Discussion
The curl of a gradient, \( \nabla \times \nabla\phi \), is
Answer: D
Writing out any component gives a difference of mixed second derivatives, such as \( \phi_{zy} - \phi_{yz} \), which vanishes whenever those derivatives are continuous. So
\[ \nabla \times \nabla\phi \;=\; \mathbf{0}. \]
This is why every gradient field is irrotational, and conversely why an irrotational field can be written as a gradient
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