I have a functional of following form,where u(x,y,z),v(x,y,z)and w(x,y,z) are three dimensional function. $$\int \mathcal{L}(u,v,w,u_{x},u_{y},u_{z},v_{x},v_{y},v_{z},w_{x},w_{y},w_{z},x,y,z)\, dz$$
i want it to minimize for u,v,w
Then please let me know if following euler equations are right or not....
\begin{aligned} \frac{\partial \mathcal{L}}{\partial u} - \frac{\mathrm{d} }{\mathrm{d} z} \bigg( \frac{\partial \mathcal{L}}{\partial u_{z}} \bigg) = 0 \\ \frac{\partial \mathcal{L}}{\partial v} - \frac{\mathrm{d} }{\mathrm{d} z} \bigg( \frac{\partial \mathcal{L}}{\partial v_{z}} \bigg) = 0 \\ \frac{\partial \mathcal{L}}{\partial w} - \frac{\mathrm{d} }{\mathrm{d} z} \bigg( \frac{\partial \mathcal{L}}{\partial w_{z}} \bigg) = 0 \end{aligned}
or Should it be in the following form \begin{aligned} \frac{\partial \mathcal{L}}{\partial u} - \frac{\mathrm{d} }{\mathrm{d} z} \bigg( \frac{\partial \mathcal{L}}{\partial u_{x}}\bigg)-\frac{\mathrm{d} }{\mathrm{d} z} \bigg( \frac{\partial \mathcal{L}}{\partial u_{y}} \bigg)-\frac{\mathrm{d} }{\mathrm{d} z} \bigg( \frac{\partial \mathcal{L}}{\partial u_{z}}\bigg) = 0 \end{aligned}