The curl of the unit vector \(\vec{e}_{\varphi}\) in cylindrical coordinates is not a zero vector due to its dependence on the spatial derivatives of the vector fields involved. The calculation utilizes the formula \(\nabla \times \mathbf{e}_\varphi = \mathbf{e}_r \times \frac{\partial \mathbf{e}_\varphi}{\partial r} + \frac{1}{r}\mathbf{e}_\varphi \times \frac{\partial \mathbf{e}_\varphi}{\partial \varphi} + \mathbf{e}_z \times \frac{\partial \mathbf{e}_\varphi}{\partial z}\). Notably, the derivative \(\frac{\partial \mathbf{e}_\varphi}{\partial \varphi} = -\mathbf{e}_r\) plays a crucial role in this calculation, confirming that the curl is indeed non-zero.
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