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% vim:ts=2 sw=2 et spell tw=80:
\section{Applications}
As suggested in the previous section, the fact that it is possible to write a
Fourier style expansion of any function on the surface of the sphere is very
useful in many fields of physics and engineering. Here we will present a few of
the most interesting applications we came across during our research.
\subsection{Electroencephalography}
\begin{figure}
\centering
\subfigure[EEG Electrodes \label{kugel:fig:eeg-electrodes}]%
{\kugelplaceholderfig{.4\linewidth}{5cm}}
\qquad
\subfigure[Gauss' Law \label{kugel:fig:eeg-flux}]%
{\includegraphics[width=.4\linewidth]{papers/kugel/figures/flux}}
\caption{
\label{kugel:fig:eeg}
}
\end{figure}
To start, we will look at an application that is from the field of medicine:
electroencephalography. The \emph{electroencephalogram} (EEG) is a measurement
of the electrical field on the scalp, which shows the brain's activity, and is
used in many fields of research such as neurology and cognitive psychology. The
measurement is done by wearing a cap that contains a number of evenly
distributed electrodes, each of which measures the electric potential (voltage)
at their location (figure \ref{kugel:fig:eeg-electrodes}). To see how this will
relate to the spherical harmonics, we will first quickly recap a bit of physics,
electrodynamics to be precise.
In section \ref{kugel:sec:construction:eigenvalue} we have shown that the
spherical harmonics arise from the surface spherical Laplacian operator, whose
origin we did not consider too much, which is how mathematicians do their work.
On the contrary, physicists usually do the opposite and start by discussing what
is happening, since variables and functions represent physical quantities. So,
we will quickly remind that the Laplacian operator does the following to an
electric potential $\phi$:
\begin{equation*}
\nabla^2 \phi
= \nabla \cdot \nabla \phi
= \nabla \cdot \mathbf{E}
= \rho / \varepsilon
\iff
\iiint_\Omega \nabla \cdot \mathbf{E} \, dv
= \iint_{\partial \Omega} \mathbf{E} \cdot d\mathbf{s}
= \Phi / \varepsilon.
\end{equation*}
Put into words: on the left we have the differential form, where we recall that
the Laplacian (which is a second derivative) is the divergence of the gradient.
Unpacking the notation we first see that we have the gradient of the potential,
which is just the electric field $\mathbf{E}$, and then the divergence of said
electric field is proportional to the charge density $\rho$. So, the Laplacian
of the electric potential is the charge density! For those that are more
familiar with the integral form of Maxwell's equation, we have also included an
additional step using the divergence theorem, which brings us to the electric
Flux, which by Gauss' law (shown in the iconic\footnote{Every electrical
engineer has seen this picture so many times that is probably burnt in their
eyes.} figure \ref{kugel:fig:eeg-flux}) is the net electric charge.
\subsection{Measuring Gravitational Fields}
\subsection{Quantisation of Angular Momentum}
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