Find the flux of F across the surface sigma by expressing sigma parametrically.
F(x,y,z)=xi+yj+zk; sigma is the portion of the cylinder x^2+z^2=1 between the planes y=1 and y=-2, oriented by outward unit normals.
Find the flux of F across the surface sigma by expressing sigma parametrically.
F(x,y,z)=xi+yj+zk; sigma is the portion of the cylinder x^2+z^2=1 between the planes y=1 and y=-2, oriented by outward unit normals.
Parametrize the given surface as $\displaystyle x = 1 cos \theta , y = t$ and $\displaystyle z = 1 sin \theta$. So we have the limits as $\displaystyle 0 \leq \theta \leq 2\pi$ and $\displaystyle 1 \leq t \leq -2$.
The surface is now $\displaystyle \Phi(\theta,t) = (cos\theta, t ,sin\theta)$.
To calculate the normal to the surface, we have, $\displaystyle \left(\frac{d\Phi}{d\theta}(\theta,t) \times \frac{d\Phi}{dt}(\theta,t)\right)$.
$\displaystyle \frac{d\Phi}{d\theta}(\theta,t) = (-sin\theta, 0 , cos\theta)$.
$\displaystyle \frac{d\Phi}{dt}(\theta,t) = (0, 1, 0)$
$\displaystyle \frac{d\Phi}{d\theta}(\theta,t) \times \frac{d\Phi}{dt}(\theta,t) = \begin{array}{|l c r|}i&j&k\\-sin\theta&0&cos\theta\\0&1&0\end{array} = (-cos\theta, 0 , -sin\theta)$
The flux is now calculated as $\displaystyle \int_1^{-2}\int_0^{2\pi} F(\Phi(\theta,t)) . \left(\frac{d\Phi}{d\theta}(\theta,t) \times \frac{d\Phi}{dt}(\theta,t)\right) d\theta \ dt$
Could you substitute and complete this?
The term
has to be in the same dirction os the orientation vector n . since n is outward we would use [cos(theta),0, sin(theta)]
Also the outer limit of integration shoud be -2 to 1
Of course these cancel and you would have the correct answer --if thats what you had in mind I apologize for my remarks.