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 0a7bc7b99..70645a097 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
@@ -247,11 +247,23 @@ and it writes :<br>
 <br>$`=\overrightarrow{MM'}=d\overrightarrow{OM}=\overrightarrow{dr}=\overrightarrow{dl}`$
 $`=\partial\overrightarrow{OM}_x+\partial\overrightarrow{OM}_y+\partial\overrightarrow{OM}_z`$
 $`=\overrightarrow{dl_x}+\overrightarrow{dl_y}+\overrightarrow{dl_z}`$
-$`=l_x\;\overrightarrow{e_x}+l_y\;\overrightarrow{e_y}+l_z\;\overrightarrow{e_z}`$<br>
-[ES] y su norma es el elemento scalar de linea :<br>
+$`=l_x\;\overrightarrow{e_x}+l_y\;\overrightarrow{e_y}+l_z\;\overrightarrow{e_z}`$
+$`=dx\;\overrightarrow{e_x}+dy\;\overrightarrow{e_y}+dz\;\overrightarrow{e_z}`$<br>
+<br>[ES] y su norma es el elemento scalar de linea :<br>
 [FR] et sa norme el l'élément de longueur :<br>
 [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>$`||\overrightarrow{dl}||=\sqrt{dl_x^2+dl_y^2+dl_z^2}=\sqrt{dx^2+dy^2+dz^2}`$<br>
+<br>$`||\overrightarrow{dl}||=\sqrt{\overrightarrow{dl}\cdot\overrightarrow{dl}}`$
+$`=\sqrt{(dx\;\overrightarrow{e_x}+dy\;\overrightarrow{e_y}+dz\;\overrightarrow{e_z})\cdot
+(dx\;\overrightarrow{e_x}+dy\;\overrightarrow{e_y}+dz\;\overrightarrow{e_z})}`$
+$`=\sqrt{dx^2\;(\overrightarrow{e_x}\cdot\overrightarrow{e_x})
++dy^2\;(\overrightarrow{e_y}\cdot\overrightarrow{e_y})   
++dz^2\;(\overrightarrow{e_z}\cdot\overrightarrow{e_z})`$
+$`+2\,dx\,dy\,x(\overrightarrow{e_x}\cdot\overrightarrow{e_y})`$
+$`+2\,dx\,dz\,x(\overrightarrow{e_x}\cdot\overrightarrow{e_z})`$
+$`+2\,dy\,dz\,x(\overrightarrow{e_y}\cdot\overrightarrow{e_z})}`$
+$`=\sqrt{dl_x^2+dl_y^2+dl_z^2}`$
+
 
 * **N3 ($`\rightarrow`$ N4)**<br>
 [ES] Los 3 vectores $`\partial\overrightarrow{OM}_x=\overrightarrow{dl_x}\quad`$,