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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?

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A bit of context: When you read Special Relativity, the rules you learn don't ensure that a well-formed "tensorial" expression in any coordinate system defines the same tensor, but just in any inertial frame (and these are related to each other specifically by Lorentz transformations). Only when you move on to General Relativity do you typically learn how to write expressions that are valid in any coordinate system (this is the basics of differential geometry). The Maxwell equations you write are Lorentz covariant but not generally covariant.

As you have (impressively!) already guessed, the presence of derivatives is typically the main problem. To get something generally covariant, you need to replace derivatives $\partial_\alpha$ by covariant derivatives $\nabla_\alpha$.

However, the derivatives are actually not the problem in your particular case. Covariant derivatives come into play on a curved spacetime or when dealing with nonlinear coordinate transformations, neither of which you have. I usually don't think about "Galilean covariance", so don't trust me 100%, but I think the equations you write down will indeed be preserved under a Galilean transformation. What will break is the relationship between $F$ and $G$, namely the equation $$ G^{\alpha\beta} = \frac{1}{2} \epsilon^{\alpha\beta\gamma\delta} F_{\gamma\delta}. $$ The $\epsilon$ tensor is defined explicitly in terms of the spacetime metric $g_{\alpha\beta}$, which is a concept that doesn't make sense in Galilean spacetime, so this equation cannot be preserved by Galilean transformations.

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