Easy way to prove $|\text{curl}\ \mathbf n|^2=(\mathbf n\cdot \text{curl}\ \mathbf n)^2+|\mathbf n \wedge\text{curl}\ \mathbf n|^2$?

Question

Let $\mathbf n$ be a unit vector field. I would like to show that $$|\text{curl}\ \mathbf n|^2=(\mathbf n\cdot \text{curl}\ \mathbf n)^2+|\mathbf n \wedge\text{curl}\ \mathbf n|^2$$ holds.

Expanding in coordinate is straightforward but looks ugly and doesn't provide much insight. Could anyone provide a better proof of this based on some properties of curl?

Answer 1

You don't need any properties of curl - for any vectors $u,v$ we have $|u|^2|v|^2 = (u \cdot v)^2 + |u \wedge v|^2,$ so your formula follows from the fact that $|n|=1.$