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M3P2
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6f0f2d01
Commit
6f0f2d01
authored
Aug 24, 2022
by
Claude Meny
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Update cheatshhet.fr.md
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12.temporary_ins/08.conservative-vector-fields/20.conservative-vector-fields-properties/20.overview/cheatshhet.fr.md
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6f0f2d01
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@@ -112,14 +112,20 @@ CHAMP VECTORIEL CONSERVATIF<br>_" du champ vectoriel (conservatif) aux champs sc
*Propriété d'un champ vectoriel conservatif*
La circulation d'un champ vectoriel conservatif le long d'un chemin $
`\Gamma`
$
$
`\displaystyle\mathbf{\int_{M_1}^{M_2}\overrightarrow{X}\cdot\overrightarrow{dl}}=\int_{M_1}^{M_2} \overrightarrow{grad}(\phi)\cdot\overrightarrow{dl}`
$
$
`\quad\quad\quad\quad = \displaystyle\int_{M_1}^{M_2} d\phi\mathbf{\;=\phi(M_2)-\phi(M_1)}`
$
La circulation d'un champ vectoriel conservatif $
`\overrightarrow{X}=\overrightarrow{grad}(\phi)`
$
`$entre deux points $`
M_1
`$ et $`
M_2
`$ ne dépend que
égale à $`
\p
hi(M_2)-
\p
hi(M_1)
`$, quelque-soit le chemin suivi entre ces deux points :
$`
\d
isplaystyle
\b
egin{align}
\m
athbf{
\i
nt_{M_1}^{M_2}
\o
verrightarrow{X}
\c
dot
\o
verrightarrow{dl}}&=
\i
nt_{M_1}^{M_2}
\o
verrightarrow{grad}(
\p
hi)
\c
dot
\o
verrightarrow{dl}
\\
& = \displaystyle\int_{M_1}^{M_2} d\phi\mathbf{\;=\phi(M_2)-\phi(M_1)}
& =
\d
isplaystyle
\i
nt_{M_1}^{M_2} d
\p
hi
\m
athbf{
\;
\;
=
\p
hi(M_2)-
\p
hi(M_1)}
\e
nd{align}
`$
$`
\L
ongrightarrow
`$ La circulation d'un champ vectoriel conservatif le long d'un contour (chemin fermé) est nulle.
---
*Intérêt en physique*
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