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*)

**] for the total curvature of the second kind**

*P**K*of the field

*τ*. For the field

**, an invariant representation of the form**

*P***= -rot**

*P**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.