Some formulas for families of curves and surfaces and their applications



A unit vector field τ in the Euclidean space E3 is considered. Let P be the vector field from the first Aminov divergent representation K = div[(r•τ)P] for the total curvature of the second kind K of the field τ . For the field P, an invariant representation of the form P = -rot R* is obtained, where the field R* is expressed in terms of the Frenet basis (τ, ν, β) and the first curvature k and the second curvature χ of the streamlines Lτ of the field τ . Formulas relating to the quantities K (or P), χ, τ , ν, and β are derived.

Three-dimensional analogs to the conservation law div Sp* = 0, which is valid for a family of plane curves Lτ , are obtained, where Sp* is the sum of the curvature vectors of the plane curves Lτ and their orthogonal curves Lν. It is shown that if the field τ is holonomic: 1) the vector field S(τ) from the second Aminov divergent representation K = –1/2 div S(τ) can be interpreted as the sum of three curvature vectors of three curves related to surfaces Sτ with the normal τ ; 2) the non-holonomicity values of the fields of the principal directions l1 and l2 are equal. Applications of the obtained geometric formulas to the equations of mathematical physics are discussed.