  An analytic approach and description are presented for the moduli cotangent
sheaf for suitable stable curve families including noded fibers. For sections
of the square of the relative dualizing sheaf, the residue map at a node gives
rise to an exact sequence. The residue kernel defines the vanishing residue
subsheaf. For suitable stable curve families, the direct image sheaf on the
base is locally free and the sequence of direct images is exact. Recent work of
Hubbard-Koch and a formal argument provide that the direct image sheaf is
naturally identified with the moduli cotangent sheaf. The result generalizes
the role of holomorphic quadratic differentials as cotangents for smooth curve
families. Formulas are developed for the pairing of an infinitesimal opening of
a node and a section of the direct image sheaf. Applications include an
analytic description of the conormal sheaf for the locus of noded stable curves
and a formula comparing infinitesimal openings of a node. The moduli action of
the automorphism group of a stable curve is described. An example of plumbing
an Abelian differential and the corresponding period variation is presented.
