$\mathbf{m}^{\mathfrak{a}}$. Thus, any function of $\mathbf{x}$ can be expressed as function of ${\boldsymbol{\xi}}^{\mathfrak{a}}$ and [*vice versa*]{}. For the plastic slip $\beta ^\mathfrak{a}$ caused by dislocations of the slip system ${\mathfrak{a}}$ we assume that their lines lie completely in the slip planes parallel to the $(\xi^{\mathfrak{a}}_1,\xi^{\mathfrak{a}}_2)$-plane. To describe the latter we introduce the loop function $l^{\mathfrak{a}}(\xi^{\mathfrak{a}}_1,\xi^{\mathfrak{a}}_2, \xi^{\mathfrak{a}}_3)$ such that its level curves $$\label{eq:2.18}
l^{\mathfrak{a}}(\xi ^{\mathfrak{a}}_1,\xi ^{\mathfrak{a}}_2,c _3)=c,$$ where $c_3$ and $c$ are constants, coincide with the dislocation lines. We denote by ${\boldsymbol{\nu}}^{\mathfrak{a}}$ and ${\boldsymbol{\tau}}^{\mathfrak{a}}$ the plane unit vectors normal and tangential to the dislocation line. From equation follow $${\boldsymbol{\nu}}^{\mathfrak{a}}=\frac{1}{\sqrt{(l^{\mathfrak{a}}_{;1})^2+(l^{\mathfrak{a}}_{;2})^2}}(l^{\mathfrak{a}}_{;1}\mathbf{s}^{\mathfrak{a}}+l^{\mathfrak{a}}_{;2}\mathbf{p}^{\mathfrak{a}}), \quad {\boldsymbol{\tau}}^{\mathfrak{a}}=\frac{1}{\sqrt{(l^{\mathfrak{a}}_{;1})^2+(l^{\mathfrak{a}}_{;2})^2}}(-l^{\mathfrak{a}}_{;2}\mathbf{s}^{\mathfrak{a}}+l^{\mathfrak{a}}_{;1}\mathbf{p}^{\mathfrak{a}}),$$ where the semicolon in indices denotes the partial derivatives of the loop function with respect to $\xi ^{\mathfrak{a}}_1$, $\xi ^{\mathfrak{a}}_2$, so these vectors lie in the slip planes parallel to the $(\xi _1,\xi _2)$-plane as expected.

For the plastic deformation the dislocation density tensor becomes $$\label{eq:2.20}
\mathbf{T}=-\mathbf{F}^p\times \nabla =\sum_{\mathfrak{a}=1}^n \mathbf{s}^{\mathfrak{a}}\otimes (\nabla \beta ^{\mathfrak{a}}\times \mathbf{m}^{\mathfrak{a}}).$$ To characterize the geometrically necessary dislocations belonging to one slip system we consider one term $\mathbf{T}^{\mathfrak{a}}=\mathbf{s}^{\mathfrak{a}}\otimes (\nabla \beta ^{\mathfrak{a}}\times \mathbf{m}^{\mathfrak{a}})$ in the sum . Let us choose an infinitesimal area $da$ with the unit normal vector ${\boldsymbol{\tau}}^{\mathfrak{a}}$ and compute the resultant Burgers vector of all excess 