On the Proof of the Capelli Identities

Using exterior calculus, we present detailed proofs of the classical Capelli identities in a purely computational manner. The proof of the fact that the Capelli elements are central is also given in a similar way. In the course of proofs of these two facts, one can easily see the mechanism of the multiplication formula of determinant with non-commutative entries. Simple treatments of the related facts from the Appendix of [HU] are also given.