${\displaystyle \sum _{i=1}^{N_{1}}\left(I_{\mathrm {m} }\right)_{i}=\sum _{j=1}^{N_{2}}\left(I_{\mathrm {m} }\right)_{j}=0\,\!}$

${\displaystyle \sum _{i=1}^{N_{1}}\left(\mathbf {j} _{\mathrm {m} }\right)_{i}=\sum _{j=1}^{N_{2}}\left(\mathbf {j} _{\mathrm {m} }\right)_{j}=\mathbf {0} \,\!}$

${\displaystyle \sum _{i=1}^{N_{1}}\mathbf {p} _{i}=\sum _{i=1}^{N_{2}}\mathbf {p} _{j}\,\!}$

which can be written in equivalent ways, most useful forms are:

${\displaystyle \sum _{i=1}^{N_{1}}m_{i}\mathbf {v} _{i}=\sum _{j=1}^{N_{2}}m_{j}\mathbf {v} _{j}}$

${\displaystyle \sum _{i=1}^{N_{1}}\left(I_{\mathrm {p} }\right)_{i}=\sum _{j=1}^{N_{2}}\left(I_{\mathrm {p} }\right)_{j}=0\,\!}$

Momentum current density conservation

${\displaystyle \sum _{i=1}^{N_{1}}\left(\mathbf {j} _{\mathrm {p} }\right)_{i}=\sum _{j=1}^{N_{2}}\left(\mathbf {j} _{\mathrm {p} }\right)_{j}=\mathbf {0} \,\!}$

${\displaystyle \mathbf {L} _{\mathrm {system} }=\sum _{i=1}^{N_{1}}\mathbf {L} _{i}=\sum _{j=1}^{N_{2}}\mathbf {L} _{j}\,\!}$

which can be written in equivalent ways, most useful forms are:

${\displaystyle \mathbf {L} _{\mathrm {system} }=\sum _{i=1}^{N_{1}}\left(\mathbf {I} _{\mathrm {ab} }{\boldsymbol {\omega }}_{\mathrm {b} }\right)_{i}=\sum _{j=1}^{N_{2}}\left(\mathbf {I} _{\mathrm {ab} }{\boldsymbol {\omega }}_{\mathrm {b} }\right)_{j}\,\!}$

${\displaystyle \mathbf {L} _{\mathrm {system} }=\sum _{i=1}^{N_{1}}\mathbf {r} _{i}\times \mathbf {p} _{i}=\sum _{j=1}^{N_{2}}\mathbf {r} _{j}\times \mathbf {p} _{j}\,\!}$

${\displaystyle \mathbf {L} _{\mathrm {system} }=\sum _{i=1}^{N_{1}}m_{i}\left(\mathbf {r} _{i}\times \mathbf {v} _{i}\right)=\sum _{j=1}^{N_{2}}m_{j}\left(\mathbf {r} _{j}\times \mathbf {v} _{j}\right)\,\!}$

${\displaystyle E_{\mathrm {system} }=\sum _{i}T_{i}+\sum _{j}V_{j}\,\!}$

or simply

${\displaystyle E=T+V\,\!}$

${\displaystyle E_{\mathrm {system} }=\sum _{i=1}^{N_{1}}\left(T_{i}+V_{i}\right)=\sum _{j=1}^{N_{2}}\left(T_{j}+V_{j}\right)\,\!}$

${\displaystyle \sum _{i}P_{i}+\sum _{j}P_{j}=0\,\!}$

Intensity conservation

${\displaystyle \sum _{i}I_{i}+\sum _{j}I_{j}=0\,\!}$

${\displaystyle \sum _{i=1}^{N_{1}}I_{i}=\sum _{j=1}^{N_{2}}I_{j}=0\,\!}$

Electric current density conservation

${\displaystyle \sum _{i=1}^{N_{1}}\mathbf {J} _{i}=\sum _{j=1}^{N_{2}}\mathbf {J} _{j}=\mathbf {0} \,\!}$

${\displaystyle j_{\mathrm {m} }\,\!}$

${\displaystyle \rho \,\!}$ = Volume mass density

${\displaystyle \mathbf {u} \,\!}$ = velocity field of fluid

${\displaystyle \mathbf {A} \,\!}$ = cross-section

${\displaystyle I\,\!}$

${\displaystyle \mathbf {J} \,\!}$ = Electric current density

${\displaystyle \rho \,\!}$ = Volume electric charge density

${\displaystyle \mathbf {u} \,\!}$ = velocity of charge carriers

${\displaystyle \mathbf {A} \,\!}$ = cross-section

${\displaystyle \mathbf {j} \,\!}$

${\displaystyle P=P(x,t)\,\!}$ = probability density function