On the Correct Formulation of the First Law of Thermodynamics

The critical analysis of the generally accepted formulation of the first law of thermodynamics is proposed. The purpose of the analysis is to prove that the standard formulation contains a mathematical error and to offer the correct formulation. The correct formulation is based on the concepts of function and differential of function. Really, if internal energy $U$of a system is a function of two independent variables $Q=Q(t)$ (describing of the thermal form of energy) and $R=R(t)$ (describing non-thermal form of energy), then the correct formulation of the first law of thermodynamics is: $\frac{dU(Q,R)}{dt}=\left( {\frac{\partial U}{\partial Q}} \right)_R \frac{dQ}{dt}+\left( {\frac{\partial U}{\partial R}} \right)_Q \frac{dR}{dt}$, where $t$ and $-{\left( {\frac{\partial U}{\partial R}} \right)_Q } \mathord{\left/ {\vphantom {{\left( {\frac{\partial U}{\partial R}} \right)_Q } {\left( {\frac{\partial U}{\partial Q}} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\frac{\partial U}{\partial Q}} \right)}_R $ are time and measure of mutual transformation of forms of energy, correspondingly. General conclusion: standard thermodynamics is incorrect.