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12.temporary_ins/05.coordinates-systems/20.cartesian-coordinates/20.overview/cheatsheet.en.md

+---
+title: Cartesian coordinates
+published: true
+routable: false
+visible: false
+lessons:
+    - slug: cartesian-cylindrical-spherical-coordinates
+      order: 2
+    - slug: cartesian-coordinates-linear
+      order: 2
+---
+
+
+<!--caligraphie de l'intégrale double curviligne-->
+$`\def\oiint{\displaystyle\mathop{{\iint}\mkern-18mu \scriptsize \bigcirc}}`$
+$`\def\Ltau{\Large{\tau}\normalsize}`$
+$`\def\Sopen{\mathscr{S}_{\smile}}`$
+$`\def\Sclosed{\mathscr{S}_{\displaystyle\tiny\bigcirc}}`$
+$`\def\Ssclosed{\mathscr{S}_{\scriptsize\bigcirc}}`$
+$`\def\PSopen{\mathscr{S}_{\smile}}`$
+$`\def\PSclosed{\mathscr{S}_{\displaystyle\tiny\bigcirc}}`$
+
+!!!! *LESSON  IN  CONSTRUCTION :* <br>
+!!!! Published but invisible: does not appear in the tree structure of the m3p2.com site. This course is *under construction*, it is *not approved by the pedagogical team* at this stage. <br>
+!!!! Working document intended only for the pedagogical team.
+
+<!--MétaDonnée : INS-1°année_-->
+
+
+#### What are ... ? 
+
+* 3 coordinates
+
+* 
+
+* **$`\mathbf{\rho}`$** and **$`\mathbf{z}`$** are
+
+* **$`\mathbf{\varphi}`$** is an *angle* expr...  *($`\mathbf{rad}`$)*.
+
+----
+
+![](cylindrical_coordinates_definition_L1200.gif)
+
+-----
+ 
+#### What are ... ? 
+
+-----
+
+![](cylindrical_coordinates_variation_range_L1200_v2.gif)
+
+-----
+
+#### How ... ?
+
+* Method : ... $`\overrightarrow{OM}`$ ... $`Oz`$, ... $`xOy`$ ... $`M_{xOy}`$ 
+* ... $`Ox`$ et $`Oy`$, *...* ... *sine* y *cosine*.
+
+----
+
+![](cylindrical_coordinates_projection.png)
+
+------
+
+* $`\Longrightarrow`$
+**$`\quad\mathbf{}\left\{\begin{array}{l} \mathbf{ x=\rho\cdot\cos\varphi} \\\mathbf{ y=\rho\cdot\sin\varphi} \\\mathbf{ z=z} \\ \end{array}\right. `$**
+
+#### How ... ?
+
+* ... $`\overrightarrow{e_{\alpha}}`$ ... **...** ... $`M`$ ... *s... $`\alpha`$* ... $`M`$ *... $`d\alpha^+`$*.
+
+##### Vectors ...  $`\overrightarrow{e_{\varphi}}`$
+
+---------
+
+![](cylindrical_coordinates_unit_vector_phi_definition_L1200_v3.gif)
+
+--------
+
+* D... **$`\mathbf{M(\rho,\varphi,z) \longrightarrow M"(\rho,\varphi+\Delta\varphi^+,z)}`$**<br>
+ (with $`\Delta\varphi^+=\Delta\varphi>0`$)<br>
+<br>**$`\Longrightarrow`$ ...** ... **$`\mathbf{\overrightarrow{e_{\varphi}}}`$**<br>
+$`\Longrightarrow\overrightarrow{e_{\varphi}}`$ : ... $`M`$ ... $`\rho_M`$ ... $`z_M=const`$, ... $`\varphi`$ ....
+
+* ... : $`l_{\Delta\varphi}`$<br>
+ ... : $`\overrightarrow{MM''}`$
+
+* ... *... :  $`\mathbf{l_{\Delta\varphi} \ne\, ||\overrightarrow{MM''}||}`$*.
+* ... **infinitesimal : $`\mathbf{dl_{\varphi}=\,||\overrightarrow{MM''}||}`$**.
+ 
+* ... ($`d\varphi=d\varphi^+>0`$  o $`d\varphi^-<0`$) :<br>
+<br>**$`\mathbf{\overrightarrow{dl_{\varphi}}}`$** *$`\displaystyle=\lim_{\Delta\varphi\rightarrow 0} \overrightarrow{MM''}`$* **$`\mathbf{=\rho_M\cdot d\varphi\cdot\overrightarrow{e_{\varphi}}}`$**<br>
+
+
+
+##### Vectors ... $`\overrightarrow{e_{\rho}}`$ y $`\overrightarrow{e_z}`$
+
+---------
+
+![](cylindrical_coordinates_e-z_e-rho_unit_vector_L1200.gif)
+
+--------
+
+* **$`\mathbf{M(\rho,\varphi,z) \longrightarrow M'(\rho+\Delta\rho^+,\varphi,z)}`$**<br>
+**$`\mathbf{M(\rho,\varphi,z) \longrightarrow M'''(\rho,\varphi,z+\Delta z^+)}`$** <br>
+(con $`\Delta\rho^+=\Delta\rho>0`$ y $`\Delta z^+=\Delta z>0`$)<br>
+<br>**$`\Longrightarrow`$ ...** ... <br>
+**$`\quad\overrightarrow{e_{\rho}}`$** : ... $`Om_{xOy}`$.<br>
+**$`\quad\overrightarrow{e_z}`$** : ... $`Oz`$.
+
+* ... $`M`$ : ... <br>
+$`\Longrightarrow`$ ... = ....<br>
+$`\Longrightarrow`$ $`l_{\Delta\rho}=||\overrightarrow{MM'}||\quad`$ et $`\quad l_{\Delta z}=||\overrightarrow{MM'''}||`$
+
+
+* ... ($`d\rho\;, dz >0\;\text{ou}<0`$) :<br>
+**$`\mathbf{\overrightarrow{dl_{\rho}}}`$** $`\displaystyle=\lim_{\Delta\rho\rightarrow 0} \overrightarrow{MM'}`$ **$`\mathbf{ = d\rho \cdot \overrightarrow{e_{\rho}}}`$**.<br>
+ **$`\mathbf{\overrightarrow{dl_z}}`$** $`\displaystyle=\lim_{\Delta z \rightarrow 0} \overrightarrow{MM'''}`$
+ **$`\mathbf{=dz \cdot \overrightarrow{e_z}}`$**.
+
+#### La base $`(\overrightarrow{e_{\rho}}, \overrightarrow{e_{\varphi}}, \overrightarrow{e_z})`$ esta ortonormada.
+
+----
+
+![](cylindrical_coordinates_orthogonal_base_L1200.jpg)
+
+---
+
+* $`(\overrightarrow{e_{\rho}}, \overrightarrow{e_{\varphi}}, \overrightarrow{e_z})`$ ... *... $`M(\rho_M,\varphi_M,z_M)`$*.
+
+*  **$`(\overrightarrow{e_{\rho}}, \overrightarrow{e_{\varphi}}, \overrightarrow{e_z})`$** ... **directa si $`(\overrightarrow{e_x}, \overrightarrow{e_y}, \overrightarrow{e_z})`$** ... **direc...**, y *...*.
+
+* **$`\left\{ \begin{array}{l}\mathbf{\overrightarrow{e_{\rho}}=\cos\varphi\cdot\overrightarrow{e_x}+\sin\varphi\cdot\overrightarrow{e_y}} \\\mathbf{\overrightarrow{e_{\varphi}}=-\sin\varphi\cdot\overrightarrow{e_x}+\cos\varphi\cdot\overrightarrow{e_y}} \end{array}\right.`$**
+
+* ...  $`(O,\overrightarrow{e_x}, \overrightarrow{e_y}, \overrightarrow{e_z},t)`$, .. *... $`(\overrightarrow{e_x}, \overrightarrow{e_y}, \overrightarrow{e_z})`$* :<br>
+\- ... **...**.<br>
+\- **...** *cuando $`\varphi_M`$ ...*.
+
+#### How ... $`\overrightarrow{OM}`$ ?
+
+----
+
+![](cylindrical_coordinates_vector_OM_L1200.gif)
+
+---
+
+*  **$`\mathbf{\overrightarrow{OM}=\rho_M\cdot\overrightarrow{e_{\rho}}+z_M\cdot\overrightarrow{e_z}}`$**
+
+#### What are ... $`dl`$ and ... $`\overrightarrow{dl}`$ ?
+
+* A point **$`M(\rho,\varphi,z)`$** ... **...** ... $`M'(\rho+d\rho,\varphi+d\varphi,z+dz)`$, with *$`d\rho`$, $`d\varphi`$ y $`dz`$ ..., ...*, ... $`\rho\;,\;\varphi\;,\;z`$.
+
+##### Vector ... $`\overrightarrow{dl}`$
+
+* vector ... = *...c* [Norme IEC](http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=102-05-02)
+
+* The **vector ...** ...
+**$`\overrightarrow{dl}`$** $`\;=dl_{\rho}\cdot\overrightarrow{e_{\rho}}+dl_{\varphi}\cdot\overrightarrow{e_{\varphi}}+dl_z\cdot\overrightarrow{e_z}`$
+**$`\quad=dl_{\rho}\cdot\overrightarrow{e_{\rho}}+\rho\,d\varphi\cdot\overrightarrow{e_{\varphi}}+dl_z\cdot\overrightarrow{e_z}`$**
+
+*  enables to calculate the vectors ... $`\overrightarrow{v}(t)`$ y ... $`\overrightarrow{a}(t)`$ of a point M at each instant t :<br>
+**$`\overrightarrow{v}(t)`$**$`\;=\dfrac{\overrightarrow{dOM}}{dt}`$**$`\;=\dfrac{\overrightarrow{dl}}{dt}`$**<br>
+**$`\overrightarrow{a}(t)`$**$`\;=\dfrac{\overrightarrow{d^2 OM}}{dt^2}`$**$`\;=\dfrac{d}{dt}\left(\dfrac{\overrightarrow{dl}}{dt}\right)`$**
+
+##### ... $`dl`$
+
+* ... = *...* [Norme IEC](http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=102-05-01)
+
+* ... **... $`dl`$** ... *...* ... $`M`$ y $`M'`$ :<br>
+**$`dl`$**$`\;=\sqrt{dl_{\rho}^2+dl_{\varphi}^2+dl_z^2}`$**$`\;=\sqrt{d\rho^2+\rho^2\,d\varphi^2+dz^2}`$**
+
+* Enables to calculate the length $`\mathscr{l}`$ of a trajectory $`L`$ ... $`\rho(t)`$, $`\varphi(t)`$ y $`z(t)`$ ...s :<br>
+**$`\displaystyle\mathbf{\mathscr{l}=\int_L dl}`$**
+
+#### What is the ... ?
+
+---
+
+![](cylindrical_coordinates_surface_4_L1200.jpg)<br>
+<br>
+![](cylindrical_coordinates_surface_2_L1200.jpg)<br>
+<br>
+![](cylindrical_coordinates_surface_3_L1200.jpg)<br>
+
+---
+
+#### What is ... ?
+
+---
+
+![](cylindrical_coordinates_volume_L1200.jpg)<br>
+
+
+     
+
+