Step 1

The equation relating to rectangular (x, y, z) and cylindrical \(\displaystyle{\left({r},\theta,{z}\right)}\) coordinates are, \(\displaystyle{x}={r} \cos{\theta},{y}= \sin{\theta},{z}={z}\)

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

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

Hence , the solution is \(\displaystyle \tan{\theta}=\frac{y}{{x}}.\)

The equation relating spherical coordinates to Cartesian and cylindrical coordinates is, \(\displaystyle{r}=\delta \sin{\theta},{x}={r} \cos{\theta}=\delta \sin{\cancel{\circ}} \cos{\theta}\)

\(\displaystyle{z}=\delta \cos{\cancel{\circ}},{y}={r} \sin{\theta}=\delta \sin{\cancel{\circ}} \sin{\theta}\)

Step 2

Here, \(\displaystyle\delta=\sqrt{{{x}^{2}+{y}^{2}+{z}^{2}}}\)

From equation 1, we know \(\displaystyle{x}^{2}+{y}^{2}={r}^{2}.\)

Therefore,

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

Hence, the solution is \(\displaystyle\delta=\sqrt{{{r}^{2}+{x}^{2}}}.\)

The diagram regarding the cylindrical coordinates \(\displaystyle{\left({r},\theta,{z}\right)}\) is given below:

The equation relating to rectangular (x, y, z) and cylindrical \(\displaystyle{\left({r},\theta,{z}\right)}\) coordinates are, \(\displaystyle{x}={r} \cos{\theta},{y}= \sin{\theta},{z}={z}\)

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

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

Hence , the solution is \(\displaystyle \tan{\theta}=\frac{y}{{x}}.\)

The equation relating spherical coordinates to Cartesian and cylindrical coordinates is, \(\displaystyle{r}=\delta \sin{\theta},{x}={r} \cos{\theta}=\delta \sin{\cancel{\circ}} \cos{\theta}\)

\(\displaystyle{z}=\delta \cos{\cancel{\circ}},{y}={r} \sin{\theta}=\delta \sin{\cancel{\circ}} \sin{\theta}\)

Step 2

Here, \(\displaystyle\delta=\sqrt{{{x}^{2}+{y}^{2}+{z}^{2}}}\)

From equation 1, we know \(\displaystyle{x}^{2}+{y}^{2}={r}^{2}.\)

Therefore,

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

Hence, the solution is \(\displaystyle\delta=\sqrt{{{r}^{2}+{x}^{2}}}.\)

The diagram regarding the cylindrical coordinates \(\displaystyle{\left({r},\theta,{z}\right)}\) is given below: