I am helping a friend to solve a calculus list. I have solved all the questions, except, this one.
Should you know how to solve, please, let me know!
Solve $\int_C \vec F \cdot dr$ by Stokes theorem, where $\vec F =(xz,xy,y^2)$. $C$ is the border of the surface of the cylinder $z = 4-x^2$, bounded by the planes $x=2, y=3$ on the first octant.
I know that:
Now I am facing problems to find a normal vector for making this integral easy to solve.
I know that most likely it is $r = i + j + k$.
However, I have no idea how to confirm that. Any help is appreciated!
You can parametrize the parabolic cylinder surface as,
$r(x,y) = (x, y, 4-x^2)$ and $r'_x \times r'_y = (2x, 0, 1)$
$Curl \vec F = \nabla \times \vec F = (2y, x, y)$
So applying Stokes' thoerem,
$\displaystyle \int_C \vec F \cdot dr = \iint_S (\nabla \times \vec F) \cdot \vec n \ dS \ = \int_0^3 \int_0^2 (2y, x, y) \cdot (2x, 0, 1) \ dx \ dy $