The new meaning or a property of the Gaussian and of the mean curvature of surfaces forming a family in terms of the vector analysis has been discovered. The divergence representations were found for the mean curvature H = H(x; y; z) and the Gaussian curvature of K = K(x; y; z) of the surfaces Sα, given either by the equation u(x; y; z) = α (α is a parameter, u is a scalar function) or parametrically; or the surfaces Sτ that are described by some general properties. The surfaces Sα and Sτ continuously fill the domain D, forming a family of {Sα} or {Sα} in D with the unit normal field τ = τ(x; y; z). Thus, the formulas of the form H = div SH, K = div SK are obtained, and three-dimensional vector fields SH, SK are expressed in terms of the normal field τ: SH = –1/2τ, SK = –1/2S(τ), S(τ) = rot τ × τ – τ div τ and have a clear geometric meaning. In the case when the surface is given by the graph z = f(x; y), the above formulas lead to divergent representations for the mean and Gaussian curvatures obtained earlier.
The applications of these general geometric formulas to the equations of mathematical physics: the eikonal equation, the Poisson equation, Euler's hydrodynamic equations are given. Furthermore, in the plane case a simple geometric interpretation of the conservation laws obtained earlier for a family of plane curves and for solutions to the eikonal and Euler's hydrodynamic equations is given.