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/spherical_coordinates.org | 39 +++++++++++++++++++++++++++++++++++++++
 1 file changed, 39 insertions(+)
 create mode 100644 math/spherical_coordinates.org

(limited to 'math/spherical_coordinates.org')

diff --git a/math/spherical_coordinates.org b/math/spherical_coordinates.org
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+++ b/math/spherical_coordinates.org
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+#+TITLE:Spherical 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 \sin\theta \cos\varphi\\
+y &= r \sin\theta \sin\varphi\\
+z &= r \cos\theta
+\end{align*}
+
+* Local unit vectors
+\begin{align*}
+\bm{\hat{e}}_r &= \sin\theta \cos\varphi \bm{\hat{e}}_x + \sin\theta \sin\varphi \bm{\hat{e}}_y + \cos\theta \bm{\hat{e}}_z\\
+\bm{\hat{e}}_\theta &= \cos\theta \cos\varphi \bm{\hat{e}}_x + \cos\theta \sin\varphi \bm{\hat{e}}_y - \sin\theta \bm{\hat{e}}_z\\
+\bm{\hat{e}}_\varphi &= - \sin\theta \sin\varphi \bm{\hat{e}}_x + \sin\theta \cos\varphi \bm{\hat{e}}_y
+\end{align*}
+* Kinematic in spherical coordinates
+** Time derivatives of the local unit vectors
+\begin{align*}
+\deriv{\unitv{r}}{t} &= \dot{\theta}\unitv{\theta} + \dot{\varphi}\sin\theta \unitv{\varphi}\\
+\deriv{\unitv{\theta}}{t} &= -\dot{\theta}\unitv{r} + \dot{\varphi}\cos\theta \unitv{\varphi}\\
+\deriv{\unitv{\varphi}}{t} &= -\dot{\varphi} \left(\sin\theta\unitv{r} + \cos\theta\unitv{\theta}\right)
+\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} + r\dot\varphi\sin\theta\unitv{\varphi}\\
+\bm{a} &= \left(\ddot{r} - r\dot\theta^2 - r\dot\varphi^2\sin^2\theta\right)\unitv{r}
++ \left(2\dot{r}\dot\theta + r\ddot\theta - r\dot\varphi^2\sin\theta\cos\theta\right)\unitv{\theta}
++ \left(2\dot{r}\dot\varphi\sin\theta + 2r\dot\theta\dot\varphi\cos\theta + r\ddot\varphi\sin\theta\right)\unitv{\varphi}
+\end{align*}
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