= Geiringer velocity relations
{c}
{title2=$du-v\,d\phi=0\ \text{on }\alpha,\qquad dv+u\,d\phi=0\ \text{on }\beta$}
For an <associated flow rule> in the reduced plane-strain yield coordinates, the <rate-of-strain tensor> has zero extension along both <slip line> directions. Write velocity as $u e_\alpha+v e_\beta$, with the local orthonormal axes turning through angle $\phi$. Differentiating along an $\alpha$ curve gives longitudinal extension $du-v\,d\phi$; along a $\beta$ curve it gives $dv+u\,d\phi$. Setting these extensions to zero proves the relations for the corresponding anisotropic <yield surface> parameterization.
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