Concept: The curl is a vector field (editing)

In any coordinate system, there exists a formula for the curl. Examples include:\[\vec\nabla\times\vec v = \left(\frac{\partial v_z}{\partial y}-\frac{\partial v_y}{\partial z}\right)\hat x + \left(\frac{\partial v_x}{\partial z}-\frac{\partial v_z}{\partial x}\right)\hat y + \left(\frac{\partial v_y}{\partial x}-\frac{\partial v_x}{\partial y}\right)\hat z \text{ in rectangular coodinates}\]\[\vec\nabla\times\vec v = \left(\frac{1}{s}\frac{\partial v_z}{\partial \phi} - \frac{\partial v_\phi}{\partial z}\right) \hat s + \left(\frac{\partial v_s}{\partial z} - \frac{\partial v_z}{\partial s}\right) \hat \phi + \frac{1}{s}\left(\frac{\partial \left(s\ v_\phi\right)}{\partial s} - \frac{\partial v_s}{\partial \phi}\right) \hat z \text{ in cylindrical coodinates}\]\[\vec\nabla\times\vec v = \frac{1}{r \sin(\theta)}\left(\frac{\partial \left(\sin(\theta)\ v_\phi\right)}{\partial \theta} - \frac{\partial v_\theta}{\partial \phi}\right) \hat r + \frac{1}{r}\left(\frac{1}{\sin(\theta)}\frac{\partial v_r}{\partial \phi} - \frac{\partial \left(r\ v_\phi\right)}{\partial r}\right)\hat \theta +\frac{1}{r} \left(\frac{\partial \left(r\ v_\theta\right)}{\partial r} - \frac{\partial v_r}{\partial \theta}\right) \hat \phi \text{ in spherical coodinates}\]