\delta A(\delta \Sigma_x) = \delta A(\Sigma_{x+\delta x }) = \delta A_{yz} = \delta y \cdot \delta z

\delta A(\delta \Sigma_y) = \delta  A(\Sigma_{y+\delta y }) = \delta A_{xz} = \delta x \cdot \delta z

\delta A(\delta \Sigma_z) = \delta A(\Sigma_{z+\delta z }) = \delta A_{xy} = \delta x \cdot \delta y

\frac{dm}{dt} \Big|_{\delta \Sigma} = {\bf j} \, {\bf \delta A} 

\frac{dm}{dt} \Big|_{\delta \Omega} =  \sum_{\alpha} j_{\alpha}A_{\alpha} + \delta \dot m_q

\frac{dm}{dt} \Big|_{\delta \Omega} =  
j_x|_{x}\cdot \delta A_{yz} - j_x|_{x+\delta x}\cdot \delta A_{yz} +j_y|_{y}\cdot \delta A_{xz} - j_y|_{y+\delta y}\cdot \delta A_{xz} +
j_z|_{z}\cdot \delta A_{xy} - j_z|_{z+\delta z}\cdot \delta A_{xy}  + \delta \dot m_q

\frac{dm}{dt \, \delta V} \Big|_{\delta \Omega} =  \frac {\partial  (\rho \, \phi)}{\partial t} = \frac{j_x|_x - j_x|_{x+\delta x}}{\delta x} + \frac{j_y|_y - j_y|_{y+\delta y}}{\delta y} + \frac{j_z|_z - j_z|_{z+\delta z}}{\delta z} + \frac{\delta \dot m_q}{\delta V}

\frac{\partial (\rho \phi)}{\partial t} = - \nabla \, {\bf j} + \frac{\delta \dot m_q}{\delta V}

\frac{\partial (\rho \phi)}{\partial t} + \nabla \, {\bf j} =  \frac{\delta \dot m_q}{\delta V} 

\frac{\delta \dot m_q}{\delta V} = \sum_k \rho_k \cdot q_k(t) \cdot \delta({\bf r}-{\bf r}_k)

\frac{\delta \dot m_q}{\delta V} = \sum_k \rho_k \cdot q_k(t) \cdot \delta({\bf r}-{\bf r}_k) =  \rho(t,{\rm r}) \cdot \sum_k q_k(t) \cdot \delta({\bf r}-{\bf r}_k)

\frac{\partial (\rho \phi)}{\partial t} + \nabla \, {\bf j} = \rho \cdot  \sum_k q_k(t) \cdot \delta({\bf r}-{\bf r}_k)

\frac{\partial (\rho \phi)}{\partial t} + \nabla \, ( \rho \, {\bf u}) = \rho \cdot  \sum_k q_k(t) \cdot \delta({\bf r}-{\bf r}_k)