  We generalize the Poincare-Hopf theorem sum_v i(v) = X(G) to vector fields on
a finite simple graph (V,E) with Whitney complex G. To do so, we define a
directed simplicial complex as a finite abstract simplicial complex equipped
with a bundle map F: G to V telling which vertex T(x) in x dominates the
simplex x. The index i(v) of a vertex v is defined as X(F^-1(v)), where X is
the Euler characteristic. We get a flow by adding a section map F: V to G. The
resulting map T on G is a discrete model for a differential equation x'=F(x) on
a compact manifold. Examples of directed complexes are defined by Whitney
complexes defined by digraphs with no cyclic triangles or gradient fields on
finite simple graphs defined by a locally injective function. The result
extends to simplicial complexes equipped with an energy function H:G to Z that
implements a divisor. The index sum is then the total energy.

Combinatorics;Discrete Mathematics;Dynamical Systems