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Commit c96604e9 authored by Claude Meny's avatar Claude Meny
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Update cheatsheet.fr.md

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    12.temporary_ins/08.grad-div-rot/70.combinaisons-of-operators/20.overview/cheatsheet.fr.md
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    ...@@ -182,8 +182,7 @@ $`\quad = \left( ...@@ -182,8 +182,7 @@ $`\quad = \left(
          et nous obtenons l'expression **en coordonnées cartésiennes** :       et nous obtenons l'expression **en coordonnées cartésiennes** :
    $`\overrightarrow{grad}\big(div\,\overrightarrow{U}\big)`$ $`\overrightarrow{grad}\big(div\,\overrightarrow{U}\big)`$
    $`\quad =`$ **$`\quad = \left(
    **$` \left(
    \begin{array}{l} \begin{array}{l}
    \dfrac{\partial^2 U_x}{\partial x^2}+\dfrac{\partial^2 U_y}{\partial x\, \partial y}+\dfrac{\partial^2 U_z}{\partial x \,\partial z}\\ \dfrac{\partial^2 U_x}{\partial x^2}+\dfrac{\partial^2 U_y}{\partial x\, \partial y}+\dfrac{\partial^2 U_z}{\partial x \,\partial z}\\
    \dfrac{\partial^2 U_x}{\partial y \,\partial x}+\dfrac{\partial^2 U_y}{\partial y^2}+\dfrac{\partial^2 U_z}{\partial y \,\partial z}\\ \dfrac{\partial^2 U_x}{\partial y \,\partial x}+\dfrac{\partial^2 U_y}{\partial y^2}+\dfrac{\partial^2 U_z}{\partial y \,\partial z}\\
    ...@@ -240,8 +239,7 @@ $`\quad = ...@@ -240,8 +239,7 @@ $`\quad =
         Nous obtenons alors l'expression **en coordonnées cartésiennes** :      Nous obtenons alors l'expression **en coordonnées cartésiennes** :
    $`\overrightarrow{rot}\big(\overrightarrow{rot}\,\overrightarrow{U}\big)`$ $`\overrightarrow{rot}\big(\overrightarrow{rot}\,\overrightarrow{U}\big)`$
    $`\quad =`$ **$`\quad = \left(\begin{array}{l}
    **$`\left(\begin{array}{l}
    \dfrac{\partial^2 U_y}{\partial y\,\partial x} \dfrac{\partial^2 U_y}{\partial y\,\partial x}
    -\dfrac{\partial^2 U_x}{\partial y^2} -\dfrac{\partial^2 U_x}{\partial y^2}
    -\dfrac{\partial^2 U_x}{\partial z^2} -\dfrac{\partial^2 U_x}{\partial z^2}
    ...@@ -341,8 +339,7 @@ $`\require{cancel}\quad = \left(\begin{array}{l} ...@@ -341,8 +339,7 @@ $`\require{cancel}\quad = \left(\begin{array}{l}
    $`\overrightarrow{grad}\big(div\;\overrightarrow{U}\big) $`\overrightarrow{grad}\big(div\;\overrightarrow{U}\big)
    -\overrightarrow{rot}\big(\overrightarrow{rot}\,\overrightarrow{U}\big)`$ -\overrightarrow{rot}\big(\overrightarrow{rot}\,\overrightarrow{U}\big)`$
    $`\quad =`$ **$`\quad =\left(\begin{array}{l}
    **$`\left(\begin{array}{l}
    \dfrac{\partial^2 U_x}{\partial x^2}+\dfrac{\partial^2 U_x}{\partial y^2}+\dfrac{\partial^2 U_x}{\partial z^2}\\ \dfrac{\partial^2 U_x}{\partial x^2}+\dfrac{\partial^2 U_x}{\partial y^2}+\dfrac{\partial^2 U_x}{\partial z^2}\\
    \dfrac{\partial^2 U_y}{\partial x^2}+\dfrac{\partial^2 U_y}{\partial y^2}+\dfrac{\partial^2 U_y}{\partial z^2}\\ \dfrac{\partial^2 U_y}{\partial x^2}+\dfrac{\partial^2 U_y}{\partial y^2}+\dfrac{\partial^2 U_y}{\partial z^2}\\
    \dfrac{\partial^2 U_z}{\partial x^2}+\dfrac{\partial^2 U_z}{\partial y^2}+\dfrac{\partial^2 U_z}{\partial z^2} \dfrac{\partial^2 U_z}{\partial x^2}+\dfrac{\partial^2 U_z}{\partial y^2}+\dfrac{\partial^2 U_z}{\partial z^2}
    ...@@ -366,15 +363,15 @@ possède son champ $`\overrightarrow{grad}\big(div\;\overrightarrow{U}\big) ...@@ -366,15 +363,15 @@ possède son champ $`\overrightarrow{grad}\big(div\;\overrightarrow{U}\big)
    * *Si $`\overrightarrow{grad}\big(div\;\overrightarrow{U}\big) * *Si $`\overrightarrow{grad}\big(div\;\overrightarrow{U}\big)
    -\overrightarrow{rot}\big(\overrightarrow{rot}\,\overrightarrow{U}\big)`$ vérifie l'* **équation d'onde** : -\overrightarrow{rot}\big(\overrightarrow{rot}\,\overrightarrow{U}\big)`$ vérifie l'* **équation d'onde** :
    <br> <br><br>
    *$`\large{\overrightarrow{grad}\big(div\;\overrightarrow{U}\big) *$`\large{\overrightarrow{grad}\big(div\;\overrightarrow{U}\big)
    -\overrightarrow{rot}\big(\overrightarrow{rot}\,\overrightarrow{U}\big)}`$ -\overrightarrow{rot}\big(\overrightarrow{rot}\,\overrightarrow{U}\big)}`$
    $`\;\;\large{-\dfrac{1}{\mathscr{v}^2}\dfrac{\partial^2 \overrightarrow{U}}{\partial t^2}=0}`$* $`\;\;\large{-\dfrac{1}{\mathscr{v}^2}\dfrac{\partial^2 \overrightarrow{U}}{\partial t^2}=0}`$*
    <br> <br><br>
    ou écrit avec le laplacien scalaire : ou écrit avec le laplacien scalaire :
    <br> <br><br>
    **$`\large{\overrightarrow{\Delta}\,\overrightarrow{U}-\dfrac{1}{\mathscr{v}^2}\dfrac{\partial^2 \overrightarrow{U}}{\partial t^2}=0}`$** **$`\large{\overrightarrow{\Delta}\,\overrightarrow{U}-\dfrac{1}{\mathscr{v}^2}\dfrac{\partial^2 \overrightarrow{U}}{\partial t^2}=0}`$**
    <br> <<br><br>
    **alors le champ vectoriel $`f`$ se propage** *à la célérité $`\mathscr{v}`$*. **alors le champ vectoriel $`f`$ se propage** *à la célérité $`\mathscr{v}`$*.
    ......
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