Given:

Point is represented as \(\displaystyle{\left({r},\theta,{z}\right)}.\)

Proof:

A point in cylindrical coordinate is represented by \(\displaystyle{\left({r},\theta.{z}\right)}.\)

\(\displaystyle{\left({r},\theta\right)}\) are polar coordinates

z is the usual - coordinates

The rectangular (x, y, z) and the spherical coordinates \(\displaystyle{\left({r},\theta,{z}\right)}\) are related as

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

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

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

These are used to convert from cylindrical to rectangular coordinates.

Rectangular to cylindrical coordinates:

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

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

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

A point in spherical coordinate is represented by \(\displaystyle{\left(\rho,\theta,\varphi\right)}.\)

The rectangular (x, y, z) and the spherical coordinates \(\displaystyle{\left(\rho,\theta,\varphi\right)}\) are related as

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

\(\displaystyle{y}=\rho \sin{\varphi} \sin{\theta}\)

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

These are used to convert from spherical to rectangular coordinates.

Rectangular to spherical coordinates:

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

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

\(\displaystyle\varphi={{\cos}^{ -{{1}}}{\left(\frac{z}{\sqrt{{{x}^{2}+{y}^{2}+{z}^{2}}}}\right)}}\)

Spherical to rectangular coordinates:

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

\(\displaystyle\theta=\varphi\)

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

Cylindricalto spherical coordinates:

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

\(\displaystyle\theta=\theta\)

\(\displaystyle\varphi={{\cos}^{ -{{1}}}{\left(\frac{z}{\sqrt{{{r}^{2}+{z}^{2}}}}\right)}}\)

Point is represented as \(\displaystyle{\left({r},\theta,{z}\right)}.\)

Proof:

A point in cylindrical coordinate is represented by \(\displaystyle{\left({r},\theta.{z}\right)}.\)

\(\displaystyle{\left({r},\theta\right)}\) are polar coordinates

z is the usual - coordinates

The rectangular (x, y, z) and the spherical coordinates \(\displaystyle{\left({r},\theta,{z}\right)}\) are related as

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

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

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

These are used to convert from cylindrical to rectangular coordinates.

Rectangular to cylindrical coordinates:

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

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

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

A point in spherical coordinate is represented by \(\displaystyle{\left(\rho,\theta,\varphi\right)}.\)

The rectangular (x, y, z) and the spherical coordinates \(\displaystyle{\left(\rho,\theta,\varphi\right)}\) are related as

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

\(\displaystyle{y}=\rho \sin{\varphi} \sin{\theta}\)

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

These are used to convert from spherical to rectangular coordinates.

Rectangular to spherical coordinates:

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

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

\(\displaystyle\varphi={{\cos}^{ -{{1}}}{\left(\frac{z}{\sqrt{{{x}^{2}+{y}^{2}+{z}^{2}}}}\right)}}\)

Spherical to rectangular coordinates:

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

\(\displaystyle\theta=\varphi\)

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

Cylindricalto spherical coordinates:

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

\(\displaystyle\theta=\theta\)

\(\displaystyle\varphi={{\cos}^{ -{{1}}}{\left(\frac{z}{\sqrt{{{r}^{2}+{z}^{2}}}}\right)}}\)