  We use the scattering matrix approach to derive generalized Bardeen-like
formulae for the conductances between the contacts of a phase-coherent
multiprobe conductor and a tunneling tip which probes its surface. These
conductances are proportional to local partial densities of states, called
injectivities and emissivities. The current and the current fluctuations
measured at the tip are related to an effective local non-equilibrium
distribution function. This distribution function contains the
quantum-mechanical phase-coherence of the charge carriers in the conductor and
is given as products of injectivities and the Fermi distribution functions in
the electron reservoirs. The results are illustrated for measurements on
ballistic conductors with barriers and for diffusive conductors.
