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In my QFT course we are supposed to vary the action of a for a scalar field coupled to an electromagnetic field with the following Lagrangian density:

$$\mathcal{L} = [D_\mu\phi(x)]^*D^\mu\phi(x)-m^2\phi(x)^*\phi(x) -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$$

where ($D_\mu\phi(x) = (\partial_\mu+ieA_\mu(x))\phi(x)$ and $x$ is a 4-vector) with respect to $A_\mu$. But what is meant by only varying the action w.r.t to one quantity? For Example when we derived the Euler-Lagrange-Equations for a free scalar field we varied w.r.t $\phi$ and $\partial_\mu\phi$.

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    $\begingroup$ This is a more sophisticated variant of this question. $\endgroup$ – knzhou Apr 11 at 20:20
  • $\begingroup$ And fittingly, the answers there are more sophisticated variants of my answer. OP, i really recommend the linl in knzhou's comment. $\endgroup$ – Herr_Mitesch Apr 11 at 21:20
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Varying with respect to $A_\mu$ automatically induces a variation in $\partial_\mu A_\nu$ given by

$$ \delta(\partial_\mu A_\nu) = \partial_\mu \delta A_\nu.$$

Thus, unless the text of your excercise says explicitly that you are supposed to hold $\partial_\mu A_\nu$ fixed (which i find highly unlikely, but will revise my answer in that case), I believe that you are supposed to vary both $A_\mu$ and $\partial_\mu A_\nu$.

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