From 61b5ce20f25c5785e41574998a12c6d06eb05a5e Mon Sep 17 00:00:00 2001
From: Thomas Albers <thomas@thomaslabs.org>
Date: Wed, 8 Mar 2023 23:43:00 +0100
Subject: Restructure build system and directory structures

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

(limited to 'src/math/cylindrical_coordinates.org')

diff --git a/src/math/cylindrical_coordinates.org b/src/math/cylindrical_coordinates.org
new file mode 100644
index 0000000..bfb6d57
--- /dev/null
+++ b/src/math/cylindrical_coordinates.org
@@ -0,0 +1,36 @@
+#+title:Cylindrical Coordinates
+#+setupfile: ../../math_options.org
+
+* 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*}
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