Electric field by a surface

Question

I have tried to demonstrate the classic result that afirms that the electric field in a charged surface is proportional to the surface densitity in each point. In the books I read they assume, when using Gauss theorem with a cylinder, that the field in the lateral is orthogonal to the normal vector. It is clear to me that the field is normal to the surface, but I think that is false in the case of the cylinder, so I dont get to solve the problem.

Answer 1

I think the assumption has to be that this is an electrostatic field and that the surface is the surface of a conductor. The electric potential is constant over the conductor, otherwise the charges would move. The electric field, which is the gradient of the potential, is orthogonal to surfaces of constant potential.