Calculate curl of e_φ: Non-Zero Vector

[LagrangeEuler]

Why
$$curl(\vec{e}_{\varphi})$$ is not zero vector? And how to calculate this. Vector $\vec{e}_{\varphi}$ is unit vector in cylindrical coordinates.

 

[pasmith]

Use $$\nabla \times \mathbf{e}_\varphi= \mathbf{e}_r \times \frac{\partial \mathbf{e}_\varphi}{\partial r} <br /> + \frac1r\mathbf{e}_\varphi \times \frac{\partial \mathbf{e}_\varphi}{\partial \varphi} <br /> + \mathbf{e}_z \times \frac{\partial \mathbf{e}_\varphi}{\partial z}$$ and remember that $$\frac{\partial \mathbf{e}_\varphi}{\partial \varphi} = -\mathbf{e}_r.$$