The curl of the unit vector \(\vec{e}_{\varphi}\) in cylindrical coordinates is not zero due to its dependence on the angular coordinate \(\varphi\). To calculate it, the formula \(\nabla \times \mathbf{e}_\varphi = \mathbf{e}_r \times \frac{\partial \mathbf{e}_\varphi}{\partial r} + \frac{1}{r}\mathbf{e}_\varphi \times \frac{\partial \mathbf{e}_\varphi}{\partial \varphi} + \mathbf{e}_z \times \frac{\partial \mathbf{e}_\varphi}{\partial z}\) is used. The key point is that \(\frac{\partial \mathbf{e}_\varphi}{\partial \varphi} = -\mathbf{e}_r\), which contributes to the non-zero result. This calculation highlights the importance of considering the variations of the unit vectors in cylindrical coordinates. Understanding this concept is crucial for applications in vector calculus and physics.