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Update textbook.fr.md

keep-around/1a6afde624b8468d1772d8a8b2bbc652ee5a324c
Claude Meny 5 years ago
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1a6afde624
  1. 17
      00.brainstorming-pedagogical-teams/40.collection-existing-pedagogical-content/04.reference-frames-coordinate-systems/textbook.fr.md

17
00.brainstorming-pedagogical-teams/40.collection-existing-pedagogical-content/04.reference-frames-coordinate-systems/textbook.fr.md

@ -254,15 +254,16 @@ $`=dx\;\overrightarrow{e_x}+dy\;\overrightarrow{e_y}+dz\;\overrightarrow{e_z}`$<
[EN] y its norm (or length) is thescalar line element :<br>
<br>$`||\overrightarrow{dl}||=\sqrt{dl_x^2+dl_y^2+dl_z^2}=\sqrt{dx^2+dy^2+dz^2}`$<br>
<br>$`||\overrightarrow{dl}||=\sqrt{\overrightarrow{dl}\cdot\overrightarrow{dl}}`$
$`=\sqrt{(dx\;\overrightarrow{e_x}+dy\;\overrightarrow{e_y}+dz\;\overrightarrow{e_z})\cdot
(dx\;\overrightarrow{e_x}+dy\;\overrightarrow{e_y}+dz\;\overrightarrow{e_z})}`$
$`=\sqrt{dx^2\;(\overrightarrow{e_x}\cdot\overrightarrow{e_x})
+dy^2\;(\overrightarrow{e_y}\cdot\overrightarrow{e_y})
+dz^2\;(\overrightarrow{e_z}\cdot\overrightarrow{e_z})`$
$`+2\,dx\,dy\,x(\overrightarrow{e_x}\cdot\overrightarrow{e_y})`$
$`+2\,dx\,dz\,x(\overrightarrow{e_x}\cdot\overrightarrow{e_z})`$
$`+2\,dy\,dz\,x(\overrightarrow{e_y}\cdot\overrightarrow{e_z})}`$
$`=\sqrt{(dl_x\;\overrightarrow{e_x}+dl_y\;\overrightarrow{e_y}+dl_z\;\overrightarrow{e_z})\cdot
(dl_x\;\overrightarrow{e_x}+dl_y\;\overrightarrow{e_y}+dl_z\;\overrightarrow{e_z})}`$
$`=\sqrt{dl_x^2\;(\overrightarrow{e_x}\cdot\overrightarrow{e_x})
+dl_y^2\;(\overrightarrow{e_y}\cdot\overrightarrow{e_y})
+dl_z^2\;(\overrightarrow{e_z}\cdot\overrightarrow{e_z})`$
$`+2\,dl_x\,dl_y\,x(\overrightarrow{e_x}\cdot\overrightarrow{e_y})`$
$`+2\,dl_x\,dl_z\,x(\overrightarrow{e_x}\cdot\overrightarrow{e_z})`$
$`+2\,dl_y\,dl_z\,x(\overrightarrow{e_y}\cdot\overrightarrow{e_z})}`$
$`=\sqrt{dl_x^2+dl_y^2+dl_z^2}`$
$`=\sqrt{dx^2+dy^2+dz^2}`$
* **N3 ($`\rightarrow`$ N4)**<br>

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