dislocations of the system ${\mathfrak{a}}$, whose dislocation lines cross this area at right angle $$\mathbf{b}^{\mathfrak{a}}_r=\mathbf{T}^{\mathfrak{a}}\cdot {\boldsymbol{\tau}}^{\mathfrak{a}}\, da=-\mathbf{s}^{\mathfrak{a}}\, (\nabla \beta ^{\mathfrak{a}}\cdot {\boldsymbol{\nu}}^{\mathfrak{a}})da=-\mathbf{s}^{\mathfrak{a}}\, \partial _{\nu ^{\mathfrak{a}}}\beta ^{\mathfrak{a}}\, da.$$ This resultant Burgers vector can be decomposed into the sum of two vectors $$\mathbf{b}^{\mathfrak{a}}_r=\mathbf{b}^{\mathfrak{a}}_{r\perp}+\mathbf{b}^{\mathfrak{a}}_{r\parallel}=-({\boldsymbol{\nu}}^{\mathfrak{a}}s^{\mathfrak{a}}_\nu +{\boldsymbol{\tau}}^{\mathfrak{a}}s^{\mathfrak{a}}_\tau )\partial _{\nu ^{\mathfrak{a}}}\beta^{\mathfrak{a}}\, da,$$ where $s^{\mathfrak{a}}_\nu =\mathbf{s}^{\mathfrak{a}}\cdot {\boldsymbol{\nu}}^{\mathfrak{a}}$ and $s^{\mathfrak{a}}_\tau =\mathbf{s}^{\mathfrak{a}}\cdot {\boldsymbol{\tau}}^{\mathfrak{a}}$ are the projections of the slip vector onto the normal and tangential direction to the dislocation line, respectively. This allows us to define two scalar densities of edge and screw dislocations of the corresponding slip system $$\label{eq:2.21}
\begin{split}
\rho^{\mathfrak{a}}_\perp =\frac{|\mathbf{b}^{\mathfrak{a}}_{r\perp}|}{b} =\frac{1}{b}|s^{\mathfrak{a}}_\nu \partial _{\nu ^{\mathfrak{a}}}\beta^{\mathfrak{a}}| =\frac{1}{b}\left| \frac{l^{\mathfrak{a}}_{;1}(\beta ^{\mathfrak{a}}_{;1}l^{\mathfrak{a}}_{;1}+\beta ^{\mathfrak{a}}_{;2}l^{\mathfrak{a}}_{;2})}{(l^{\mathfrak{a}}_{;1})^2+(l^{\mathfrak{a}}_{;2})^2}\right| , 
\\
\rho^{\mathfrak{a}}_\parallel =\frac{|\mathbf{b}^{\mathfrak{a}}_{r\parallel }|}{b} =\frac{1}{b}|s^{\mathfrak{a}}_\tau \partial _{\nu ^{\mathfrak{a}}}\beta^{\mathfrak{a}}|=\frac{1}{b}\left| \frac{l^{\mathfrak{a}}_{;2}(\beta ^{\mathfrak{a}}_{;1}l^{\mathfrak{a}}_{;1}+\beta ^{\mathfrak{a}}_{;2}l^{\mathfrak{a}}_{;2})}{(l^{\mathfrak{a}}_{;1})^2+(l^{\mathfrak{a}}_{;2})^2}\right| ,
\end{split}$$ We see that the dislocation densities $\rho^{\mathfrak{a}}_\perp $ and $\rho^{\mathfrak{a}}_\parallel $ depend only on the partial derivatives $\beta ^{\mathfrak{a}}_{;\alpha }$ and $l^{\mathfrak{a}}_{;\alpha }$, $\alpha =1,2$. For the moving dislocations we allow the loop functions to depend on time $t$ such that the level curves $$l^{\mathfrak{a}}(\xi ^{\mathfrak{a}}_1,\xi ^{\mathfrak{a}}_2,c _3,t)=c,$$ with $c_3$ and $c$ being constants, coincide with the dislocation lines during their motion. Similar to the single slip we introduce the normal velocities of dislocation lines as follows $$v^{\mathfrak{a}}_\nu =\mathbf{v}^{\mathfrak{a}}\cdot {\boldsymbol{\nu}}^{\mathfrak{a}}=-\frac{\dot{l}^{\mathfrak{a}}}{\sqrt{(l^{\mathfrak{a}}_{;1})^2+(l^{\mathfrak{a}}_{;2})^2}},$$ with $\dot{l}=l_{,t}$. These kinematic quantities 