Proof of Divergence of a Triple Cross Product Identity
\[\mathbf{\nabla} \cdot (\mathbf{a} \times (\mathbf{b} \times \mathbf{c})) =( \mathbf{a} \cdot \mathbf{c}) \mathbf{\nabla} \mathbf{b} - ( \mathbf{a} \cdot \mathbf{b}) \mathbf{\nabla} \mathbf{c} + \mathbf{\nabla} (\mathbf{a} \cdot \mathbf{c} ) \cdot \mathbf{b} - \mathbf{\nabla} (\mathbf{a} \cdot \mathbf{b} ) \cdot \mathbf{c}\]
Proof
\[\begin{equation} \begin{aligned} \mathbf{\nabla} \cdot (\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) &= \mathbf{\nabla} \cdot ((\mathbf{a} \cdot \mathbf{c} ) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b} ) \mathbf{c})) \\&= \mathbf{\nabla} \cdot (\mathbf{a} \cdot \mathbf{c} ) \mathbf{b} - \mathbf{\nabla} \cdot (\mathbf{a} \cdot \mathbf{b} \mathbf{c})) \\&= ( \mathbf{a}) \cdot \mathbf{c} ) \mathbf{\nabla} \mathbf{b} + ( (\mathbf{a} \cdot \mathbf{c} )) \mathbf{\nabla} \mathbf{b} - ( \mathbf{a}) \cdot \mathbf{b} ) \mathbf{\nabla} \mathbf{c} - ( (\mathbf{a} \cdot \mathbf{b} )) \mathbf{\nabla} \mathbf{c} \\ &= \mathbf{\nabla} (\mathbf{a} \cdot \mathbf{c} ) \cdot \mathbf{b} - \mathbf{\nabla} (\mathbf{a} \cdot \mathbf{b} ) \cdot \mathbf{c} \end{aligned} \end{equation} \]
