Update textbook.en.md

12.temporary_ins/05.coordinates-systems/30.cylindrical-coordinates/10.main/textbook.en.md

@@ -26,4 +26,76 @@ $`\def\PSclosed{\mathscr{S}_{\displaystyle\tiny\bigcirc}}`$
 
 <!--MétaDonnée : INS-1°année_-->
 
-Main Part to be done
+! *Suggested method:*
+!
+! Each in his own language adapts with his own words, his own sentences, the content of the little ones
+! numbered elements
+! of jointly developed courses. So it's not a word-for-word translation, but
+! the course elements being small, there is a very high corespondance on the content.
+! We can really display the courses in parallel in 2 or in all 3 languages,
+! it really makes sense to the student.
+! If we use different mathematical notations in the 3 languages, each language
+! keep its rating. The course display in "exchange" mode allows the student to compare
+! vocabulary, and mathematical notations.
+
+
+### Cylindrical coordinates
+
+#### Definition of coordinates and definition domains
+
+! For example, this course element denoted * CS300 *:
+
+* * CS300 *:
+
+Reference frame: Cartesian coordinate system $ `(O, x, y, z)` $
+\ - **1 point $`O`$ origin** of the space. <br>
+\ - **3 axes** named **$`Ox,Oy,Oz`$**, intersecting at $`O`$, **orthogonal 2 to 2**. <br>
+\ - **1 unit of length**. <br>
+
+! can give :
+
+The cylindrical coordinates are defined from a Cartesian coordinate system, i.e.
+\- 1 point $`O`  origin of space. <br>
+\- 3 axes named $`Ox, Oy, Oz`$, intersecting at $`O`$, orthogonal 2 to 2. <br>
+\- 1 unit of length. <br>
+
+! The following element * CS310 *:
+
+* * CS310 *:
+
+Cylindrical coordinates $`(\rho,\varphi,z)`$:
+
+\- Any point $`M`$ of space is orthogonally projected onto the plane $`xOy`$ leading 
+to the point $`m_{xy}`$, and on the $`Oz`$ axis leading to the point $`m_z`$.
+
+\- The **coordinate $`\ rho_M`$** of the point $`M`$ is the *nonalgebraic distance $`Om_{xy}`$* 
+between the point $`O`$ and the point $`m_ {xy}`$.<br>
+\- The **coordinate $`\varphi_M`$** of the point $`M`$ is the *nonalgebraic angle 
+$`\widehat{xOm_ {xy}}`$* between the axis $`Ox`$ and the half-line $`Om_{xy}`$,
+the direction of rotation being such that the trihedron *$`(Ox,Om_ {xy},Oz)`$* is 
+a *direct trihedron*.<br>
+\- The **coordinate $`z_M`$** of the point $`M`$ is the *algebraic distance $`\overline {Om_z}`$*
+between the point $`O`$ and the point $`m_z`$.
+
+**$`\rho_M=\overline{Om_ {xy}}`$, $`\varphi_M = \widehat{xOm_y}`, $`z_M =Om_z`$**
+
+! can give :
+
+The cylindrical coordinates are ordered and noted $`(\rho,\varphi,z)`$.
+
+For any point $`M`$ in space:
+
+\- The $`\ rho_M`$ coordinate of the point $`M`$ is the nonalgebraic distance $`Om_{xy}`$
+between point $`O`$ and point $ m_{xy}`$. <br>
+\- The coordinate $`\varphi_M`$ of the point $`M`$ is the nonalgebraic angle
+$`\widehat{xOm_{xy}}`$ between the axis $`Ox`$ and the half-line $`Om_ {xy}`$,
+the direction of rotation being such that the trihedron $`(Ox,Om_{xy},Oz)`$ is a direct trihedron. <br>
+\- The $`z_M`$ coordinate of the point $`M` $ is the algebraic distance $`\overline{Om_z}`$ 
+between the point $`O`$ and the point $`m_z`$.
+
+A same point $`M`$ located in $`z_M`$ on the axis $`Oz`$ can be represented by any triplet
+$`(z_M, 0, \varphi)`$ where $`\varphi`$ can take all possible values. By convention,
+the value $`\varphi`$ is set to 0, and the cylindrical coordinates of any point $`M`$ located
+in $`z_M`$ on the $`Oz`$ axis will be $`(z_M, 0, 0)`$.
+
+! and we continue on the sequence of course elements decided jointly:
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