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'laminar field'
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| Title of object: |
laminar field |
| Canonical Name: |
LaminarField |
| Type: |
Definition |
| Created on: |
2004-10-11 14:13:38 |
| Modified on: |
2005-01-17 18:12:16 |
| Classification: |
msc:26B12 |
| Defines: |
scalar potential, potential |
| Synonyms: |
laminar field=vortexless
laminar field=irrotational |
Preamble:
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Content:
A vector field \,$\vec{F} = \vec{F}(x,\,y,\,z)$, defined in an open set $D$ of $\mathbb{R}^3$, is {\em laminar} iff the condition
$$\nabla\times\vec{F} = \vec{0}$$
is satisfied in every point $(x,\,y,\,z)$ of $D$.
Here, $\nabla\times\vec{F}$ is the curl or rotor of $\vec{F}$. \,The condition is equivalent with both of the following:
\begin{itemize}
\item The line integrals
$$\oint_s \vec{F}\cdot d\vec{s}$$
taken around every \PMlinkescapetext{closed} contractible curve $s$ vanish.
\item The vector field has a \PMlinkescapetext{{\em scalar potential}} \,$u = u(x,\,y,\,z)$\, which has continuous partial derivatives and which is up to a \PMlinkescapetext{constant term} unique in a simply connected domain; the scalar potential means that
$$\vec{F} = \nabla u.$$
\end{itemize}
The scalar potential has the expression
$$u = \int_{P_0}^P\vec{F}\cdot d\vec{s},$$
where the point $P_0$ may be chosen freely, \,$P = (x,\,y,\,z)$.
\textbf{Note.} \,In physics, $u$ is in general replaced with \,$V = -u$. \,If the $\vec{F}$ is interpreted as a \PMlinkescapetext{force}, then the potential $V$ is equal to the work made by the \PMlinkescapetext{force} when its point of application is deplaced from $P_0$ to infinity. |
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