\begin{align} \frac{d}{d\epsilon} I[\bar{u}+\epsilon\delta]  =& \left[ \delta_i(x)\frac{\partial F}{\partial m_{i,\mu}}  (u_\epsilon, \partial u_\epsilon, x) \right]_{\partial_\Omega} \\  &+\int_\Omega \delta_i(x) \left\{  \frac{\partial F}{\partial v_i}(u_\epsilon, \partial u_\epsilon, x)  -\frac{\partial}{\partial x^\mu} \left(  \frac{\partial F}{\partial m_{i,\mu}}(u_\epsilon, \partial u_\epsilon, x)  \right) \right\}\, d^dx \\ \end{align}
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