For any Hermitian operator formula_33 under an inner product formula_43, we have formula_44 by definition, and the Rayleigh-Carson reciprocity theorem is merely the vectorial version of this statement for this particular operator formula_45: that is, formula_46. 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 formula_39 and formula_40, integration by parts (or the divergence theorem) over a volume "V" enclosed by a surface "S" gives the identity: