Is the given unit vector derivation valid for any coordinate system?

[LagrangeEuler]

$\vec{r}=\rho \cos \varphi \vec{i}+\rho \sin \varphi \vec{j}+z\vec{k}$
we get
$$\vec{e}_{\rho}=\frac{\frac{\partial \vec{r}}{\partial \rho}}{|\frac{\partial \vec{r}}{\partial \rho}|}$$
$$\vec{e}_{\varphi}=\frac{\frac{\partial \vec{r}}{\partial \varphi}}{|\frac{\partial \vec{r}}{\partial \varphi}|}$$
Is it correct for any orthogonal system or maybe for any system? Why you can use this relation?

 

[mathman]

It would help if you would define the symbols, $$\vec{e}_{\varphi}, \vec{e}_{\rho}$$

 

[LagrangeEuler]

Unit vectors in cylindrical system.

 

[mathman]

If you divide any vector by its magnitude you get a unit vector. I am not sure what you are asking?