From 61b5ce20f25c5785e41574998a12c6d06eb05a5e Mon Sep 17 00:00:00 2001 From: Thomas Albers Date: Wed, 8 Mar 2023 23:43:00 +0100 Subject: Restructure build system and directory structures --- math/cylindrical_coordinates.org | 37 ---------------- math/derivatives.org | 31 -------------- math/index.org | 13 ------ math/integrals.org | 20 --------- math/levi_cevita.org | 26 ----------- math/math.css | 85 ------------------------------------ math/matrices.org | 32 -------------- math/orbit.org | 93 ---------------------------------------- math/polar_coordinates.org | 36 ---------------- math/spherical_coordinates.org | 39 ----------------- math/trigonometry.org | 51 ---------------------- math/vectors.org | 58 ------------------------- 12 files changed, 521 deletions(-) delete mode 100644 math/cylindrical_coordinates.org delete mode 100644 math/derivatives.org delete mode 100644 math/index.org delete mode 100644 math/integrals.org delete mode 100644 math/levi_cevita.org delete mode 100644 math/math.css delete mode 100644 math/matrices.org delete mode 100644 math/orbit.org delete mode 100644 math/polar_coordinates.org delete mode 100644 math/spherical_coordinates.org delete mode 100644 math/trigonometry.org delete mode 100644 math/vectors.org (limited to 'math') diff --git a/math/cylindrical_coordinates.org b/math/cylindrical_coordinates.org deleted file mode 100644 index fab185b..0000000 --- a/math/cylindrical_coordinates.org +++ /dev/null @@ -1,37 +0,0 @@ -#+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*} diff --git a/math/derivatives.org b/math/derivatives.org deleted file mode 100644 index 4d5d4be..0000000 --- a/math/derivatives.org +++ /dev/null @@ -1,31 +0,0 @@ -#+TITLE:Table of Derivatives -#+SETUPFILE: ../math_options.org -#+LATEX_HEADER: \usepackage{bm} -#+LATEX_HEADER: \usepackage{mathtools} -#+LATEX_HEADER: \newcommand{\dcoff}[1]{\frac{\text{d}}{\text{d}x} #1} -#+LATEX_HEADER: \newcommand{\sdcoff}[1]{\frac{\text{d}#1}{\text{d}x}} -#+LATEX_HEADER: \newcommand{\deriv}[2]{\frac{\text{d}}{\text{d}x} #1 &= #2} - -* 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) - -* General Properties of the Derivative -Let $f$ and $g$ be real valued functions and $c$ some real constant: -\begin{align*} -\dcoff{(cf)} &= c\sdcoff{f}\\ -\dcoff{(f \pm g)} &= \sdcoff{f} \pm \sdcoff{g}\\ -\dcoff{(fg)} &= \sdcoff{f}g + f\sdcoff{g}\\ -\dcoff{\left(\frac{f}{g}\right)} &= \frac{\sdcoff{f}g - f\sdcoff{g}}{g^2} -\end{align*} - -* Trigonometric Funtions -\begin{align*} -\deriv{\sin(x)}{\cos(x)}\\ -\deriv{\cos(x)}{-\sin(x)}\\ -\deriv{\tan(x)}{\sec^2(x)}\\ -\deriv{\sec(x)}{\sec(x)\tan(x)}\\ -\deriv{\csc(x)}{\csc(x)\cot(x)}\\ -\deriv{\csc(x)}{-\csc^2(x)} -\end{align*} diff --git a/math/index.org b/math/index.org deleted file mode 100644 index 069c4b6..0000000 --- a/math/index.org +++ /dev/null @@ -1,13 +0,0 @@ -#+TITLE: Math and Physics articles -#+SETUPFILE: ../math_options.org - -- [[file:vectors.org][Cross Product]] -- [[file:cylindrical_coordinates.org][Cylindrical Coordinates]] -- [[file:matrices.org][Matrix Properties]] -- [[file:orbit.org][Orbit]] -- [[file:polar_coordinates.org][Polar Coordinates]] -- [[file:spherical_coordinates.org][Spherical Coordinates]] -- [[file:derivatives.org][Table of Derivatives]] -- [[file:integrals.org][Table of Integrals]] -- [[file:levi_cevita.org][The Levi Cevita Symbol]] -- [[file:trigonometry.org][Trigonometric identities]] \ No newline at end of file diff --git a/math/integrals.org b/math/integrals.org deleted file mode 100644 index 6a32903..0000000 --- a/math/integrals.org +++ /dev/null @@ -1,20 +0,0 @@ -#+TITLE:Table of Integrals -#+SETUPFILE: ../math_options.org -#+LATEX_HEADER: \usepackage{bm} -#+LATEX_HEADER: \usepackage{mathtools} -#+LATEX_HEADER: \newcommand{\intg}[2]{\int #1 \text{d}x &= #2 + C} - -* 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) - -* Trigonometric Funtions -\begin{align*} -\intg{\sin(x)}{-\cos(x)}\\ -\intg{\cos(x)}{\sin(x)}\\ -\intg{\tan(x)}{-\ln(\cos(x))}\\ -\intg{\sec(x)}{\ln(\sec(x) + \tan(x))}\\ -\intg{\csc(x)}{-\ln(\csc(x) + \cot(x))}\\ -\intg{\cot(x)}{\ln(\sin(x))} -\end{align*} diff --git a/math/levi_cevita.org b/math/levi_cevita.org deleted file mode 100644 index aaa2074..0000000 --- a/math/levi_cevita.org +++ /dev/null @@ -1,26 +0,0 @@ -#+TITLE: The Levi Cevita Symbol -#+SETUPFILE: ../math_options.org -#+LATEX_HEADER: \usepackage{bm} -#+LATEX_HEADER: \usepackage{mathtools} - -* 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) -* Levi Cevita symbol -\begin{equation*} -\varepsilon_{ijk} = \left\{\begin{array}{rl} -1,&(i,j,k) \in \{(1,2,3), (2,3,1), (3,1,2)\}\\ --1,&(i,j,k) \in \{(3,2,1), (2,1,3), (1,3,2)\}\\ -0,&\text{otherwise} -\right{} -\end{array} -\end{equation*} - -\begin{equation*} -\varepsilon_{ijk} = \varepsilon_{jki} = \varepsilon_{kij} = - \varepsilon_{ikj} = - \varepsilon_{jik} = - \varepsilon_{kji} = 1 -\end{equation*} - -\begin{align*} -\sum_{k=1}^3 \varepsilon_{ijk}\varepsilon_{kmn} = \delta_{im} \delta_{jn} - \delta_{in} \delta_{jm} -\end{align*} diff --git a/math/math.css b/math/math.css deleted file mode 100644 index d4e1054..0000000 --- a/math/math.css +++ /dev/null @@ -1,85 +0,0 @@ -/* -p.author -p.date -p.creator -.title -.subtitle -.todo -.done -.WAITING -.timestamp -.timestamp-kwd -.timestamp-wrapper -.tag -._HOME -.target -.linenr -.code-highlighted -div.outline-N -div.outline-text-N -.section-number-N -.figure-number -.table-number -.listing-number -div.figure -pre.src -pre.example -p.verse -div.footnotes -p.footnote -.footref -.footnum -.org-svg -*/ - -html { - height: 100%; - color: #cccccc; - background-color: hsl(240, 30%, 10%); -} - -body { - margin: auto; - max-width: 64em; - min-height: 100%; - display: flex; - flex-direction: column; - background-color: hsl(240, 30%, 15%); -} - -main { - padding: 2em 2em 2em 2em; - flex-grow: 1; -} - -h1 { - text-align: center; -} - -#content { - max-width: 50em; - margin: auto; -} - -div.outline-2:first-of-type { - padding: 0.25em 1em; - background-color: hsl(60,30%,30%); -} - -.equation-container { - display: block; - padding: 1em 2em; - background-color: hsl(240, 30%, 20%); - overflow: auto; - border: 1px solid hsl(240, 30%, 50%) -} - -img { - max-width: unset; -} - -.equation-container img { - object-fit: contain; - display: block; - margin: auto; -} diff --git a/math/matrices.org b/math/matrices.org deleted file mode 100644 index a7ba58b..0000000 --- a/math/matrices.org +++ /dev/null @@ -1,32 +0,0 @@ -#+TITLE:Matrix Properties -#+SETUPFILE: ../math_options.org -#+LATEX_HEADER: \usepackage{bm} -#+LATEX_HEADER: \usepackage{mathtools} - -* 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) -* Basic properties -\begin{align*} -A + B = B + A -\end{align*} -* Dot product -\begin{align*} -(A^\text{T})^\text{T} &= A\\ -(A + B)^\text{T} &= A^\text{T} + B^\text{T}\\ -(AB)^\text{T} &= B^\text{T}A^\text{T} -\end{align*} -* Transpose -\begin{align*} -\bm{a} \cdot \bm{b} &= \overline{\bm{b}} \cdot \bm{a}\\ -\bm{a} \cdot \bm{b} &= \bm{a}^\text{T} \bm{b} -\end{align*} - -* Hermitian transpose -\begin{align*} -A^\ast &= \left[\overline{a_{ij}}\right]\\ -(\lambda A)^\ast &= \left[\overline{\lambda a_{ij}}\right] = \overline{\lambda} \left[a_{ij}\right]^\ast = \overline{\lambda}\,\overline{A}\\ -A^\dag &= (A^\ast)^\text{T}\\ -A^{\dag\dag} &= A -\end{align*} diff --git a/math/orbit.org b/math/orbit.org deleted file mode 100644 index 88079a6..0000000 --- a/math/orbit.org +++ /dev/null @@ -1,93 +0,0 @@ -#+TITLE:Orbit -#+SETUPFILE: ../math_options.org -#+LATEX_HEADER: \usepackage{bm} -#+LATEX_HEADER: \usepackage{mathtools} -#+LATEX_HEADER: \usepackage{amssymb} -#+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) -* Deriving Kepler's first law from Newton's law of universal gravitation - -The movement of an object with mass $m$ orbiting another body with mass $M$ is -given by Newton's law of gravitation. If $m \ll M$ it is possible to consider -the position of the larger object constant and use it as the origin of our -coordinate system. Then the following equation applies for movement of the -smaller object: - -\begin{equation*} -m\ddot{\bm{r}} = - \frac{GMm}{r^3}\bm{r} \Leftrightarrow \ddot{\bm{r}} = - \frac{GM}{r^3}\bm{r} -\end{equation*} - -In order to solve this differential equation we first consider the angular -momentum of or object around its orbit. - -\begin{equation*} -\bm{L} = \bm{r} \times m \dot{\bm{r}} -\end{equation*} - -In the abscense of external toques, because the only force acting on the object -is parallel to its position, the angular momentum is conserved. - -\begin{equation*} -\deriv{\bm{L}}{t} = \dot{\bm{r}} \times m \dot{\bm{r}} + \bm{r} \times m \ddot{\bm{r}} = 0 -\end{equation*} - -We now multiply both sides of our equation from the right by the angular -momentum and develop the right side of the equation using vector identities. - -\begin{align*} -\ddot{\bm{r}} \times \bm{L} &= -GM\frac{\bm{r} \times \left(\bm{r} \times m \dot{\bm{r}}\right)}{r^3}\\ -&= - \frac{GMm}{r^3} \left(\left(\bm{r} \cdot \dot{\bm{r}}\right)\bm{r} - \left(\bm{r} \cdot \bm{r}\right)\dot{\bm{r}}\right)\\ -&= GMm\left(\frac{\left(\bm{r} \cdot \bm{r}\right)\dot{\bm{r}}}{r^3} - \frac{\left(\bm{r} \cdot \dot{\bm{r}}\right)\bm{r}}{r^3}\right)\\ -&= GMm\left(\frac{\dot{\bm{r}}}{r} - \frac{\left(\bm{r} \cdot \dot{\bm{r}}\right)\bm{r}}{r^3}\right)\\ -&= GMm\left(\frac{1}{r} \deriv{\bm{r}}{t} + \deriv{}{t}\left(\frac{1}{r}\right)\bm{r}\right)\\ -&= GMm\deriv{}{t}\left(\frac{\bm{r}}{r}\right) -\end{align*} - -We observe that each side of our equation is a derivative of a quantity. We know -integrate both sides and take the integrations constant into account. - -\begin{align*} -& \deriv{}{t} \left(\dot{\bm{r}} \times \bm{L}\right) = GM \deriv{}{t}\left(\frac{\bm{r}}{r}\right)\\ -\Leftrightarrow \quad & \dot{\bm{r}} \times \bm{L} = GM \frac{\bm{r}}{r} + \bm{a} -\end{align*} - -Our objective is now to solve the equation for $r$, so we multiply both sides by $\bm{r}$: - -\begin{align*} -\dot{\bm{r}} \times \left(\bm{r} \times \dot{\bm{r}}\right) &= GM \frac{\bm{r}}{r} + \bm{a}\\ -\bm{r} \cdot \left(\dot{\bm{r}} \times \left(\bm{r} \times \dot{\bm{r}}\right)\right) &= GMr + \bm{r} \cdot \bm{a} -\end{align*} - -By applying a cyclic permutation of the resulting triple product and using the -known property of the scalar product we now express the equation only in terms -of the magnitudes of the vectors. - -\begin{align*} -\left(\bm{r} \times \dot{\bm{r}}\right) \cdot \left(\bm{r} \times \dot{\bm{r}}\right) &= GMr + \bm{r} \cdot \bm{a}\\ -\left(\frac{L}{m}\right)^2 &= GMr + \bm{r} \cdot \bm{a}\\ -\left(\frac{L}{m}\right)^2 &= GMr + r a \cos\theta -\end{align*} - -The last steps are to solve for $r$ - -\begin{align*} -r &= \left(\frac{L}{m}\right)^2 \frac{1}{GM + a \cos\theta}\\ -&= \left(\frac{L}{m}\right)^2 \frac{1}{GM} \frac{1}{1 + \frac{a}{GM} \cos\theta}\\ -&= \left(\frac{L}{m}\right)^2 \frac{1}{GM} \frac{1}{1 + e\cos\theta} -\end{align*} - -Finally we reach our result. Objects orbiting according to Newton's Law of -Gravitation follow paths that correspond to the conic sections. Here is Kepler's -first Law a special case, where our object has a stable orbit around the larger -body. - -\begin{equation*} -r &= \frac{L^2}{GM m^2} \frac{1}{1 + e\cos\theta} -\end{equation*} - -* Deriving a physical interpretation of the excentricity of the orbit diff --git a/math/polar_coordinates.org b/math/polar_coordinates.org deleted file mode 100644 index ff07be3..0000000 --- a/math/polar_coordinates.org +++ /dev/null @@ -1,36 +0,0 @@ -#+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*} diff --git a/math/spherical_coordinates.org b/math/spherical_coordinates.org deleted file mode 100644 index 1c7fd9c..0000000 --- a/math/spherical_coordinates.org +++ /dev/null @@ -1,39 +0,0 @@ -#+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*} diff --git a/math/trigonometry.org b/math/trigonometry.org deleted file mode 100644 index 3e81eeb..0000000 --- a/math/trigonometry.org +++ /dev/null @@ -1,51 +0,0 @@ -#+TITLE:Trigonometric identities -#+SETUPFILE: ../math_options.org -#+LATEX_HEADER: \usepackage{bm} -#+LATEX_HEADER: \usepackage{mathtools} - -* 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) - -* Pythagorean identities -\begin{align*} -&\cos^2\left(x\right) + \sin^2\left(x\right) = 1\\ -&\tan^2\left(x\right) + 1 = \sec^2\left(x\right)\\ -&1 + \cot^2\left(x\right) = \csc^2\left(x\right) -\end{align*} -* Sum of angles -\begin{align*} -&\sin\left(a \pm b\right) = \sin\left(a\right)\cos\left(b\right) \pm \cos\left(a\right)\sin\left(b\right)\\ -&\cos\left(a \pm b\right) = \cos\left(a\right)\cos\left(b\right) \mp \sin\left(a\right)\sin\left(b\right)\\ -&\tan(a \pm b) = \frac{\tan(a) \pm \tan(b)}{1 \mp \tan(a) \tan(b)} -\end{align*} -* Multiple angles -\begin{align*} -&\sin(2\theta) = 2\sin(\theta)\cos(\theta)\\ -&\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta -\end{align*} -* Half-angle formulae -\begin{align*} -&\sin \frac{\theta}{2} = \pm \sqrt{\frac{1-\cos\theta}{2}}\\ -&\cos \frac{\theta}{2} = \pm \sqrt{\frac{1+\cos\theta}{2}} -\end{align*} -* Power-reduction formulae -\begin{align*} -&\sin^2\theta = \frac{1 - \cos(2\theta)}{2}\\ -&\cos^2\theta = \frac{1 + \cos(2\theta)}{2} -\end{align*} -* Product-to-sum formulae -\begin{align*} -&2 \cos\theta \cos\varphi = \cos(\theta-\varphi) + \cos(\theta+\varphi)\\ -&2 \sin\theta \sin\varphi = \cos(\theta-\varphi) - \cos(\theta+\varphi)\\ -&2 \sin\theta \cos\varphi = \sin(\theta+\varphi) + \sin(\theta-\varphi)\\ -&2\cos\theta\sin\varphi = \sin(\theta+\varphi) - \sin(\theta-\varphi) -\end{align*} -* Sum-to-product formulae -\begin{align*} -&\sin\theta \pm \sin\varphi = 2 \sin\left(\frac{\theta\pm\varphi}{2}\right) \cos\left(\frac{\theta\mp\varphi}{2}\right)\\ -&\cos\theta + \cos\varphi = 2 \cos\left(\frac{\theta+\varphi}{2}\right) \cos\left(\frac{\theta-\varphi}{2}\right)\\ -&\tan\theta \pm \tan\varphi = \frac{\sin(\theta\pm\varphi)}{\cos\theta \cos\varphi}\\ -&\cos\theta - \cos\varphi = - 2 \sin\left(\frac{\theta+\varphi}{2}\right) \sin\left(\frac{\theta-\varphi}{2}\right) -\end{align*} diff --git a/math/vectors.org b/math/vectors.org deleted file mode 100644 index 49d8132..0000000 --- a/math/vectors.org +++ /dev/null @@ -1,58 +0,0 @@ -#+TITLE:Cross Product -#+SETUPFILE: ../math_options.org -#+LATEX_HEADER: \usepackage{bm} -#+LATEX_HEADER: \usepackage{mathtools} - -* 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) - -* Definition -We shall define the cross product in terms of two properties we are -interested in: - -- Distributive: $\bm{a} \times \left(\bm{b} + \bm{c}\right) = \bm{a} \times \bm{b} + \bm{a} \times \bm{c}$ -- Orthogonal: $\bm{a} \times \bm{b} = \bm{c} \implies \bm{c} \cdot \bm{a} = \bm{0} \land \bm{c} \cdot \bm{b} = \bm{0}$ - -It is worth mentioning that given a pair of vectors in $\mathbb{R}^3$ -there exist an infinite amount of vectors that satisfy these -properties, so it is also necessary to introduce the following -relations between the basis vectors to properly define the cross -product. - -\begin{align*} -\bm{e}_1 &= \bm{e}_2 \times \bm{e}_3\\ -\bm{e}_2 &= \bm{e}_3 \times \bm{e}_1\\ -\bm{e}_3 &= \bm{e}_1 \times \bm{e}_2\\ -\bm{e}_i \times \bm{e}_i &= \bm{0},\qquad\text{For}\quad i = 1,2,3 -\end{align*} - -We introduce the Levi-Civita symbol to condense our calculations. - -\begin{equation*} -\epsilon_{ijk} \coloneqq \bm{e}_i \cdot \left(\bm{e}_j \times \bm{e}_k\right) -\end{equation*} - -Based on this we may now derive a way to compute the cross product of -two vectors - -\begin{align*} -\left[\bm{a} \times \bm{b}\right]_i &= \left[\left(\sum_j a_j \bm{e}_j\right) \times \left(\sum_k b_k \bm{e_k}\right)\right]_i\\ -&= \left[\sum_{jk} a_j b_k \left(\bm{e}_j \times \bm{e}_k\right)\right]_i\\ -&= \sum_{jk} a_j b_k \bm{e}_i \cdot \left(\bm{e}_j \times \bm{e}_k\right)\\ -&= \sum_{jk} \epsilon_{ijk} a_j b_k\\ -\bm{a} \times \bm{b} &= \sum_{ijk} \epsilon_{ijk} a_i b_j \bm{e}_k -\end{align*} - -* Properties -\begin{align*} -&\bm{a} \times \bm{b} = - \bm{b} \times \bm{a}\\ -&\bm{a} \cdot \left(\bm{b} \times \bm{c}\right) -= \bm{b} \cdot \left(\bm{c} \times \bm{a}\right) -= \bm{c} \cdot \left(\bm{a} \times \bm{b}\right)\\ -&\bm{a} \times \left(\bm{b} \times \bm{c}\right) -= \left(\bm{a} \cdot \bm{c}\right)\bm{b} - \left(\bm{a} \cdot \bm{b}\right)\bm{c}\\ -&\left(\bm{a} \times \bm{b}\right) \times \left(\bm{c} \times \bm{d}\right) -= \left(\bm{a}\cdot\bm{c}\right) \left(\bm{b}\cdot\bm{d}\right) - \left(\bm{a}\cdot\bm{d}\right) \left(\bm{b}\cdot\bm{c}\right) -\end{align*} -- cgit v1.2.3