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 formula_1 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 formula_1 induced by the metric of "M", and "H" is the mean curvature of formula_1. The normal vector is parallel to formula_5 where formula_6 is the tangent vector. The mean curvature is parallel to the normal vector.