Cylindrical coordinate $\theta$ when $r=0$ When we use the cylindrical coordinate system $(r, \theta, z)$ where $r$ is the distance from the point in the $xy$-plane, $\theta$ is the angle with the $x$ axis and $z$ is the height. As can been seen in the picture

I have a vector field described by $(0,U_{\theta}(r),U_z)$ but how can the angle differ when $r$ is always zero?
When $r$ is zero, then there is no difference between different angles. Or am I wrong?
 A: Writing $ (x,y,z)=(r,\theta,z)$ is a bit of abuse of notation. This equation is not meant literally, in the sence that if two triples of numbers are the same then each number in the triplet must be the same. Rather it means that you can describe the same point in space either by giving the cartesian coordinates: x,y,z or by giving the zylindrical coordinates $r,\theta,z $. 
The notation for the vectorfield, $(0,U_{\theta}(r),U_z)$, again means something different. It means that at each point in space, parameterized by the zylindrical coordinates (thats why the U is written as a function of r), you assign a vector, which is a linear combination of the three zylindrical basis vectors, $e_r,e_\theta,e_z. $ These point from each point in space in the direction of increasing radius, theta angle or z respectively. So we have
$$(0,U_{\theta}(r),U_z)=0*e_r(\theta)+U_{\theta}(r)e_\theta(r,\theta)+U_z e_z$$
Where i have highlighted the fact that the basis vectors also depend on the position in space, and can be parameterized as a function of the coordinates.
This vectorfield is defined for every value of $ \theta, z$ and for all r except $r=0$, since as you wrote, in zylindrical coordinates the $\theta$ coordinate is not defined there.
A: The crucial point is that it is a vector field, which associates to every point in space, a vector. In this case, we have that, 
$$\vec u = U_\theta(r) \hat\theta + U_z \hat z.$$
Thus, we can draw the conclusions:


*

*The field does not care about the $z$ or $\theta$ coordinates of a point when assigning a vector $\vec u$.

*The radial $\hat r$ component of $\vec u$ is always zero, for all points.


So, if I pick a point, say $(r,\theta,z) = (2, \pi/9, 0)$, the vector field would assign to that point the vector,
$${\left.\vec{u}\right|}_{(2,\,\pi/9,\,0)} = U_\theta(2) \hat\theta + U_z \hat z.$$
The $\hat z$ is easy to interpret, it is simply a vector along the $z$ axis. As for $\hat \theta$, if you pick a point in space, and draw a circle which intersects that point, the tangent at that point has direction $\hat\theta$. So for example, the tangential velocity of a rotating object is $\Omega r \hat\theta$.
