Important recent research by Samuel Preston and Ansley Coale (1982) extends the Lotka system of stable population equations (Lotka, 1939) to any population. Here we present an alternative general system and describe its duality with the Preston-Coale system. We derive these results by considering the calculus of change on the surface of population density defined over age and time. We show that analysis of this Lexis surface leads to all the known fundamental relationships of the dynamics of single-region human populations, as well as to some interesting new relationships.