In the wiki article of Noether's Theorem (https://en.wikipedia.org/wiki/Noether%27s_theorem#Field-theoretic_derivation) and many other places the $\delta$ (variation) and $\partial$ (partial derivative) operators commute when acting on the fields.
Is this always the case, or just for scalar fields? Isn't $\partial$ ill-defined for tensor fields and instead be replaced by $\nabla$? (and in that case the lie derivative won't commute with covariant derivatie).