  The aim of the present paper is to introduce a first order approach to the
abstract concept of boundary triples for Laplace operators. Our main
application is the Laplace operator on a manifold with boundary; a case in
which the ordinary concept of boundary triples does not apply directly. In our
first order approach, we show that we can use the usual boundary operators also
in the abstract Green's formula. Another motivation for the first order
approach is to give an intrinsic definition of the Dirichlet-to-Neumann map and
intrinsic norms on the corresponding boundary spaces. We also show how the
first order boundary triples can be used to define a usual boundary triple
leading to a Dirac operator.
