Curl of the Gradient of a Scalar | Quantum Science Philippines

Curl of the Gradient of a Scalar

proof that the curl of the gradient of a scalar function is equal to zero
$\vec{\nabla}\times\vec{\nabla}\psi=0$
let
$\vec{\nabla}_j= \partial_j\hat{e}_j$
$\vec{\nabla}_k= \partial_k\hat{e}_k$
and
$\triangledown\psi= \partial_k\psi\hat{e}_k$

$\vec{\nabla}_j \times \vec{\nabla}_k \psi=\in_{ijk} \partial_j \partial_k \psi\hat{e}_j \times \hat{e}_k$
$= \in_{ijk}\hat{e}_i\partial_j\partial_k\psi$
$= \hat{e}_1(\partial_2\partial_3\psi - \partial_3\partial_2\psi) + \hat{e}_2(\partial_3\partial_1\psi - \partial_1\partial_3\psi) + \hat{e}_3(\partial_1\partial_2\psi - \partial_2\partial_1\psi)$
$= \hat{e}_1(\partial^2\psi - \partial^2\psi) + \hat{e}_2(\partial^2\psi - \partial^2\psi) + \hat{e}_3(\partial^2\psi - \partial^2\psi)$
$= 0$