Let $x_{1}, ..., x_{n}$ be several independent variables. Let $u = u(x_{1}, ..., x_{n})$ be a function. Let $L(x_{1}, ..., x_{n},u,\partial_{1}u,...,\partial_{n}u)$ be the Lagrangian whose action we want to extremize.
Euler Lagrange equation is stated as $$\frac{\partial L}{\partial u} ~=~ \sum_{i=1}^{n}\frac{\partial}{\partial x_{i}}\frac{\partial L}{\partial (\partial_{i}u)} .$$
The problem is that the partial derivative with respect to $x_i$ is not really a partial derivative, rather it includes both implicit and explicit dependence on $x_i$. The notation is really unclear on this issue. Is there a way of indicating this dependency in the notation? Or is there another way of writing Euler Lagrange Equation in which this issue is bypassed altogether?