On a surface Σ with an area form, the flux homomorphism measures the flow of a nullhomotopic area-preserving diffeomorphism across first homology cycles of Σ. We consider three different ways of extending the flux homomorphism to a map defined on all area-preserving diffeomorpisms, not just the nullhomotopic ones. The differences of these constructions descend to maps on the mapping class group, and are closely related to the Johnson homomorphism.