In N dimension,

$$\begin{aligned} \bm f\times\bm g&\rightarrow f_a\times g_b=[f_ag_b]_{ab}\\ \bm f\cdot\bm g\times\bm h&\rightarrow(f_a, g_b, h_c)=[f_ag_bh_c]_{abc} \end{aligned}$$

Suppose \(\bm f(\bm r)=f_i\hat e_i\), and a constraint \(f_i\dd r_i=0\)

Even if the \(\nabla\times\bm f\neq 0\), we may find some function \(\varphi\) which is nonzero at the nonzero points of \(\bm f\) s.t.

$$\nabla\times(\varphi\bm f)=\nabla\varphi\times\bm f+\varphi\nabla\times\bm f=0$$

Dotted by \(\bm f\), we find

$$\varphi\bm f\cdot\nabla \times\bm f=(\bm f,\nabla\varphi, \bm f)\propto [f_i\pp_j f_k]_{ijk} =0$$

E.g. For a 2D function, \(A_{ijk}\equiv 0\), so there is always a solution.


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