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Benedict Irwin edited Sketchy Generation.tex
over 9 years ago
Commit id: e1d3560864c4f897225afabd5370100b9218a16a
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\begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \\ \textbf{i} & \textbf{j} & \textbf{k} \\ \textbf{i} & \textbf{j} & \textbf{k} \end{vmatrix}= \epsilon_{ijk}
\end{equation}
The first operation takes a vector and generates a skew symmetric matrix. The third operation defines the Levi-Civita symbol, a rank $3$ tensor which is essential to vector calculus. A use of the first operation would be to define the electromagnetic tensor which looks like \begin{equation}
F_{\mu\nu}=\begin{bmatrix} 0 & E_x/c & E_y/c & E_z/c \\ -E_x/c & 0 & B_z & -B_y \\ -E_y/c & -B_z & 0 & B_x \\ -E_z/c & B_y & -B_x & 0 \end{bmatrix}
\end{equation}