  A multiple integral functional is equivalent to a curvilinear integral
functional, if the domain is a hyper-parallelepiped, but equivalence is only
theoretical. The introduction of this kind of functionals in multitime optimal
control problems, particularly in multitime differential games, is due to
recent works of Udriste research group. The purpose of this paper is to
introduce those ingredients that are necessary to formulate and to prove
theorems about multitime differential games based on a multiple integral
functional and an $m$-flow as constraint. The most important idea is to use a
generating vector field for basic functions. The original results include:
fundamental properties of multitime upper and lower values, viscosity solutions
of multitime (dHJIU) PDEs, representation formula of viscosity solutions for a
multitime (dHJ) PDE, and max-min representations.
