Cylindrical coordinates prefer represent a point P in space by ordered triples \(\displaystyle{\left({r},\theta,\ {z}\right)}\) in which
r and \(\displaystyle\theta\) are polar coordinates for the vertical projection of P on the \(\displaystyle{x}{y}={p}{l}{a}\ne,\) with \(\displaystyle{r}\ \geq\ {0}\), and
x is the rectangular vertical coordinate.

The equations relating rectangular \(\displaystyle{\left({x},\ {y},\ {z}\right)}\) and cylindrical \(\displaystyle{\left({r},\theta,\ {z}\right)}\) coordinates are,

\(\displaystyle{x}={r}{\cos{\theta}}\)

\(\displaystyle{y}={r}{\sin{\theta}}\)

\(\displaystyle{z}={z}\)

\(\displaystyle{r}^{{{2}}}={x}^{{{2}}}\ +\ {y}^{{{2}}}\)

\(\displaystyle{\tan{\theta}}={\frac{{{y}}}{{{x}}}}\)

The spherical coordinates represent a point P in by ordered triples \(\displaystyle{\left(\rho,\phi,\theta\right)}\) in which,

\(\displaystyle\rho\) is the distance from P to the origin \(\displaystyle{\left(\rho\ \geq\ {0}\right)}\)

. \(\displaystyle\phi\) is the angle \(\displaystyle{o}{v}{e}{r}\rightarrow{\left\lbrace{O}{P}\right\rbrace}\) makes with the positive z - axis \(\displaystyle{\left({0}\ \leq\ \phi\ \leq\ \pi\right)}\)

. \(\displaystyle\theta\) is the angle from cylindrical coordinates.

The equations relating spherical coordinates to Cartesian and cylindrical coordinates are,

\(\displaystyle{r}=\rho{\sin{\phi}}\)

\(\displaystyle{x}={r}{\cos{\theta}}=\rho{\sin{\phi}}{\cos{\theta}}\)

\(\displaystyle{z}=\rho{\cos{\phi}}\)

\(\displaystyle{y}={r}{\sin{\theta}}=\rho{\sin{\phi}}{\sin{\theta}}\)

\(\displaystyle\rho=\sqrt{{{x}^{{{2}}}\ +\ {y}^{{{2}}}\ +\ {z}^{{{2}}}}}=\sqrt{{{r}^{{{2}}}\ +\ {z}^{{{2}}}}}\)

\(\displaystyle{\tan{\theta}}={\frac{{{y}}}{{{x}}}}\)

Cylindrical coordinates are good for describing cylinders whose axes run along the z - axis and planes that either contain the z - axis or lie perpendicular to the z -axis.

Surfaces like these have equations of constant coordinate value.

The equations relating rectangular \(\displaystyle{\left({x},\ {y},\ {z}\right)}\) and cylindrical \(\displaystyle{\left({r},\theta,\ {z}\right)}\) coordinates are,

\(\displaystyle{x}={r}{\cos{\theta}}\)

\(\displaystyle{y}={r}{\sin{\theta}}\)

\(\displaystyle{z}={z}\)

\(\displaystyle{r}^{{{2}}}={x}^{{{2}}}\ +\ {y}^{{{2}}}\)

\(\displaystyle{\tan{\theta}}={\frac{{{y}}}{{{x}}}}\)

The spherical coordinates represent a point P in by ordered triples \(\displaystyle{\left(\rho,\phi,\theta\right)}\) in which,

\(\displaystyle\rho\) is the distance from P to the origin \(\displaystyle{\left(\rho\ \geq\ {0}\right)}\)

. \(\displaystyle\phi\) is the angle \(\displaystyle{o}{v}{e}{r}\rightarrow{\left\lbrace{O}{P}\right\rbrace}\) makes with the positive z - axis \(\displaystyle{\left({0}\ \leq\ \phi\ \leq\ \pi\right)}\)

. \(\displaystyle\theta\) is the angle from cylindrical coordinates.

The equations relating spherical coordinates to Cartesian and cylindrical coordinates are,

\(\displaystyle{r}=\rho{\sin{\phi}}\)

\(\displaystyle{x}={r}{\cos{\theta}}=\rho{\sin{\phi}}{\cos{\theta}}\)

\(\displaystyle{z}=\rho{\cos{\phi}}\)

\(\displaystyle{y}={r}{\sin{\theta}}=\rho{\sin{\phi}}{\sin{\theta}}\)

\(\displaystyle\rho=\sqrt{{{x}^{{{2}}}\ +\ {y}^{{{2}}}\ +\ {z}^{{{2}}}}}=\sqrt{{{r}^{{{2}}}\ +\ {z}^{{{2}}}}}\)

\(\displaystyle{\tan{\theta}}={\frac{{{y}}}{{{x}}}}\)

Cylindrical coordinates are good for describing cylinders whose axes run along the z - axis and planes that either contain the z - axis or lie perpendicular to the z -axis.

Surfaces like these have equations of constant coordinate value.