diff --git a/00.brainstorming-pedagogical-teams/40.collection-existing-pedagogical-content/04.reference-frames-coordinate-systems/textbook.fr.md b/00.brainstorming-pedagogical-teams/40.collection-existing-pedagogical-content/04.reference-frames-coordinate-systems/textbook.fr.md index 9bd79cbcd..8e748931f 100644 --- a/00.brainstorming-pedagogical-teams/40.collection-existing-pedagogical-content/04.reference-frames-coordinate-systems/textbook.fr.md +++ b/00.brainstorming-pedagogical-teams/40.collection-existing-pedagogical-content/04.reference-frames-coordinate-systems/textbook.fr.md @@ -254,11 +254,12 @@ $`=dx\;\overrightarrow{e_x}+dy\;\overrightarrow{e_y}+dz\;\overrightarrow{e_z}`$< [EN] y its norm (or length) is thescalar line element :

$`||\overrightarrow{dl}||=\sqrt{dl_x^2+dl_y^2+dl_z^2}=\sqrt{dx^2+dy^2+dz^2}`$

$`||\overrightarrow{dl}||=\sqrt{\overrightarrow{dl}\cdot\overrightarrow{dl}}`$ -$`=\sqrt{(dl_x\;\overrightarrow{e_x}+dl_y\;\overrightarrow{e_y}+dl_z\;\overrightarrow{e_z})\cdot -(dl_x\;\overrightarrow{e_x}+dl_y\;\overrightarrow{e_y}+dl_z\;\overrightarrow{e_z})}`$ +$`=\left[(dl_x\;\overrightarrow{e_x}+dl_y\;\overrightarrow{e_y}+dl_z\;\overrightarrow{e_z})\cdot +(dl_x\;\overrightarrow{e_x}\rigth.`$ +$`\left.+dl_y\;\overrightarrow{e_y}+dl_z\;\overrightarrow{e_z})\right]^{1/2}`$ $`=\left[(dl_x)^2\;(\overrightarrow{e_x}\cdot\overrightarrow{e_x})\right.`$ $`+(dl_y)^2\;(\overrightarrow{e_y}\cdot\overrightarrow{e_y})`$ -$`+(dl_z)^2\;(\overrightarrow{e_z}\cdot\overrightarrow{e_z})}`$ +$`+(dl_z)^2\;(\overrightarrow{e_z}\cdot\overrightarrow{e_z})`$ $`+(2\,dl_x\,dl_y)\,(\overrightarrow{e_x}\cdot\overrightarrow{e_y})`$ $`+(2\,dl_x\,dl_z)\,(\overrightarrow{e_x}\cdot\overrightarrow{e_z})`$ $`\left.+(2\,dl_y\,dl_z)\,(\overrightarrow{e_y}\cdot\overrightarrow{e_z})\right]^{1/2}`$