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[tex]\frac{\delta f^\alpha}{\delta x^\alpha} = 0[/tex] (2.36 in Mandl & Shaw)

In other textbooks that I have consulted, I usually find the following expression for a conserved quantity :

[tex]\frac{\partial f^\alpha}{\partial x^\alpha} = 0[/tex]

I would like to know why is Mandl & Shaw using what seems to be a functional derivative instead of a simple partial derivative.

Furthermore, does anyone know how to work with the functional derivative used by Mandl & Shaw and how to obtain equation (2.36a) with it? Can I treat it as a simple derivative? It seems wrong to do so...