I have been asked to find the Euler-Lagrange equation for the variational problem with the fundamental integral:
$$
\iiint_G[x^2 u_x^2(x,y,z)-y^2u_y^2(x,y,z)+z^2u_z^2(x,y,z)+f(x,y,z)]dx dy dz
$$
where $u=u(x,y,z)$ and $u_x=\frac{\partial u}{\partial x}, u_y=\frac{\partial u}{\partial y}, u_z=\frac{\partial u}{\partial z}$ and $f$ is a given function.
The Euler-Lagrange equation I have for variational Calculus is:
$$
\left(\sum^m_{\alpha=1}\frac{d}{dt_\alpha}\frac{\partial L}{\partial (x_k)_{t_\alpha}}\right)-\frac{\partial L}{\partial u}=0
$$
with $t=[x,y,z]$; $X=u,\therefore k=1;\alpha =3$
Thus the equation becomes:
$$\frac{d}{dx}\frac{\partial L}{\partial u_x}+\frac{d}{dy}\frac{\partial L}{\partial u_y}+\frac{d}{dz}\frac{\partial L}{\partial u_z}+\frac{\partial L}{\partial u}=0$$
Solving for the partial derivatives gives me:
$$
\frac{d}{dx}(2x^2u_x)-\frac{d}{dy}(2y^2u_y)+\frac{d}{dz}(2z^2u_z)+0=0
$$
This is where I am unsure of my calculations.
My gut tells me I need to use the chain rule and (using the first part as an example here):
$$
\frac{d}{dx}(x^2u_x)=4xu_x+2x^2u_{xx}
$$
Am I correct in this calculation?