Let $F: M^m \to (N^n,g)$ be a smooth immersion of a smooth $m$-manifold into a Riemannian manifold. Let $\varepsilon > 0$ and let $\mathcal{F}: (-\varepsilon,\varepsilon) \times M \to N$ be a smooth map. For every $t \in (-\varepsilon,\varepsilon)$, define $F_t:M\to N$ by $F_t(p):=\mathcal{F}(t,p)$, and assume that each $F_t$ is a smooth immersion and $F_0=F$. Define the variation vector field $X:M\to F^*TN$ by

\begin{align*} X(p):=\left.\frac{\partial}{\partial t}\right|_{t=0}\mathcal{F}(t,p). \end{align*}

Let $K \subset M$ be a compact domain with smooth boundary, and assume that $\operatorname{supp} X \subset \operatorname{int} K$.

Let $\bar g := F^*g$ and, for each $t$, let $\mu_t := \mu_{F_t^*g}$ denote the Riemannian volume measure on $K$ induced by $F_t^*g$. Define the area functional over $K$ by

\begin{align*} \mathcal A_K[F_t] := \int_K 1\,d\mu_t. \end{align*}

Let $H: M \to (F^*TN)^\perp$ be the mean curvature vector field of the immersion $F$, defined as the trace of the second fundamental form with respect to $\bar g$. Then

\begin{align*} \left.\frac{d}{dt}\right|_{t=0}\mathcal A_K[F_t] = -\int_K g(H,X)\,d\mu_{\bar g}. \end{align*}

In particular, if $M$ is compact without boundary and $X$ is any smooth variation vector field, then one may take $K=M$.