\[\nabla = \begin{bmatrix} \partial_x \\ \partial_y \\ \partial_z \end{bmatrix}\\ \\ \nabla\cdot = \begin{bmatrix} \partial_x & \partial_y & \partial_z \end{bmatrix} \\ \\ \nabla \times = \begin{bmatrix} 0 & -\partial_z & \partial_y \\ \partial_z & 0 & -\partial_x \\ -\partial_y & \partial_x & 0 \end{bmatrix} \\ \\ \nabla(\nabla \cdot)= \begin{bmatrix} \partial_x^2 & \partial_y\partial_x & \partial_z\partial_x \\ \partial_x\partial_y & \partial_y^2 & \partial_z\partial_y \\ \partial_x\partial_z & \partial_y\partial_z & \partial_z^2 \end{bmatrix} = \partial_i\partial_j \\ \\ \nabla(\nabla(\nabla \cdot))=\partial_i\partial_j\partial_k ????\\ \\ \nabla^2= \begin{bmatrix} \partial_x^2 + \partial_y^2 + \partial_z^2 & 0 & 0 \\ 0 & \partial_x^2 + \partial_y^2 + \partial_z^2 & 0 \\ 0 & 0 & \partial_x^2 + \partial_y^2 + \partial_z^2 \end{bmatrix}= \nabla\cdot\nabla I\\ \\ \nabla \times (\nabla \times)= \begin{bmatrix} -\partial_z^2-\partial_y^2 & \partial_y\partial_x & \partial_z\partial_x \\ \partial_x\partial_y & -\partial_z^2-\partial_x^2 & \partial_z\partial_y \\ \partial_x\partial_z & \partial_y\partial_z & -\partial_y^2-\partial_x^2 \end{bmatrix} \\ \\ \nabla^2_{scalar} = \nabla\cdot\nabla =\partial_x^2 + \partial_y^2 + \partial_z^2 = \frac{1}{3}Tr(\nabla^2) \\\\ \nabla^4_{scalar} = (\partial_x^2 + \partial_y^2 + \partial_z^2)^2 \\\\ \nabla^T\nabla= \nabla\cdot\nabla = \partial_x^2 + \partial_y^2 + \partial_z^2 \\\\ \nabla\cdot(\nabla(\nabla\cdot)) = \begin{bmatrix} ... \end{bmatrix}^T \\\\ \nabla \times (\nabla \times (\nabla \times))= \begin{bmatrix} 0 & \partial_z(\nabla\cdot\nabla) & -\partial_y(\nabla\cdot\nabla) \\ -\partial_z(\nabla\cdot\nabla) & 0 & \partial_x(\nabla\cdot\nabla) \\ \partial_y(\nabla\cdot\nabla) & -\partial_x(\nabla\cdot\nabla) & 0 \end{bmatrix} =-(\nabla\cdot\nabla)\nabla \times\]