If a continuously differentiable vector function is the gradient of a scalar function, then its curl is

Correct option is D

​$$\text{Gradient of } \phi: \\[6pt]\vec{\nabla} \phi = \hat{i} \frac{\partial \phi}{\partial x} + \hat{j} \frac{\partial \phi}{\partial y} + \hat{k} \frac{\partial \phi}{\partial z} \\[10pt]\text{Curl of the gradient (i.e., } \nabla \times (\nabla \phi) \text{):} \\[6pt]\nabla \times (\nabla \phi) =\begin{vmatrix}\hat{i} & \hat{j} & \hat{k} \\\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\\frac{\partial \phi}{\partial x} & \frac{\partial \phi}{\partial y} & \frac{\partial \phi}{\partial z}\end{vmatrix} \\[10pt]\text{Evaluating the determinant:} \\[6pt]\nabla \times (\nabla \phi) = \hat{i}(0) + \hat{j}(0) + \hat{k}(0) = \vec{0}$$​​