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'sources and sinks of vector field'
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| Title of object: |
sources and sinks of vector field |
| Canonical Name: |
SourcesAndSinksOfVectorField |
| Type: |
Definition |
| Created on: |
2009-01-28 17:38:27 |
| Modified on: |
2009-01-28 17:38:27 |
| Classification: |
msc:26B12, msc:26B15 |
| Defines: |
source, sink, source of vector field, sink of vector field, productivity |
Revision comment (for changes between this and next version):
Preamble:
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Content:
\PMlinkescapeword{flow} \PMlinkescapeword{field}
Let the vector field $\vec{U}$ of $\mathbb{R}^3$ be interpreted, as in the remark of the \PMlinkname{parent entry}{Flux}, 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 \PMlinkescapetext{closed} surface $a$ expresses how much more liquid it comes from inside of $a$ to outside than contrarily.\, Since for a usual non-compressible 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 contractible or that there are inside the surface some {\em souces} 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} of the souces inside $a$.
For example, the souces 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}.\\
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