Find the flux of the vector field \({\bf F}= \bigg\langle \f…

Find the flux of the vector field \({\bf F}= \bigg\langle \frac{-y}{x^2+y^2},\frac{x}{x^2+y^2},1 \bigg\rangle\) over the cylinder of radius 1, centered around the \(z\)-axis and between \(z=0\) and \(z=1\) with the disks at the top and bottom included so that the surface is closed.

Compute \(\oint_{C}{\bf F}\cdot d{\bf r}\) where \(C\) is th…

Compute \(\oint_{C}{\bf F}\cdot d{\bf r}\) where \(C\) is the curve parametrized by \({\bf r}(t)= \langle \cos t,\sin t, \cos t+\sin t \rangle, \quad 0\leq t\leq 2\pi\) and where \({\bf F}=(xy+z){\bf i}+z^2{\bf j}+xyz{\bf k}\) [Hint: The curve lies on the surface \(z=x+y\)]