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 *S*α, given
either by the equation *u(x; y; z)* = α (α is a parameter, *u* is a scalar function) or
parametrically; or the surfaces *S*τ that are described by some general properties.
The surfaces *S*α and *S*τ continuously fill the domain *D*, forming a family of {*S*α}
or {*S*α} in *D* with the unit normal field **τ** = **τ***(x; y; z)*. Thus, the formulas of
the form *H* = div * S*H,

*K*= div

*K are obtained, and three-dimensional vector fields*

**S***H,*

**S***K are expressed in terms of the normal field*

**S****τ**:

*H = –1/2*

**S****τ**,

*K = –1/2*

**S****S(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.