## Proof That the Cylindrical Coordinate System is Orthonormal

Theorem
The cylindrical coordinate system is orthonormal.
Proof
In cylindrical coordinates
$(r, \theta , z)$
we can write
$x= r cos \theta$

$y= r sin \theta$

$z=z$

Hence
$\mathbf{r}= r cos \theta \mathbf{i} + r sin \theta \mathbf{j} + z \mathbf{k}$

Then
$\frac{\partial \mathbf{r}}{\partial r}= cos \theta \mathbf{i} + sin \theta \mathbf{j}$

$\frac{\partial \mathbf{r}}{\partial \theta}= -r sin \theta \mathbf{i} + r cos \theta \mathbf{j}$

$\frac{\partial \mathbf{r}}{\partial z}= \mathbf{k}$

$\mathbf{e_1} = \frac{ \partial \mathbf{r} / \partial r}{| \partial \mathbf{r} / \partial r} = \frac{cos \theta \mathbf{i} + sin \theta \mathbf{j}}{ \sqrt{cos^2 \theta + sin^2 \theta}} = cos \theta \mathbf{i} + sin \theta \mathbf{j}$

$\mathbf{e_2} = \frac{ \partial \mathbf{r} / \partial \theta}{| \partial \mathbf{r} / \partial \theta}= \frac{-r sin \theta \mathbf{i} + r cos \theta \mathbf{j}}{ \sqrt{ r^2 cos^2 \theta + r^2 sin^2 \theta}} = - sin \theta \mathbf{i} + cos \theta \mathbf{j}$

$\mathbf{e_3} = \frac{ \partial \mathbf{r} / \partial z}{| \partial \mathbf{r} / \partial z |}= \frac{\mathbf{k}}{1} = \mathbf{k}$

Then
$\mathbf{e_1} \cdot \mathbf{e_1} = \mathbf{e_2} \cdot \mathbf{e_2} = \mathbf{e_3} \cdot \mathbf{e_3}=1$

and
$\mathbf{e_1} \cdot \mathbf{e_2} = \mathbf{e_1} \cdot \mathbf{e_3} = \mathbf{e_2} \cdot \mathbf{e_3}=\mathbf{e_2} \cdot \mathbf{e_1} = \mathbf{e_3} \cdot \mathbf{e_1} = \mathbf{e_3} \cdot \mathbf{e_2}=0$