 # To compare and contrast: the rectangular, cylindrical and spherical coordinates systems. Tazmin Horton 2020-12-28 Answered
To compare and contrast: the rectangular, cylindrical and spherical coordinates systems.
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Given:
Point is represented as $\left(r,\theta ,z\right).$
Proof:
A point in cylindrical coordinate is represented by $\left(r,\theta .z\right).$
$\left(r,\theta \right)$ are polar coordinates
z is the usual - coordinates
The rectangular (x, y, z) and the spherical coordinates $\left(r,\theta ,z\right)$ are related as
$x=r\mathrm{cos}\theta$
$y=r\mathrm{sin}\theta$
$z=z$
These are used to convert from cylindrical to rectangular coordinates.
Rectangular to cylindrical coordinates:
${r}^{2}={x}^{2}+{y}^{2}$
$\mathrm{tan}\theta =\frac{y}{x}$
$z=z$
A point in spherical coordinate is represented by $\left(\rho ,\theta ,\phi \right).$
The rectangular (x, y, z) and the spherical coordinates $\left(\rho ,\theta ,\phi \right)$ are related as
$x=\rho \mathrm{sin}\phi \mathrm{cos}\theta$
$y=\rho \mathrm{sin}\phi \mathrm{sin}\theta$
$z=\rho \mathrm{cos}\phi$
These are used to convert from spherical to rectangular coordinates.
Rectangular to spherical coordinates:
${\rho }^{2}={x}^{2}+{y}^{2}+{z}^{2}$
$\mathrm{tan}\theta =\frac{y}{x}$
$\phi ={\mathrm{cos}}^{-1}\left(\frac{z}{\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}}\right)$
Spherical to rectangular coordinates:
$r=\rho \mathrm{sin}\phi$
$\theta =\phi$
$z=\rho \mathrm{cos}\phi$
Cylindricalto spherical coordinates:
${\rho }^{2}=\sqrt{{r}^{2}+{z}^{2}}$
$\theta =\theta$
$\phi ={\mathrm{cos}}^{-1}\left(\frac{z}{\sqrt{{r}^{2}+{z}^{2}}}\right)$