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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 Σ {\displaystyle \Sigma } 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
where dA is the area form on Σ {\displaystyle \Sigma } induced by the metric of M, and H is the mean curvature of Σ {\displaystyle \Sigma }. The normal vector is parallel to D α e → β {\displaystyle D_{\alpha }{\vec {e}}_{\beta }} where e → β {\displaystyle {\vec {e}}_{\beta }} is the tangent vector. The mean curvature is parallel to the normal vector.
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