I want to check my understanding of the symmetry arguments that allow for E to come out Gauss's Law and the symmetry arguments that allow for E vector *dA to become EdA. Specifically for an infinitely long cylinder.
The Attempt at a Solution
So, for an infinitely long cylinder, we know that the orientation does not change when translated or rotated about an axis. The E field also is perpendicular to the curved surface of the cylinder, and so there is cylindrical symmetry (obviously). We know that the E field always points in the same direction, so we can essentially 'get rid of' the vector. Next, we assume that the E field is the same magnitude at all points (constant) on the cylinder (there is nothing to suggest otherwise), we can pull it out of the integral, so the integral becomes E∫dA.
But why can the E field at the ends of the cylinder be ignored (which allows us to make all the previous assumptions)? Is it because they are in different directions, so they cancel out?