# Variation in field theory with respect to one quantity

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 $$\partial_\mu\phi$$.

• This is a more sophisticated variant of this question. – knzhou Apr 11 at 20:20
• And fittingly, the answers there are more sophisticated variants of my answer. OP, i really recommend the linl in knzhou's comment. – Herr_Mitesch Apr 11 at 21:20

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 $$\partial_\mu A_\nu$$.