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Benedict Irwin edited Sketchy Generation.tex
over 9 years ago
Commit id: 434a39c4e3641757bda47daa82895c560c19b672
deletions | additions
diff --git a/Sketchy Generation.tex b/Sketchy Generation.tex
index 8eccb5f..e7fc3aa 100644
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\end{bmatrix}
=\begin{bmatrix}\partial_x^{-1}-\partial_x^{-1}&\partial_y^{-1}-\partial_y^{-1}&\partial_z^{-1}-\partial_z^{-1}\end{bmatrix}
\end{equation}
Knowing this we require the gradient operator to be a column vector such that when it left operates on the divergence row operator we may create the matrix form above. This gives the form \begin{equation}
\begin{bmatrix} a \\ b \\ c \end{bmatrix}
\begin{bmatrix} \partial_z^{-1}\partial_y^{-1} & \partial_z^{-1}\partial_x^{-1} & \partial_y^{-1}\partial_x^{-1} \end{bmatrix}
=
\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}
\end{equation}