dr$$ where $ F=(ye^x,e^x+x^3,z^5)$ and $ \gamma\ $ is the intersection between $x^2+y^2=1 $ and $z=2xy$ oriented in such a way that the orthogonal projection on the $xy$ plane is oriented counterclockwise.
Here is how I have tried to solve this problem: I used Stokes and first calculated the curl of F which is $(0,0,3x^2)$ and then doted this with unit normal but it become zero, however the answer should not become zero.
What am I doing wrong ? Any suggestion would be great , Thanks
Parameterize the surface as $ \ r(\rho, t) = (\rho \cos t, \rho \sin t, \rho^2 \sin2t)$
$0 \leq \rho \leq 1, 0 \leq t \leq 2\pi$.
$r_{\rho} \times r_{t} = (2 \rho^2 \sin t, 2 \rho^2 \cos t, \rho)$
$\nabla \times \vec{F} = (0, 0, 3 \rho^2 \cos^2t)$
So the integral should be,
$\displaystyle \int_0^{2\pi} \int_0^1 (0,0,3 \rho^2 \cos^2t) \cdot (2 \rho^2 \sin t, 2 \rho^2 \cos t, \rho) \ d\rho \ dt$
$ = \displaystyle \int_0^{2\pi} \int_0^1 3 \rho^3 \cos^2t \ d\rho \ dt = \frac{3 \pi}{4}$