First variation of area formula

In Riemannian geometry, the first variation of area formula relates the mean curvature of a hypersurface to the rate of change of its area as it evolves in the outward normal direction.

Let ${\displaystyle \Sigma (t)}$ be a smooth family of oriented hypersurfaces in a Riemannian manifold M such that the velocity of each point is given by the outward unit normal at that point. The first variation of area formula is

${\displaystyle {\frac {d}{dt}}\,dA=H\,dA,}$

where dA is the area form on ${\displaystyle \Sigma (t)}$ induced by the metric of M, and H is the mean curvature of ${\displaystyle \Sigma (t)}$. The normal vector is parallel to ${\displaystyle D_{\alpha }{\vec {e}}_{\beta }}$ where ${\displaystyle {\vec {e}}_{\beta }}$ is the tangent vector. The mean curvature is parallel to the normal vector.