I'm really struggling to understand the use of tensors when we want to have a covariant equation.
From what I understand, if we write an equation using tensors only, then the physics behind it will be independent of the choice of the coordinate system.
For example, written in a covariant form, the Maxwell's equations are: \begin{aligned}&{\frac {\partial F^{\alpha \beta }}{\partial x^{\alpha }}}=\mu _{0}J^{\beta }\\&{\frac {\partial G^{\alpha \beta }}{\partial x^{\alpha }}}=0\end{aligned}
Maxwell's equations are not covariant under Galilean transformations. But since they can be written as tensorial equations, shouldn't they also be covariant under Galilean transformations (or any change of coordinate system)?
Is it because there is a partial derivative? Then, why does Wikipedia say that these equations are manifestly-covariant?
In this case, any equation written only with "propers tensors" (so no partial derivative for example) will be covariant under any choice of frame transformation (Galilean or Lorentz)? Is it the presence of a derivative that determines if an equation is covariant under Galilean or Lorentz transformation?