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1.1 Coordinate Transformations

Transformations between Spherical and Cartesian Coordinate Systems

From Cartesian to Spherical

$\displaystyle \rho_{0}=\sqrt{x_{0}^{2}+y_{0}^{2}+z_{0}^{2}},$

 

$\displaystyle \theta_{0}=\arccos{\frac{z_{0}}{\sqrt{x_{0}^{2}+y_{0}^{2}+z_{0}^{2}}}},$

 

$\displaystyle \phi_{0}=\arccos{\frac{x_{0}}{\sqrt{x_{0}^{2}+y_{0}^{2}}}}$

 

(1.1.1)

 

 

(1.1.2)

 

 

(1.1.3)


$\displaystyle \vec{r}_{0}$ $\displaystyle =$ $\displaystyle [\,\rho_{0},\theta_{0},\phi_{0}]$ (1.1.4)
$\displaystyle \hat{\rho}_{0}$ $\displaystyle =$ $\displaystyle [\,\sin{\theta_{0}}\cos{\phi_{0}},\,\sin{\theta_{0}}\sin{\phi_{0}},\,\cos{\theta_{0}}]$ (1.1.5)
$\displaystyle \hat{\theta}_{0}$ $\displaystyle =$ $\displaystyle [\,\cos{\theta_{0}}\cos{\phi_{0}},\,\cos{\theta_{0}}\sin{\phi_{0}},-\sin{\theta_{0}}]$ (1.1.6)
$\displaystyle \hat{\phi}_{0}$ $\displaystyle =$ $\displaystyle [-\sin{\phi_{0}},\,\cos{\phi_{0}},0]$ (1.1.7)

 

From Spherical to Cartesian


$\displaystyle \vec{r}_{0}$ $\displaystyle =$ $\displaystyle [\,\rho_{0}\sin{\theta_{0}}\cos{\phi_{0}},\,\rho_{0}\sin{\theta_{0}}\cos{\phi_{0}},\,\rho_{0}\cos{\theta_{0}}]$ (1.1.8)
$\displaystyle \hat{x}_{0}$ $\displaystyle =$ $\displaystyle [\,\sin{\theta_{0}}\cos{\phi_{0}},\,\cos{\theta_{0}}\sin{\phi_{0}},\,-\sin{\phi_{0}}]$ (1.1.9)
$\displaystyle \hat{y}_{0}$ $\displaystyle =$ $\displaystyle [\,\sin{\theta_{0}}\sin{\phi_{0}},\,\cos{\theta_{0}}\cos{\phi_{0}},\,\cos{\phi_{0}}]$ (1.1.10)
$\displaystyle \hat{r}_{0}$ $\displaystyle =$ $\displaystyle [\,\cos{\theta_{0}},\,-\sin{\theta_{0}},\,0]$ (1.1.11)

 

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