I'm a little puzzled by the following computation.
Suppose $\Sigma$ is a compact two-dimensional Riemannian manifold and $M$ is any Riemannian manifold. Let $\Omega$ be some space of functions from $\Sigma$ to $M$ (you can take it to be $C^{\infty}$, Sobolev, it doesn't really matter for my question).
Now consider the energy functional $E : \Omega \rightarrow \mathbb{R}$ given by
$$E(f) = \int_{\Sigma} \| df(x) \|^2_x dV.$$
My question is twofold: how do I interpret and prove the formula:
$$dE(f)V = \int_{\Sigma} \langle df(x), \nabla V(x) \rangle_x dV$$
where $V$ is a section of $f^*(TM)$? It seems that $dV(x)$ lies in the second tangent space of $M$, whereas $df(x)$ maps into the tangent space. Something is off.