For any Hermitian operator O ^ ^ under an inner product ( f , g ) , we have ( f , O ^ ^ ⁡ ⁡ g ) = ( O ^ ^ ⁡ ⁡ f , g ) by definition, and the Rayleigh-Carson reciprocity theorem is merely the vectorial version of this statement for this particular operator J = O ^ ^ ⁡ ⁡ E : that is, ( E 1 , O ^ ^ ⁡ ⁡ E 2 ) = ( O ^ ^ ⁡ ⁡ E 1 , E 2 ) . The Hermitian property of the operator here can be derived by integration by parts. For a finite integration volume, the surface terms from this integration by parts yield the more-general surface-integral theorem above. In particular, the key fact is that, for vector fields F and G , integration by parts (or the divergence theorem) over a volume V enclosed by a surface S gives the identity: ∫ ∫ V F ⋅ ⋅ ( ∇ ∇ × × G ) d V ≡ ≡ ∫ ∫ V ( ∇ ∇ × × F ) ⋅ ⋅ G d V − − ∮ ∮ S ( F × × G ) ⋅ ⋅ d A .