12 files changed, 521 insertions, 0 deletions
diff --git a/math/cylindrical_coordinates.org b/math/cylindrical_coordinates.org
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--- /dev/null
+++ b/math/cylindrical_coordinates.org
@@ -0,0 +1,37 @@
+#+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
new file mode 100644
index 0000000..4d5d4be
--- /dev/null
+++ b/math/derivatives.org
@@ -0,0 +1,31 @@
+#+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
new file mode 100644
index 0000000..069c4b6
--- /dev/null
+++ b/math/index.org
@@ -0,0 +1,13 @@
+#+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
new file mode 100644
index 0000000..6a32903
--- /dev/null
+++ b/math/integrals.org
@@ -0,0 +1,20 @@
+#+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
new file mode 100644
index 0000000..aaa2074
--- /dev/null
+++ b/math/levi_cevita.org
@@ -0,0 +1,26 @@
+#+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
new file mode 100644
index 0000000..d4e1054
--- /dev/null
+++ b/math/math.css
@@ -0,0 +1,85 @@
+/*
+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
new file mode 100644
index 0000000..a7ba58b
--- /dev/null
+++ b/math/matrices.org
@@ -0,0 +1,32 @@
+#+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
new file mode 100644
index 0000000..88079a6
--- /dev/null
+++ b/math/orbit.org
@@ -0,0 +1,93 @@
+#+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
new file mode 100644
index 0000000..ff07be3
--- /dev/null
+++ b/math/polar_coordinates.org
@@ -0,0 +1,36 @@
+#+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
new file mode 100644
index 0000000..1c7fd9c
--- /dev/null
+++ b/math/spherical_coordinates.org
@@ -0,0 +1,39 @@
+#+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
new file mode 100644
index 0000000..3e81eeb
--- /dev/null
+++ b/math/trigonometry.org
@@ -0,0 +1,51 @@
+#+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
new file mode 100644
index 0000000..49d8132
--- /dev/null
+++ b/math/vectors.org
@@ -0,0 +1,58 @@
+#+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*}