# Derivative of a Variation vs Variation of a Derivative

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• quickAndLucky

#### quickAndLucky

When a classical field is varied so that ##\phi ^{'}=\phi +\delta \phi## the spatial partial derivatives of the field is often written $$\partial _{\mu }\phi ^{'}=\partial _{\mu }(\phi +\delta \phi )=\partial _{\mu }\phi +\partial _{\mu }\delta \phi$$. Often times the next step is to switch the order of the variation and the partial derivative to get ##\partial _{\mu }\phi ^{'}=\partial _{\mu }\phi +\delta (\partial _{\mu }\phi )##. What justifies the replacement of ##\partial_{\mu }(\delta\phi )## by ##\delta (\partial _{\mu }\phi )##?

Variations and derivatives commute if you keep your coordinates fixed during the variation. In deriving the Euler Lagrange eqns e.g. this is the case: the field variations involve functional variations.

quickAndLucky
Variations and derivatives commute if you keep your coordinates fixed during the variation. In deriving the Euler Lagrange eqns e.g. this is the case: the field variations involve functional variations.
I guess my question is "why do variations and derivatives commute?"

I guess my question is "why do variations and derivatives commute?"
haushofer answered your question correctly. The variation $\delta$ measures the change in the functional form of a field at a fixed coordinate value. So, if you define the field $\psi_{\mu}(x) = \partial_{\mu}\phi (x)$, then it follows from the definition of $\delta$ that $$\delta \psi_{\mu}(x) = \psi_{\mu}^{'}(x) - \psi_{\mu}(x),$$ or
$$\delta (\partial_{\mu}\phi )(x) = \partial_{\mu}\phi^{'}(x) - \partial_{\mu}\phi(x) = \partial_{\mu}\left(\phi^{'} - \phi \right) (x) = \partial_{\mu}( \delta \phi )(x) .$$

quickAndLucky
Thinking of ##\partial _{\mu}\phi ## as an independent vector field that itself varies seemed to help! Thanks haushofer and samalkhaiat!