math/cylindrical_coordinates.org


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#+TITLE:Cylindrical Coordinates
#+SETUPFILE: ../math_options.org
#+LATEX_HEADER: \usepackage{bm}
#+LATEX_HEADER: \usepackage{mathtools}
#+LATEX_HEADER: \newcommand{\deriv}[2]{\frac{\text{d}#1}{\text{d}#2}}
#+LATEX_HEADER: \newcommand{\unitv}[1]{\bm{\hat{e}}_#1}

* Disclaimer
This site as of now just a technology demonstration and its claims
should not be taken as true (even though I myself am pretty confident
they are)
* Coordinate transformations
\begin{align*}
x &= r \cos\varphi\\
y &= r \sin\varphi\\
z &= z
\end{align*}

* Local unit vectors
\begin{align*}
\unitv{r} &= \cos\theta \unitv{x} + \sin\theta \unitv{y}\\
\unitv{\theta} &= -\sin\theta \unitv{x} + \cos\theta \unitv{y}\\
\unitv{z} &= \unitv{z}
\end{align*}
* Kinematic in cylindrical coordinates
** Time derivatives of the local unit vectors
\begin{align*}
\deriv{\unitv{r}}{t} &= \dot{\theta}\unitv{\theta}\\
\deriv{\unitv{\theta}}{t} &= -\dot{\theta}\unitv{r}\\
\deriv{\unitv{z}}{t} &= 0
\end{align*}
** Position vector and its time derivatives
\begin{align*}
\bm{r} &= r\unitv{r} + z\unitv{z}\\
\bm{v} &= \dot{r}\unitv{r} + r\dot\theta\unitv{\theta} + \dot{z}\unitv{z}\\
\bm{a} &= \left(\ddot{r}-r\dot{\theta}^2\right)\unitv{r} + \left(2\dot{r}\dot{\theta}+r\ddot{\theta}\right)\unitv{\theta} + \ddot{z}\unitv{z}
\end{align*}