How can the vortex filament equation $$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$ where $\chi(t,s)$ is a curve in $\mathbb R^3$, be derived from the Euler equation $$\partial_t \omega + v\cdot \nabla \omega = \omega \cdot \nabla v, \quad \operatorname{div} v = 0,$$ where $v:\mathbb R \times \mathbb R^3 \to \mathbb R^3$, and $\omega = \operatorname{curl}(v):\mathbb R \times \mathbb R^3 \to \mathbb R^3.$

I've asked a more general question at Survey on the vortex filament equation.