Benedict Irwin edited Summary.tex  over 9 years ago

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\section{Summary of Operators}  \begin{equation}  \nabla = \begin{bmatrix} \partial_x^2\partial_y\partial_z \partial_x  \\ \partial_x\partial_y^2\partial_z \partial_y  \\ \partial_x\partial_y\partial_z^2 \partial_z  \end{bmatrix}\\ \\ \nabla\cdot =  \begin{bmatrix} \partial_z^{-1}\partial_y^{-1} \partial_x  & \partial_z^{-1}\partial_x^{-1} \partial_y  & \partial_y^{-1}\partial_x^{-1} \partial_z  \end{bmatrix} \\ \\ \nabla \times =  \begin{bmatrix} 

\nabla^4_{scalar} = (\partial_x^2 + \partial_y^2 + \partial_z^2)^2 \\\\  \nabla^T\nabla=\partial_x^2\partial_y^2\partial_z^2( \nabla^T\nabla= \nabla\cdot\nabla =  \partial_x^2 + \partial_y^2 + \partial_z^2) \\\\  \frac{\nabla^T\nabla}{\nabla\cdot\nabla} = \partial_x^2\partial_y^2\partial_z^2  \\\\ \nabla\cdot(\nabla(\nabla\cdot)) =   \begin{bmatrix}   \partial_x^2\partial_y^{-1}\partial_z^{-1} + \partial_y\partial_z^{-1} + \partial_y^{-1}\partial_z \\  \partial_y^2\partial_x^{-1}\partial_z^{-1} + \partial_x\partial_z^{-1} + \partial_x^{-1}\partial_z \\  \partial_z^2\partial_x^{-1}\partial_y^{-1} + \partial_x\partial_y^{-1} + \partial_x^{-1}\partial_y ...  \end{bmatrix}^T \\\\  \nabla \times (\nabla \times (\nabla \times))=