How to write a Vector Field in Cylindrical Co-ordinates?

[greswd]

Let's say we have a vector field that looks similar to this. Assume that the above image is of the x-y plane.

The vector arrows circulate a central axis, you can think of them as tangents to circles.

The field does not depend on the height z.

The lengths of the arrows is a function of their radial distance from the center/axis, f(r).How do we write this vector field in terms of Cylindrical coordinates?
$A_\rho \hat{\boldsymbol \rho} + A_\varphi \hat{\boldsymbol \varphi} + A_z \hat{\mathbf z}$

How does one find $A_\rho , A_\varphi$ and $A_z$ ?

 

[Let'sthink]

Vector V = (Aφ)* unit vector φ, where Aφ = f(ρ) with Aρ = 0 and Aρ = 0 and Az = 0

 

[greswd]

Vector V = (Aφ)* unit vector φ, where Aφ = f(ρ) with Aρ = 0 and Aρ = 0 and Az = 0

thanks!

But what do you mean by "with Aρ = 0"?

Also, can you describe how the cylindrical unit vectors work? You can post a link if you like.

 

[Let'sthink]

Aρ = 0 and Az = 0 means the component of the vector field along ρ unit vector and along z unit vector both are zero.In other words vector field every where is perpendicular to ρ unit vector and z unit vector and is entirely along unit vector φ.

 

[greswd]

Aρ = 0 and Az = 0 means the component of the vector field along ρ unit vector and along z unit vector both are zero.In other words vector field every where is perpendicular to ρ unit vector and z unit vector a ρ unit vector and is entirely along unit vector φ.

thanks, that was a great simple explanation.