J. Carl F. Gauss (1777-1855)

An E field and a surface.

We can define a vector area differential dA for each point on a surface.

Since there is no net charge anywhere inside a conductor, there is no E field through any arbitrary closed Gaussian surface anywhere inside, so there cannot be an E field anywhere inside.

The electric field is zero everywhere inside a conductor; all charge must reside on its surfaces.

The electric field is everywhere perpendicular to the surface of a conductor, and cannot penetrate the surface. Induced charge is confined to the surface.