Vector Calculus Question

Question

Let Q be the region bounded by the cylinder y² + z² = 9 and the planes x = 0, x = 2. The boundary of Q is a closed surface S (positively oriented). Suppose you have a vector field F = 3xi + 3yj + 3zk.

a. Find the flux of F over S, showing work. You can do this either by setting up and evaluating the flux over each part of S, or (if appropriate) finding the flux using a vector calculus theorem.

b. Show that your parametrizations of S are correct by using them to sketch S in GeoGebra.

Answer 1

(a) Use the Divergence Theorem. The divergence of the field is elementary and the volume of integration is just a cylinder with axis of symmetry the x axis. The volume integral is very easy to take.