直角座標系(x,y,z)と円筒座標系({\rho},{\phi},{z})の間の関係式

x={\rho}\cos({\phi})
y={\rho}\sin({\phi})
z=z

{\rho}={\sqrt{x^2+y^2}}
{{\phi}}={\tan^{-1}(\frac{y}{x})
z=z

ベクトル表示だと、
直角座標系{\bfA}  {\bf{A}}={A}_{x}{\bf{i}}+{A}_{y}{\bf{j}}+{A}_{z}{\bf{k}
円筒座標系　{\bf{A}}={A}_{{\rho}}{\bf{u}_{\rho}}+{A}_{{\phi}}{\bf{u}_{\phi}}+{A}_{z}{\bf{u}_{z}}

\begin{pmatrix} A_{\rho} \\ A_{\phi} \\ A_{z} \\ \end{pmatrix}=\begin{pmatrix} \ \ \cos{\phi} \ \ \sin{\phi} \ 0 \\ -\sin{\phi} \ \cos{\phi} \ \ 0 \\ \ \ \ \ \ 0 \ \ \ \ \ \ 0 \ \ \ \ \ 1 \end{pmatrix}\begin{pmatrix} A_{x} \\ A_{y} \\ A_{z} \\ \end{pmatrix}