  Serious mathematical defect in the important kinematics theorem known in
continuum mechanics as Convection (or Transport) Theorem is reported. We claim
that the traditional demonstration does not take into account a special
constraint on integrand functions given in Lagrangian representation. Thus, we
put in doubt that the traditional procedure for the transition from integral
formulations of physical laws of some classical field theories to their
differential form is mathematically rigorous. Reconsidered formulation shows
the way how the system of fundamental differential equations of continuum
mechanics and some other field theories could be modified without any change of
the original integral formulation. The continuity equation and the differential
form of the equation of motion for continuous media are discussed as examples
of modification.
