sources and sinks of vector fieldSourcesAndSinksOfVectorField2009-01-28 17:38:272009-02-01 18:21:24Definitionflux of vector fieldsourcesinksource of vector fieldproductivitystrengthsource densitysink of vector field% this is the default PlanetMath preamble. as your knowledge
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\PMlinkescapeword{flow} \PMlinkescapeword{field} \PMlinkescapeword{closed}
\PMlinkescapeword{density}
Let the vector field $\vec{U}$ of $\mathbb{R}^3$ be interpreted, as in the remark of the \PMlinkname{parent entry}{Flux}, as the velocity \PMlinkescapetext{field of a stationary flow} of a liquid.\, Then the flux
\[
\oint_a\vec{U}\cdot d\vec{a}
\]
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 vector $\vec{U}$ through $a$ is called the {\em productivity} or the {\em strength} of the sources inside $a$.
For example, the sources and sinks of an electric field ($\vec{E}$) are the locations containing positive and negative charges, respectively.\, The gravitational field has only sinks, which are the locations containing \PMlinkescapetext{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 volume, 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 theorem
\[
\int_v\nabla\cdot\vec{U}\,dv \;=\; \oint_a\vec{U}\cdot d\vec{a}
\]
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 \PMlinkescapetext{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.