  We give a cohomological formula for the index of a fully elliptic
pseudodifferential operator on a manifold with boundary. As in the classic case
of Atiyah-Singer, we use an embedding into an euclidean space to express the
index as the integral of a cohomology class depending in this case on a
noncommutative symbol, the integral being over a $C^\infty$-manifold called the
singular normal bundle associated to the embedding. The formula is based on a
K-theoretical Atiyah-Patodi-Singer theorem for manifolds with boundary that is
drawn from Connes' tangent groupoid approach.
