Coordenadas Cilíndricas ${\xi }_k\in \left\{\mathrm{\rho ,\; \phi ,\; z}\right\}$

Um sistema de coordenadas ortogonais curvilíneas Cilíndricas consiste na definição de três parâmetros $\mathrm{\{\rho ,\; \phi ,\; z\; \}}$, únicos para cada ponto $\overrightarrow{r}\in {ℝ}^3$, ou seja duma correspondência bijectiva $\overrightarrow{r}=\; f(\rho ,\; \phi ,\; z)$ (de facto será necessário definir várias e por partes i.e. um Atlas) com $\mathrm{\rho \hspace{0.17em}\in \hspace{0.17em}]0,+\infty ]}$, $\mathrm{\phi \hspace{0.17em}\in \hspace{0.17em}[0,2\pi [}$, $\mathrm{z }\in \hspace{0.17em}$ℝ, de tal maneira que as curvas $\mathrm{u(\rho )\; =\; f(\rho ,}{\phi }_o,z_o),\mathrm{v(\phi )=f(}{\rho }_o,\phi ,z_o),w\mathrm{(}z\mathrm{)=f(}{\rho }_o,{\phi }_o,z\mathrm{)}$, obtidas pela fixação de qualquer par de parâmetros $\mathrm{\{\rho ,\; \phi ,\; z\}}$, passem por ${\overrightarrow{r}}_o\mathrm{=f(}{\rho }_o,{\phi }_o,z_o\mathrm{)=\; u(}{\rho }_o\mathrm{)=v(}{\phi }_o\mathrm{)=\; w(}{z}_o\mathrm{)}$ e sejam ortogonais entre si, qualquer que seja o ponto ${\overrightarrow{r}}_o$ escolhido.

Designamos por referencial móvel associado ao sistema de coordenadas curvilíneas indicado o triedro de vectores de módulo unitário ${\overrightarrow{e}}_{\rho }$, ${\overrightarrow{e}}_{\phi }$, ${\overrightarrow{e}}_z$ no ponto ${\overrightarrow{r}}_o$ que representam as direcções tangentes a cada uma das curvas

$u\left(\rho \right)$ - (semi-rectas horizontais passando pelo eixo dos $z$),
$v\left(\phi \right)$ - (circulos horizontais concêntricos com o eixo dos $z$),
$w\left(z\right)$ - (rectas verticais).

Vector Posição no referencial associado:

$\overrightarrow{r}=\rho {\overrightarrow{e}}_{\rho }+z{\overrightarrow{e}}_z$

Relação com coordenadas cartesianas:

$\{\begin{array}{l}r\in \left[0,\infty \right]\\ \phi \in \left[0,\pi \right]\\ z\in \left]\mathrm{-}\mathrm{\infty },\mathrm{+}\infty \right[\end{array}\begin{array}{l}\stackrel{{\Lambda }^{\mathrm{-}1}}{\to }\{\begin{array}{l}x=\rho \cos \left(\phi \right)\\ y=\rho \sin \left(\phi \right)\\ z=z\end{array}\end{array}$

$\overrightarrow{r}=\rho \left(\cos \left(\phi \right){\overrightarrow{e}}_x+\sin \left(\phi \right){\overrightarrow{e}}_y\right)+z{\overrightarrow{e}}_z$

Versores do referencial associado:

${\overrightarrow{e}}_{\rho }\left(\phi \right)=\cos \left(\phi \right){\overrightarrow{e}}_x+\sin \left(\phi \right){\overrightarrow{e}}_y$

${\overrightarrow{e}}_{\phi }\left(\phi \right)=\mathrm{-}\sin \left(\phi \right){\overrightarrow{e}}_x+\cos \left(\phi \right){\overrightarrow{e}}_y$

${\overrightarrow{e}}_z={\overrightarrow{e}}_z$

Funções Escala:

${\eta }_k={\displaystyle \left|\frac{\partial \overrightarrow{r}}{\partial {\xi }_k}\right|} \to \{\begin{array}{c}{\eta }_{\mathrm{\rho }}=1\\ {\eta }_{\mathrm{\phi }}=\rho \\ {\eta }_{\mathrm{z}}=1\end{array}$

Elemento de Linha:

$d\overrightarrow{r}=d\rho {\overrightarrow{e}}_{\rho }+\rho d\phi {\overrightarrow{e}}_{\phi }+dz{\overrightarrow{e}}_z$

Elementos de áreas coordenadas infinitesimais:

$d\overrightarrow{S}\left(\phi ,z\right)=\left({\eta }_{\phi }d\phi {\overrightarrow{e}}_{\phi }\right)\times \left({\eta }_{z}dz{\overrightarrow{e}}_z\right)=\rho d\phi dz{\overrightarrow{e}}_{\rho }$

$d\overrightarrow{S}\left(\rho ,\phi \right)=\left({\eta }_{\rho }d\rho {\overrightarrow{e}}_{\rho }\right)\times \left({\eta }_{\phi }d\phi {\overrightarrow{e}}_{\phi }\right)=\rho d\rho d\phi {\overrightarrow{e}}_z$

$d\overrightarrow{S}\left(z,\rho \right)=\left({\eta }_{z}dz{\overrightarrow{e}}_z\right)\times \left({\eta }_{\rho }d\rho {\overrightarrow{e}}_{\rho }\right)=dzd\rho {\overrightarrow{e}}_{\phi }$

Elemento de Volume infinitesimal

$dV\left(\rho ,\theta ,z\right)=\rho d\rho d\phi dz$

Gradiente:

$$\nabla f=\frac{\partial f}{\partial \rho }{\overrightarrow{e}}_{\rho }+\frac{1}{\rho }\frac{\partial f}{\partial \phi }{\overrightarrow{e}}_{\phi }+\frac{\partial f}{\partial z}{\overrightarrow{e}}_z$$

Divergência:

$$\nabla ·\overrightarrow{\mathrm{A}}\text{=}\frac{1}{\rho }\left(\frac{\partial \left(\rho A_{\rho }\right)}{\partial \rho }+\frac{\partial A_{\phi }}{\partial \phi }+\frac{\partial \left(\rho A_z\right)}{\partial z}\right)=\frac{1}{\rho }{A}_{\rho }\mathrm{+}\frac{\partial A_{\rho }}{\partial \rho }+\frac{1}{\rho }\frac{\partial A_{\phi }}{\partial \phi }+\frac{\partial A_z}{\partial z}$$

Rotacional:

$$\nabla \times \overrightarrow{\mathrm{A}}=\frac{1}{\rho }\left(\frac{\partial A_z}{\partial \phi }\mathrm{-}\frac{\partial \left(\rho A_{\phi }\right)}{\partial z}\right){\overrightarrow{e}}_{\rho }+\left(\frac{\partial A_{\rho }}{\partial z}\mathrm{-}\frac{\partial A_z}{\partial \rho }\right){\overrightarrow{e}}_{\phi }+\frac{1}{\rho }\left(\frac{\partial \left(\rho A_{\phi }\right)}{\partial \rho }\mathrm{-}\frac{\partial A_{\rho }}{\partial \phi }\right){\overrightarrow{e}}_z$$

Laplaciano:

$${\nabla }^{2}f=\frac{1}{\rho }\left(\frac{\partial }{\partial \rho }\left(\rho \frac{\partial f}{\partial \rho }\right)+\frac{\partial }{\partial \phi }\left(\frac{1}{\rho }\frac{\partial f}{\partial \phi }\right)+\frac{\partial }{\partial z}\left(\rho \frac{\partial f}{\partial z}\right)\right)=\frac{{\partial }^{2}f}{\partial {\rho }^2}+\frac{1}{\rho }\frac{\partial f}{\partial \rho }+\frac{1}{{\rho }^2}\frac{{\partial }^{2}f}{\partial {\phi }^2}+\frac{{\partial }^{2}f}{\partial z^2}$$

© Amaro Rica da Silva, Prof. Dep. Física-IST with Mathematica  (September 20, 2005)