General notation

We use Greek letters $\alpha, \beta \ldots = 0,1,2,3$ for components of 4-vectors,tensors, etc and Roman letters $i,j,k \ldots = 1,2,3$ for their spatial components. Ordinary partial derivative with respect to coordinate $x^{\alpha}$ is often denoted by comma

$$\frac{\partial}{\partial x^\alpha} A\left(x^\alpha\right) \equiv A_{,\alpha}$$ (1)

Up or down position of the index after comma is generally important.

We often differentiate with respect to vectors (note, coordinates $x^\alpha$ themselves do not constitute vectors) or even tensors, for example, the Lagrangian density

$$\frac{\partial {\cal L}}{\partial u^\alpha} = \left( \frac{\... ...\partial u^2}, \frac{\partial {\cal L}}{\partial u^3} \right)$$ (2)

Note, that it matters, whether one differentiate with respect to covariant or contravariant components, i.e

$$\frac{\partial {\cal L}}{\partial u_\alpha} = \left( \frac{\... ...\partial u_2}, \frac{\partial {\cal L}}{\partial u_3} \right)$$ (3)

is a different object. We always write the derivatives with respect to vectors explicitly. One needs some care with notation when the vector that we differentiate with respect to is itself the gradient of a scalar function. Then we get the notation like $\frac{\partial {\cal L}}{\partial \phi_{,\alpha}}$ which means $\frac{\partial {\cal L}}{\partial (\phi_{,\alpha})}$.