Curl of the Gradient of a Scalar | Quantum Science Philippines
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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

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