Issue with derivation of $\nabla \times\nabla{\phi} =0 $

Question

To prove that $\nabla \times\nabla{\phi} =0 $. The argument defines that $\nabla{\phi} = \frac{\partial}{\partial{x}} \hat{i} + \frac{\partial}{\partial{y}} \hat{j} + \frac{\partial}{\partial{z}} \hat{k}$. But shouldn't the dot product result in a scalar and not a vector? Though If it did result in a scalar, one cannot take the cross product of a scalar and a vector, so why does the dot product result in a vector here ?

Answer 1

You've conflated the gradient $\nabla\phi$ of a scalar $\phi$ with the divergence $\nabla\cdot V$ of a vector $V$. Gradients are vectors; divergences are scalars.