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diff --git a/buch/papers/kugel/applications.tex b/buch/papers/kugel/applications.tex index 40d5291..39e4cfc 100644 --- a/buch/papers/kugel/applications.tex +++ b/buch/papers/kugel/applications.tex @@ -7,11 +7,59 @@ 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 (EEG)} +\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} |