Chapter 16 - Vector Calculus - 16.5 Curl and Divergence - 16.5 Exercises - Page 1150: 29

$curl (curl F)=grad(div F) -\nabla^2 F$

We will have to show that $curl (curl F)=grad(div F) -\nabla^2 F$ $curl (curl F)=\nabla \times (\nabla \times F)$ $=\nabla (\nabla \cdot F) -F (\nabla \cdot \nabla)$ $=\nabla (div F) -F (\nabla^2)$ Hence, the result has been proved. $curl (curl F)=grad(div F) -\nabla^2 F$