Vector Identity

Question

So I have a unit vector n. In a formula in a paper I'm reading. I see that $$n\cdot \triangledown n= n \times \triangledown \times n$$ I know that for the topic I'm studying that the orientation of $n$ is not and issue so $n = -n$ which might be helpful...I tried to use the vector triple product on the RHS but ended up at a dead end. I also tried expanding both sides to see if something canceled but to no avail. I'm beggining to think that there may be a typo in the paper... Can it be done? Is there a trick?

Answer 1

I see two main issues. One is that the paper cited is not very polished; this is likely because it appears set of unpublished notes (which can be found on the author's academic website here). In particular, consider the following statement near the start of section 3.2:

"We consider a uniaxial liquid crystal and let $\mathbf{n}(\mathbf{r})$ be a unit vector giving the direction of the preferred orientation at the point $\mathbf{r}$. As we discussed before, the sign of this vector has no physical significance. Once $\mathbf{n}$ is defined at some point we assume that it varies slowly from point to point. At $\mathbf{r}$ we introduce a local right-handed Cartesian coordinate system $x, y, z$ with $z$ parallel to $\mathbf{n}$." [emphasis added]

They then proceed to define the curvature components as partial derivatives of $n_x,n_y$. But if $\mathbf{n}$ and $z$ are parallel, I'd expect these two components to vanish identically. I had initially thought that it should instead be $z$ perpendicular to $\mathbf{n}$, but their meaning seems instead to be that $n_z\approx 1$ everywhere near the origin (eq. 22). So you may be well-advised to compare with a separate reference.

The other issue is that, in the question as written, the relevant claim is presented as the vector equality $(\mathbf{n}\cdot\nabla)\mathbf{n}=\mathbf{n}\times \nabla\times \mathbf{n}$. But what appears in the free energy is not these vectors but their squares i.e. their magnitudes, and these being equal is a weaker claim (e.g. $\hat{x}\neq \hat{y}$ but both are unit vectors). So the claim as stated in the question could be false without contradicting the notes. But I haven't confirmed that this is indeed the case.