Having given a unit vector field $\textbf{v}$ in $\mathbb{R}^3$ we can determine it's vorticity $\textbf{w}=\nabla \times \textbf{v}$. Now we can determine the, so called, Lamb vector $\textbf{b}=\textbf{w}\times\textbf{v}$ of length $b=|\textbf{b}|$, the helicity $t=\textbf{v}\cdot\textbf{w}$ and the splay $s=\nabla \cdot \textbf{v}$.

At the same time it is possible to derive these quantities using directional derivatives i.e. $\textbf{b}=(\textbf{v}\cdot\nabla)\textbf{v}$.

Extending this notion and constructing a moving frame $(\textbf{v},\frac{1}{b}\textbf{b},\textbf{v}\times\frac{1}{b}\textbf{b})=(n_1,n_2,n_3)$ we reconstruct \begin{eqnarray} b=&n_2\cdot((n_1\cdot\nabla )n_1)\\ s=&n_1\cdot((n_1\cdot\nabla )n_3)+n_2\cdot((n_2\cdot\nabla )n_3)\\ t=&n_1\cdot((n_3\cdot\nabla )n_2)+n_1\cdot((n_2\cdot\nabla )n_3)\\ \end{eqnarray} It is also possible to construct a lot of other scalar functions.

The question is: Are these 3 functions enough to determine the field uniquely and are the other scalars just derivatives & linear combinations of the above? How could I prove it?

It seems that knowing $\nabla \otimes \textbf{u}$ would give enough information, which would suggest 9 minus the information supplied by the unit length of the vector field AND I was able to express $\nabla\cdot n_3$ in terms of the 3 given quantities and their derivatives,( such as quantities $n_2\cdot \nabla b$ etc.)


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