---
title: Cylindrical coordinates
published: false
routable: false
visible: false
lessons:
    - slug: cartesian-cylindrical-spherical-coordinates
      order: 2
    - slug: cylindrical-coordinates-linear
      order: 2
---

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    - slug: es-en-cylindrical-coordinates-exchange
      order: 2
-->


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$`\def\oiint{\displaystyle\mathop{{\iint}\mkern-18mu \scriptsize \bigcirc}}`$
$`\def\Ltau{\Large{\tau}\normalsize}`$
$`\def\Sopen{\mathscr{S}_{\smile}}`$
$`\def\Sclosed{\mathscr{S}_{\displaystyle\tiny\bigcirc}}`$
$`\def\Ssclosed{\mathscr{S}_{\scriptsize\bigcirc}}`$
$`\def\PSopen{\mathscr{S}_{\smile}}`$
$`\def\PSclosed{\mathscr{S}_{\displaystyle\tiny\bigcirc}}`$

!!!! *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...  *($`\mathbf{rad}`$)*.

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![](cylindrical_coordinates_definition_L1200.gif)

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#### What are ... ? 

-----

![](cylindrical_coordinates_variation_range_L1200_v2.gif)

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#### How ... ?

* Method : ... $`\overrightarrow{OM}`$ ... $`Oz`$, ... $`xOy`$ ... $`M_{xOy}`$ 
* ... $`Ox`$ et $`Oy`$, *...* ... *sine* y *cosine*.

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![](cylindrical_coordinates_projection.png)

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* $`\Longrightarrow`$
**$`\quad\mathbf{}\left\{\begin{array}{l} \mathbf{ x=\rho\cdot\cos\varphi} \\\mathbf{ y=\rho\cdot\sin\varphi} \\\mathbf{ z=z} \\ \end{array}\right. `$**

#### 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.

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![](cylindrical_coordinates_orthogonal_base_L1200.jpg)

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* $`(\overrightarrow{e_{\rho}}, \overrightarrow{e_{\varphi}}, \overrightarrow{e_z})`$ ... *... $`M(\rho_M,\varphi_M,z_M)`$*.

*  **$`(\overrightarrow{e_{\rho}}, \overrightarrow{e_{\varphi}}, \overrightarrow{e_z})`$** ... **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>


     


