How would you (initially) approach solving this (by hand):
Consider the vector field $$F(x,y,z) = \left(0, \frac{y}{\sqrt{y^2+z^2}}, \frac{z}{\sqrt{y^2+z^2}}\right).$$ In cylindrical coordinates this vector field is given $F(r,\theta,x)=(1,0,0)$.
What is the Cartesian coordinate system $(x,y,z)$ in function of $(r,\theta,x)$?