On Mathematical Defect in Demonstration of Convection Theorem:   Implications For Continuum Mechanics and Other Classical Field Theories
  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. 
R. Smirnov-Rueda