So I recently learned about flux and work of the field, but I cannot really solve this problem:
What is the flux of the field $F(x,y,z)=(x,y,z)$ through the surface of the cylinder $ \{ (x,y,z), x^2+y^2 \leq 0, |z| \leq 1 \}$
I am confused because usually in the examples i saw it was $x^2+y^2 = z^2 $ or a number. Could you please tell me how to solve this? I was thinking i can say it is less than $z$ or just say $x^2+y^2=0$.But i am not sure what to do after this.
Take a look at: $x^{2} + y^{2}\leq 0 \Leftrightarrow (x,y)=(0,0)$. Then, the surface is: $$ S= \left \{ (0,0,z)\in \mathbb{R}^{3} : -1\leq z\leq 1 \right \} $$ which is really a curve... You can parametrize it as: $$ \alpha (t) = (0,0, t), \quad -1\leq t\leq 1 $$
I'm sure you can take it from here. :)