Divergence formulas (conservation laws) for families of curves and surfaces and applications



The divergence formulas we have obtained (differential conservation laws) of the form div F = 0 for an arbitrary smooth field of unit vectors τ(x;  y;  z), for a family of spatial curves as well as {Lτ} for a family of surfaces {Sτ} continuously filling a certain domain. The solenoidal vector field F in these formulas is expressed, respectively, through the field τ(x;  y;  z), the characteristics of the curves Lτ and characteristics of the surfaces Sτ. Also, we have found the formulas connecting the surface characteristics and those of the curves orthogonal to them. In the case when the curves Lτ and the surfaces Sτ are vector lines of the vector field υ = |υ|τ with the direction τ and the surfaces orthogonal to them, the conservation laws found are equivalent to divergence formulas for the field υ. With these general geometric formulas the divergent identities (differential conservation laws) for the solutions of the eikonal equation |grad τ|2 = n2(x; y; z), the Poisson equation uxx + uyy + uzz = –4πρ(x; y; z) and for solutions of Euler's hydrodynamic equations are obtained. In the plane case, these formulas transform to the conservation laws obtained earlier.