Do these "identities" even make sense? $F$ is a 3D vector field. For the second equation, the LHS is a vector but the RHS is a scalar. $$\nabla\cdot\Delta F=\Delta(\nabla\cdot F),\,\nabla\times\Delta F=\Delta(\nabla\times F)$$
2026-03-26 22:12:08.1774563128
Vector calculus "identities"
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The first equation means $\partial_i(\partial_j\partial_j F_i)=\partial_j\partial_j(\partial_i F_i)$. The second means $\epsilon_{ijk}\partial_j(\partial_l\partial_l F_k)=\partial_l\partial_l(\epsilon_{ijk}\partial_j F_k)$.