On the conservation laws for a family of surfaces

A family {Sτ} of surfaces Sτ with the unit normal τ = τ (x, y, z) in the Euclidean space E3 is considered. The surfaces Sτ continuously fill a domain D in E3. For the family {Sτ} of surfaces Sτ, the law of conservation div F = 0 is proved, where the solenoidal vector field F is expressed in terms of the main classical characteristics of the surfaces Sτ: the unit normal, the principal directions, the principal curvatures, the mean curvature, and the Gaussian curvature.