How do I change between cartesian and cylindrical coordinate systems?

Question

I have a vector $\textbf{D}=(x,3,5)$ in cartesian coordinates $(x,y,z)$ that I want to express in cylindrical coordinates $(r,\phi,z)$. Do I just plug the values for x,y,z into the expressions to get the new coordinates and thats it? $$ \begin{align*} \phi &= \arctan \frac{x}{y} \\ r &= \sqrt{x^2+y^2} \\ z &= z \end{align*} $$

Answer 1

That is basically correct although you have to be careful with arctan. Cylindrical coordinates is the same thing as converting the $x,y$ plane to polar and keeping the $z$-axis unchanged. This wikipedia article explains the subtleties of the arctan conversion: https://en.wikipedia.org/wiki/Polar_coordinate_system#Converting_between_polar_and_Cartesian_coordinates.