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Transformations between Cylindrical and Cartesian

 

 

 

 

 

 

 


From Cartesian to Cylindrical

$\displaystyle \vec{r}_{0}$ $\displaystyle =$ $\displaystyle [\,\rho_{0},\,\phi_{0},\,z_{0}]=\left[\sqrt{x_{0}^{2}+y_{0}^{2}},\,\arctan{\left(\frac{x_{0}}{y_{0}}\right),\,z_{0}}\right]$ (1.1.12)
$\displaystyle \hat{\rho}_{0}$ $\displaystyle =$ $\displaystyle \left[\frac{x_{0}}{\sqrt{x_{0}^{2}+y_{0}^{2}}},\,\frac{y_{0}}{\sqrt{x_{0}^{2}+y_{0}^{2}}},\,0\right]$ (1.1.13)
$\displaystyle \hat{\phi}_{0}$ $\displaystyle =$ $\displaystyle \left[-\frac{y_{0}}{\sqrt{x_{0}^{2}+y_{0}^{2}}},\,\frac{x_{0}}{\sqrt{x_{0}^{2}+y_{0}^{2}}},\,0\right]$ (1.1.14)
$\displaystyle \hat{z}_{0}$ $\displaystyle =$ $\displaystyle [\,0,0,1]$ (1.1.15)

 

From Cylindrical to Cartesian


$\displaystyle \vec{r}_{0}$ $\displaystyle =$ $\displaystyle [\,x_{0},\,y_{0},\,z_{0}]=\left[\,\rho_{0}\cos{\phi_{0}},\,\rho_{0}\sin{\phi_{0}},\,z_{0}\right]$ (1.1.16)
$\displaystyle \hat{x}_{0}$ $\displaystyle =$ $\displaystyle \left[\,\cos{\phi_{0}},\,-\sin{\phi_{0}},\,0\right]$ (1.1.17)
$\displaystyle \hat{y}_{0}$ $\displaystyle =$ $\displaystyle \left[\,\sin{\phi_{0}},\,\cos{\phi_{0}},\,0\right]$ (1.1.18)
$\displaystyle \hat{z}_{0}$ $\displaystyle =$ $\displaystyle [\,0,0,1]$ (1.1.19)

 

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