Angular momentum in Cylindrical Coordinates How to calculate the angular momentum of a particle in  a cylindrical coordinates system
$$x_1 = r \cos{\theta}$$
$$x_2 = r \sin{\theta}$$
$$x_3 = z$$
Thanks.

I assume you are seeking the infinitesimal deformation operators generating rotations, as in quantum mechanics, so, then,  $\vec{p}\sim \nabla$.
In that case, since 
$$
\vec{R}= z \hat z + r \hat r ,\\
\nabla = \hat z \partial_z + \hat r \partial_r + \hat \theta \frac{1}{r} \partial _\theta,
$$
$$
\vec L = \vec{R} \times \nabla = (z \hat z + r \hat r ) \times
\left ( \hat z \partial_z + \hat r \partial_r + \hat \theta \frac{1}{r} \partial _\theta \right ) \\
= \hat z \partial_\theta- \hat r \frac{z}{r} \partial_\theta +\hat \theta (z\partial_r -r\partial_z).
$$ 
Confirm by limiting cases z=0, θ=0, etc.
