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Benedict Irwin edited Electromagnetism.tex
about 10 years ago
Commit id: 8933bdf88fdc329de0fab028ff9d3ab4756301d5
deletions | additions
diff --git a/Electromagnetism.tex b/Electromagnetism.tex
index 2521bff..576bde8 100644
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\frac{\varphi}{c}
\end{bmatrix}
k
\end{equation}
There are now hypercomplex four potentials from the real potential such that
$A_\mu \to A_\mu + B_mu i + \Gamma_\mu j + \Delta_\mu k$
Now, as usually defined we can create the tensor \begin{equation}
H_{\mu \nu} = (dA)_{\mu \nu}= \partial_\mu A_\nu - \partial_\nu A_\mu
\end{equation}
But with the hypercomplex components it is clear that $H_{\mu \nu} = F_{\mu \nu} +I_{\mu \nu}i + J_{\mu \nu}j + K_{\mu \nu}k where \begin{equation}
F_{\mu \nu}+Re[H_{\mu \nu}]=\partial_\mu A_\nu - \partial_\nu A_\mu \\
I_{\mu \nu}+Im_i[H_{\mu \nu}]=\partial_\mu B_\nu - \partial_\nu B_\mu \\
J_{\mu \nu}+Im_j[H_{\mu \nu}]=\partial_\mu \Gamma_\nu - \partial_\nu \Gamma_\mu \\
K_{\mu \nu}+Im_k[H_{\mu \nu}]=\partial_\mu \Delta_\nu - \partial_\nu \Delta_\mu
\end{equation}