From 561bac75579391c14e47eaccfabdf9eda98855da Mon Sep 17 00:00:00 2001
From: Thomas Albers <thomas@thomaslabs.org>
Date: Wed, 27 Jul 2022 18:13:20 +0200
Subject: Initial commit

---
 math/polar_coordinates.org | 36 ++++++++++++++++++++++++++++++++++++
 1 file changed, 36 insertions(+)
 create mode 100644 math/polar_coordinates.org

(limited to 'math/polar_coordinates.org')

diff --git a/math/polar_coordinates.org b/math/polar_coordinates.org
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+#+TITLE:Polar 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\theta\\
+y &= r \sin\theta
+\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}
+\end{align*}
+
+* Kinematic in polar 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}
+\end{align*}
+** Position vector and its time derivatives
+\begin{align*}
+\bm{r} &= r\unitv{r}\\
+\bm{v} &= \dot{r}\unitv{r} + r\dot\theta\unitv{\theta}\\
+\bm{a} &= \left(\ddot{r} - r\dot\theta^2\right)\unitv{r}
++ \left(2\dot{r}\dot\theta + r\ddot\theta\right)\unitv{\theta}
+\end{align*}
-- 
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