Talk:PlanetPhysics/Sources and Sinks of Vector Field

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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: sources and sinks of vector field %%% Primary Category Code: 02. %%% Filename: SourcesAndSinksOfVectorField.tex %%% Version: 1 %%% Owner: pahio %%% Author(s): pahio %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

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Let the \htmladdnormallink{vector field}{http://planetphysics.us/encyclopedia/NeutrinoRestMass.html} $\vec{U}$ of $\mathbb{R}^3$ be interpreted, as in the remark of the \htmladdnormallink{parent entry}{http://planetphysics.us/encyclopedia/Flux.html}, as the \htmladdnormallink{velocity}{http://planetphysics.us/encyclopedia/Velocity.html} field of a stationary flow of a liquid.\, Then the \htmladdnormallink{flux}{http://planetphysics.us/encyclopedia/AbsoluteMagnitude.html} \[ \oint_a\vec{U}\cdot d\vec{a} \] of $\vec{U}$ through a closed surface $a$ expresses how much more liquid per time-unit it comes from inside of $a$ to outside than contrarily.\, Since for a usual incompressible liquid, the outwards flow and the inwards flow are equal, we must think in the case that the flux differs from 0 either that the flowing liquid is suitably compressible or that there are inside the surface some {\em sources} creating liquid and {\em sinks} annihilating liquid.\, Ordinarily, one uses the latter idea.\, Both the sources and the sinks may be called sources, when the sinks are {\em negative sources}.\, The flux of the \htmladdnormallink{vector}{http://planetphysics.us/encyclopedia/Vectors.html} $\vec{U}$ through $a$ is called the {\em productivity} or the {\em strength} of the sources inside $a$.

For example, the sources and sinks of an \htmladdnormallink{Electric Field}{http://planetphysics.us/encyclopedia/ElectricField.html} ($\vec{E}$) are the locations containing positive and negative \htmladdnormallink{charges}{http://planetphysics.us/encyclopedia/Charge.html}, respectively.\, \htmladdnormallink{The Gravitational Field}{http://planetphysics.us/encyclopedia/GravitationalField.html} has only sinks, which are the locations containing mass.\\

The expression \[ \frac{1}{\Delta v}\oint_{\partial\Delta v}\vec{U}\cdot d\vec{a}, \] where $\Delta v$ means a region in the vector field and also its \htmladdnormallink{volume}{http://planetphysics.us/encyclopedia/Volume.html}, is the productivity of the sources in $\Delta v$ per a volume-unit.\, When we let $\Delta v$ to shrink towards a point $P$ in it, to an infinitesimal volume-element $dv$, we get the limiting value \begin{align} \varrho \;:=\; \frac{1}{dv}\oint_{\partial dv}\vec{U}\cdot d\vec{a}, \end{align} called the {\em source density} in $P$.\, Thus the productivity of the source in $P$ is $\varrho\,dv$.\, If\, $\varrho = 0$, there is in $P$ neither a source, nor a sink.\\

The Gauss's \htmladdnormallink{theorem}{http://planetphysics.us/encyclopedia/Formula.html} \[ \int_v\nabla\cdot\vec{U}\,dv \;=\; \oint_a\vec{U}\cdot d\vec{a} \] applied to $dv$ says that \begin{align} \nabla\cdot\vec{U} \;=\; \frac{1}{dv}\oint_{\partial dv}\vec{U}\cdot d\vec{a}. \end{align} Accordingly, \begin{align} \varrho \;=\; \nabla\cdot\vec{U} \end{align} and \begin{align} \oint_{a}\vec{U}\cdot d\vec{a} \;=\; \int_v\varrho\,dv. \end{align} This last formula can be read that\, {\em the flux of the vector through a closed surface equals to the total productivity of the sources inside the surface.}\, For example, if $\vec{U}$ is the electric flux density $\vec{D}$, (4) means that the electric flux through a closed surface equals to the total charge inside.

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