I am a little confused about how to generally go about applying Stokes's Theorem to cylinders, in order to calculate a line integral. If, for example you have a cylinder whose height is about the z axis, I get perfectly well how to parameterize the x and y components, using polar coordinates, for example, but what about z (and thus dz)? If you have a vector field that is dependent on z, such as F= (z,x,yz) (it doesn't matter which, I am just listing one for clarity) and you want to replace with polar coordinates, what approach do you take with z, when projecting your surface about a closed loop? What confuses me is if you have a sphere, or something similar, you find the shadow of the surface, which is effectively the z-value where the circumference is widest-- but with the cylinder, all z-values are at locations with equal circumference about the x-y plane. Is it the bottom? Top? Somewhere in between? Thanks!!!