Corollary of the Uniqueness Theorem in Electrostatics

[PhDeezNutz]

Following my instructor's notes the statement of the Uniqueness Theorem(s) are as follows

"If $\rho_{inside}$ and $\phi_{boundary}$ (OR $\frac{d \phi_{boundary}}{dn}$ ) are known then $\phi_{inside}$ is uniquely determined"

A few paragraphs later the notes state

"For the field inside S (a surface), knowing $\phi_{boundary}$(OR $\frac{d \phi_{boundary}}{dn}$) everywhere on S is as good as knowing all the outside charges; it carries all the same information about their effects"

I don't see how this follows from the statement of the Uniqueness Theorem. If anything it **seems to me** that the instructor is saying the converse of the Uniqueness Theorem while flipping definitions of "inside" and "outside".

"If $\phi_{boundary}$ (OR $\frac{d \phi_{boundary}}{dn}$ ) are known on surface S then $\rho_{outside}$ is uniquely determined"

Can anyone help me

1) decipher what my instructor is trying to say

2) Offer help in the way of a formal proof or a convincing physical argument

Any help would be appreciated. Thanks in advanced.

 

[kuruman]

Say you have a closed surface in space. It forms a clear boundary between two regions of space. Region I is has two boundaries, one at infinity and one at the closed surface. Region II has one boundary at the surface. I think your instructor is defining "inside" as the region in which $\rho$ is known, either I or II. BTW, if $\frac{d\phi_{boundary}}{dn}$ is known, the potential is determined to within a constant. Some people will not call that "uniquely".