What is Cylindrical coordinates and basis? — Physics

Definition 11.5 University Physics — Year 1 · Chapter 11 — Kinematics of a Point

Point $M$ is located by $r \geq 0$ (distance to the $z$ axis), the angle $\theta$ of its projection on the $xy$ plane with $\vect e_x$ , and $z$ : $\vect{OM} = r\,\vect e_r + z\,\vect e_z$ , where

\vect e_r = \cos\theta\,\vect e_x + \sin\theta\,\vect e_y, \qquad \vect e_\theta = -\sin\theta\,\vect e_x + \cos\theta\,\vect e_y,

and $\vect e_z$ form the local basis $(\vect e_r, \vect e_\theta, \vect e_z)$ , orthonormal and direct, which moves with $M$ . In the plane ( $z$ fixed) these are the polar coordinates $(r, \theta)$ .

Polar coordinates: the local basis ( e_r, e_ ) turns with M; the velocity has a radial part r and an orthoradial part r. Here the trajectory (red) spirals outward, so r > 0 and v leans outward from the tangent to the circle. — Polar coordinates: the local basis $(\vect e_r, \vect e_\theta)$ turns with $M$ ; the velocity has a radial part $\dot r$ and an orthoradial part $r\dot\theta$ . Here the trajectory (red) spirals outward, so $\dot r > 0$ and $\vect v$ leans outward from the tangent to the circle.

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