The integraltheorem of the vector field energy is derived in a covariant way, according towhich under certain conditions the potential energy of the system’s field turnsout to be half as large in the absolute value as the field’s kinetic energyassociated with the four-potential of the field and the four-current of thesystem’s particles. Thus, the integral theorem turns out to be the analogue ofthe virial theorem, but with respect to the field rather than to the particles.Using this theorem, it becomes possible to substantiate the fact thatelectrostatic energy can be calculated by two seemingly unrelated ways, eitherthrough the scalar potential of the field or through the stres energy-momentumtensor of the field. In closed systems, the theorem formulation is simplifiedfor the electromagnetic and gravitational fields, which can act at a distanceup to infinity. At the same time for the fields acting locally in the matter,such as the acceleration field and the pressure field, in the theoremformulation it is necessary to take into account the additional term withintegral taken over the system’s surface. The proof of the theorem for an idealrelativistic uniform system containing non-rotating and randomly movingparticles shows full coincidence in all significant terms, particularly for theelectromagnetic and gravitational fields, the acceleration field and the vectorpressure field.