First, for the terms, A surface is essential if it is both incompressible and boundary-incompressible.
I want to show that A surface S is essential if ans only if each components of S is essential.
My idea, for the only if part, is to suppose a component of S, S_i, is not essential. So it either has compressing disk, or not boundary-incompressible. In either case, the disk D is a compressing disk for S, or it bounds a disk on the boundary of S. But it is a contradiction, since S is essential.
Is it correct?
And, for the if part, I have no idea. Can any one help me with this?