discord-math · josh-kynaston · Dec 7, 2022 · Dec 7, 2022 · Dec 8, 2022 · Dec 9, 2022
diff --git a/_includes/extra_styles/cheatsheet.css b/_includes/extra_styles/cheatsheet.css
@@ -0,0 +1,21 @@
+.latexexample {
+  padding: .5em;
+  background: var(--background-secondary);
+  border: 1px solid var(--foreground-tertirary);
+  border-radius: 4px;
+  line-height: 1.125rem;
+}
+
+table.latex_table {
+  margin: auto;
+}
+
+td {
+  padding-left: 20px;
+  padding-right: 20px;
+}
+
+th {
+  padding-left: 20px;
+  padding-right: 20px;
+}
diff --git a/_includes/hero_style.css → _includes/extra_styles/hero.css b/_includes/hero_style.css → _includes/extra_styles/hero.css
diff --git a/_includes/head.html b/_includes/head.html
@@ -8,9 +8,9 @@
     <style>
         {%- include modern-normalize.css -%}
         {%- include style.css -%}
-        {% if include.hero %}
-            {%- include hero_style.css -%}
-        {% endif %}
+        {% for css in page.extra_styles %}
+            {%- include extra_styles/{{ css }}.css -%}
+        {% endfor %}
     </style>
 
     <title>

diff --git a/_layouts/home.html b/_layouts/home.html
@@ -1,6 +1,6 @@
 <!DOCTYPE html>
 <html lang="en">
-    {%- include head.html hero=true -%}
+    {%- include head.html -%}
     <body>
         {%- include navigation.html -%}
 

diff --git a/index.md b/index.md
@@ -1,3 +1,4 @@
 ---
 layout: home
+extra_styles: hero
 ---
diff --git a/latex-cheatsheet.md b/latex-cheatsheet.md
@@ -0,0 +1,331 @@
+---
+title: LaTeX cheatsheet
+updated_at: 2022-12-07
+mathjax: true
+extra_styles: cheatsheet
+---
+
+## Table of Contents
+{:.no_toc}
+
+* TOC
+{:toc}
+<hr>
+
+## What is $\LaTeX$?
+$\LaTeX$ is the standard system used for rendering mathematical and scientific documents. It is also how you can render mathematical expressions and formulae server-wide using the [@TeXiT](https://discord.com/users/510789298321096704){: .mention target="_blank"} bot. If you've ever wondered how scientific papers or textbooks are laid out in the way they are, this is the guide for you.
+
+Unlike other document creation systems such as Microsoft Word, documents produced using $\LaTeX$ are not written using a so-called "what-you-see-is-what-you-get" format. Instead, documents are written in plaintext `.tex` files in the $\LaTeX$ language that are then *compiled* into `.pdf` documents.
+
+Regular text will be rendered as itself, but the following punctuation symbols have special meaning: The backslash symbol `\` is used for "commands" or "macros" which insert specical symbols or notation into the text. Braces (`{` and `}`) are used to group symbols together into a block. Dollar signs (`$` and `$$`) are used to include formulae in the text.
+
+There are two general modes of operation: "math mode" is used for formulae, and "text mode" is used for text. Formulae can be surrounded by single dollars to be included in the text "inline", for example `$a + b = c$` produces $a+b=c$. Double dollars render a large fomrula in "display style", which inserts line breaks around the formula, and also has an effect on how some notation is rendered (more on this in [Examples](#examples)). 
+
+Most symbols are rendered as themselves (namely, `!`, `'`, `(`, `)`, `*`, `+`, `,`, `-`, `.`, `/`, `:`, `;`, `<`, `>`, `=`, `?`, `[`, `]`, `|`). To render other, more interesting characters, the commands given in the [Commands Reference](#command-reference) below can be used. To write literal braces and dollars, `\{`, `\}`, and `\$` must be used, respectively.
+
+## Examples
+In this section we explore some demonstrative examples of how to render mathematical expressions and formulae using $\LaTeX$, as well as point out some common mistakes.
+
+### Inline vs Display mode
+Let's look at a $\LaTeX$ snippet that renders an equation in math mode:
+```latex
+Perhaps the most famous equation in all of physics is $E=mc^2$
+```
+This is rendered as:
+<div class="latexexample">
+$\text{Perhaps the most famous equation in all of physics is } E=mc^2$
+</div>
+
+There are a few things worth noting here. First, the $\LaTeX$ rendering engine will automatically add whitespace around symbols such as `=`. Second, the math that is rendered is *inline* with the rest of the text. 
+
+Let's see what happens when we try rendering the same equation in display mode.
+```latex
+Perhaps the most famous equation in all of physics is $$E=mc^2$$
+```
+<div class="latexexample">
+$\text{Perhaps the most famous equation in all of physics is}$
+
+$$ E = mc^2 $$
+
+</div>
+
+Display mode renders your mathematical expression on its own centered line. A common use case for display mode is for rendering "large" operators in a more readable format. Let's look at another two examples where the only difference is that the first is rendered in math mode and the second in display mode.
+
+<div class="latexexample">
+$\text{Fubini's theorem gives conditions under which} \int_{X\times Y} f(x,y)\,\text{d}(x,y) = \int_X\left(\int_Y f(x,y)\,\text{d}y\right)\,\text{d}x$
+</div>
+versus
+<div class="latexexample">
+$\text{Fubini's theorem gives conditions under which}$
+
+$$\int_{X\times Y} f(x,y)\,\text{d}(x,y) = \int_X\left(\int_Y f(x,y)\,\text{d}y\right)\,\text{d}x$$
+
+</div>
+
+### Large operators
+Mathematics (and therefore $\LaTeX$) contains many large operators, such as $\sum$, $\int$, and $\prod$. Other symbols that act like large operators are $\lim$, $\sup$, $\limsup$, etc. Common between all these symbols is that they are most often seen with expressions placed below and/or above them. Thankfully, this is easy to achieve in $\LaTeX$. The most common way to achieve this is to use the subscript `_` and superscript `^` characters; for example,
+```latex
+\int_a^b
+```
+is rendered in inline mode as
+<div class="latexexample">
+$\int_a^b$
+</div>
+and in display mode as
+<div class="latexexample">
+$$\int_a^b$$
+</div>
+A common mistake people make is in trying to render more complex expressions. For example, you may try to write a sum from $n=1$ to $5$ with `\sum_n=1^5`. This, however, renders as:
+<div class="latexexample">
+$$\sum_n=1^5$$
+</div>
+The correct way to render what we want is to use braces to enclose the lower bound of the sum; i.e. `\sum_{n=1}^5`, giving
+<div class="latexexample">
+$$\sum_{n=1}^5$$
+</div>
+A special case of "large operator" is the fraction. Fractions are rendered with their own syntax[^1]; namely, `\frac{a}{b}`, which renders in inline mode as
+<div class="latexexample">
+$\frac{a}{b}$
+</div>
+and in display mode as
+<div class="latexexample">
+$$\frac{a}{b}$$
+</div>
+
+### Parentheses
+$\LaTeX$ does not automatically (at least, not by default), scale parentheses to match the vertical height of the enclosed expression. For example, `(\frac{a}{b})` is rendered as
+<div class="latexexample">
+$$(\frac{a}{b})$$
+</div>
+Thankfully, there is an easy means by which to scale parentheses (as well as braces, brackets, and other symbols); specifically, prefix the left symbol with `\left` and the right symbol with `\right`. So, rewriting our example from above, we obtain `\left( \frac{a}{b} \right)`, which is rendered as
+<div class="latexexample">
+$$\left( \frac{a}{b} \right)$$
+</div>
+Note that there is also a `\middle` command that is most often used for matching the height of the vertical bar glyph `|`. An example of this being used may be
+```tex
+\left\{ x \middle\mid x \leq \frac{1}{2} \right\}
+```
+rendered in display mode as
+<!-- BUG For some reason the vertical bar here is not rendering properly -->
+<div class="latexexample">
+$$\left\{ x \middle\lvert x \leq \frac{1}{2} \right\}$$
+</div>
+
+### Spacing
+In general, $\LaTeX$ handles the horizontal spacing of mathematical expressions automatically. Sometimes, however, this produces undesirable results. Compare the following two expressions,
+```latex
+$$ \int f(x) dx $$
+```
+<div class="latexexample">
+$$ \int f(x) dx $$
+</div>
+and
+```latex
+$$ \int f(x)\, dx$$
+```
+<div class="latexexample">
+$$ \int f(x)\, dx$$
+</div>
+In the former case the space between $f(x)$ and $dx$ may seem too small. To fix this, authors commonly add a spacing commands (in this example `\,`) to add some additional horizontal space. A list of common spacing commands is given in the [Commands Reference](#spacing) below.
+
+### Some common mistakes
+There are a number of common mistakes that $\LaTeX$ beginners make. The most obvious case of this is not using commands for standard functions such as triogonometric functions ($\sin$, $\cos$, etc) and logarithms ($\ln$ and $\log$). Compare, for example,
+```latex
+$$ \int tan(x)\, dx = ln \lvert sec(x) \rvert + c $$
+```
+<div class="latexexample">
+$$ \int tan(x)\, dx = ln \lvert sec(x) \rvert + c $$
+</div>
+with
+```latex
+$$ \int \tan(x)\, dx = \ln \lvert \sec(x) \rvert + c $$
+```
+<div class="latexexample">
+$$ \int \tan(x)\, dx = \ln \lvert \sec(x) \rvert + c $$
+</div>
+One can also render derivatives and infinitesimals in a more standard way by using the `\mathrm` command; compare
+```latex
+$$ \int \tan(x)\, dx $$
+```
+<div class="latexexample">
+$$ \int \tan(x)\, dx $$
+</div>
+and
+```latex
+$$ \int \tan(x)\, \mathrm{d}x $$
+```
+<div class="latexexample">
+$$ \int \tan(x)\, \mathrm{d}x $$
+</div>
+Another way to accomplish this is using a command such as `\dd` or `\dv` from the [physics package](https://ctan.org/pkg/physics){:target="_blank"}.
+
+## Using $\LaTeX$ on the server
+The <a class="mention">@TeXiT</a> bot can be used anywhere on the server to render mathematics using $\LaTeX$. To use it, simply write a normal message with any latex expressions that you want to be rendered enclosed in either `$ ... $` or `$$ ... $$`.
+
+## Command reference
+
+### Simple algebraic symbols
+
+| Name | Expression | $\LaTeX$ code |
+|:---|:---:|---:|
+| Fraction | $\frac{a}{b}$ | `\frac{a}{b}` |
+| Multiplication | $\times$ | `\times` |
+| Multiplication (dot) | $\cdot$ | `\cdot` |
+| Composition | $\circ$ | `\circ` |
+| Superscript | $a^{b}$ | `a^{b}`|
+| Subscript | $a_{b}$ | `a_{b}` |
+| Plus/Minus | $\pm$ | `\pm` |
+| Minus/Plus | $\mp$ | `\mp` |
+| Square root | $\sqrt{a}$ | `\sqrt{a}` |
+| Arbitrary root | $\sqrt[n]{a}$ | `\sqrt[n]{a}` |
+| Not equals | $\neq$ | `\neq` or `\not=`|
+| Approximately equals | $\approx$ | `\approx` |
+| Tilde | $\sim$ | `\sim` | 
+| Proportional to | $\propto$ | `\propto` |
+| Less than or equal to | $\leq$ | `\leq` or `\le` |
+| Greater than or equal to | $\geq$ | `geq` or `\ge` |
+| Much less than | $\ll$ | `\ll` |
+| Much greater than | $\gg$ | `\gg` |
+| Congruent to | $\cong$ | `\cong` |
+| Magnitude | $\lvert a \rvert$ | `\lvert a \rvert` |
+| Norm | $\lVert a \rVert$ | `\lVert a \rVert` |
+| Floor | $\lfloor a \rfloor$ | `\lfloor a \rfloor` |
+| Ceiling | $\lceil a \rceil$ | `\lceil a \rceil` |
+| Bar | $\bar{a}$ | `\bar{a}` |
+{: .latex_table}
+
+### Greek letters
+Below is a list of all greek letters available in $\LaTeX$. The commands given are for the lowercase versions; however, to obtain the uppercase version of a letter you simply make the first letter of the command uppercase. For example `\gamma` gives $\gamma$ whereas `\Gamma` gives $\Gamma$.
+
+| Letter | Rendered version | $\LaTeX$ code |
+|:---|:---:|---:|
+| Alpha | $\alpha$ | `\alpha` |
+| Beta | $\beta$ | `\beta` |
+| Gamma | $\gamma$ | `\gamma` |
+| Delta | $\delta$ | `\delta` |
+| Epsilon | $\epsilon$ | `\epsilon` |
+| Zeta | $\zeta$ | `\zeta` |
+| Eta | $\eta$ | `\eta` |
+| Theta | $ \theta $ | `\theta` |
+| Iota | $ \iota $ | `\iota` |
+| Kappa | $ \kappa $ | `\kappa` |
+| Lambda | $ \lambda $ | `\lambda` |
+| Mu | $ \mu $ | `\mu` |
+| Nu | $ \nu $ | `\nu` |
+| Pi | $ \pi $ | `\pi` |
+| Rho | $ \rho $ | `\rho` |
+| Sigma | $ \sigma $ | `\sigma` |
+| Tau | $ \tau $ | `\tau` |
+| Upsilon | $ \upsilon $ | `\upsilon` |
+| Phi | $ \phi $ | `\phi` |
+| Chi | $ \chi $ | `\chi` |
+| Psi | $ \psi $ | `\psi` |
+| Omega | $ \omega $ | `\omega` |
+{: .latex_table}
+
+#### Mathematical variants
+A handful of variant greek letters see widespread use in mathematics and physics. These are listed below.
+
+| Letter | Rendered version | $\LaTeX$ code |
+|:---|:---:|---:|
+| Epsilon | $\varepsilon$ | `\varepsilon` |
+| Theta | $\vartheta$ | `\vartheta` |
+| Phi | $\varphi$ | `\varphi` |
+{: .latex_table}
+
+### Other scripts
+Note that most of these fonts require the <a href="https://ctan.org/pkg/amsfonts" target="_blank">`amsfonts`</a> package.
+
+| Script | Examples | $\LaTeX$ code |
+|:---|:---:|---:|
+| Blackboard Bold | $\mathbb{N}$ and $\mathbb{R}$ | `\mathbb{N}` and  `\mathbb{R}` |
+| Fraktur | $\mathfrak{N}$ and $\mathfrak{R}$ | `\mathfrak{N}` and `\mathfrak{R}` |
+| Script | $\mathscr{N}$ and $\mathscr{R}$ | `\mathscr{N}` and `\mathscr{R}` |
+| Caligraphic | $\mathcal{N}$ and $\mathcal{R}$ | `\mathcal{N}` and `\mathcal{R}` |
+| Boldface | $\mathbf{N}$ and $\mathbf{R}$ | `\mathbf{N}` and `\mathbf{R}` |
+| Sans-serif | $\mathsf{N}$ and $\mathsf{R}$ | `\mathsf{N}` and `\mathsf{R}` |
+| Typewriter | $\mathtt{N}$ and $\mathtt{R}$ | `\mathtt{N}` and `\mathtt{R}` |
+| Italic | $\mathit{N}$ and $\mathit{R}$ | `\mathit{N}` and `\mathit{R}` |
+| Normal (default math font) | $\mathnormal{N}$ and $\mathnormal{R}$ | `\mathnormal{N}` and `\mathtt{R}` |
+{: .latex_table}
+
+### Set-theoretic and logical symbols
+
+| Name | Symbol | $\LaTeX$ code |
+|:---|:---:|---:|
+| In | $\in$ | `\in` |
+| Not in | $\notin$ | `\notin` |
+| Empty set | $\varnothing$ | `\varnothing` |
+| Proper subset | $\subset$ | `\subset` |
+| Subset | $\subseteq$ | `\subseteq` |
+| Proper superset | $\supset$ | `\supset` |
+| Superset | $\supseteq$ | `\supseteq` |
+| Union | $\cup$ | `\cup` |
+| Cap | $\cap$ | `cap` |
+| Set minus / Backslash | $\setminus$ | `\setminus` |
+| For all | $\forall$ | `\forall` |
+| There exists | $\exists$ | `\exists` |
+| Implies | $\implies$ | `\implies` |
+| If and only if | $\iff$ | `\iff`| 
+{: .latex_table}
+
+### Calculus
+
+| Name | Symbol | $\LaTeX$ code |
+|:---|:---:|---:|
+| Sum | $\sum$ | `\sum` |
+| Product | $\prod$ | `\prod` |
+| Coproduct | $\coprod$ | `\coprod` |
+| Infinity | $\infty$ | `\infty` |
+| To | $\to$ | `\to` |
+| Maps to | $\mapsto$ | `\mapsto` |
+| Prime | $\prime$ | `\prime` |
+| Partial derivative | $\partial$ | `\partial` |
+| Velocity | $\dot{x}$ | `\dot{x}` |
+| Acceleration | $\ddot{x}$ | `\ddot{x}` |
+| Integral | $\int_{a}^{b}$ | `\int_{a}^{b}` |
+| Double integral | $\iint$ | `\iint` |
+| Triple integral | $\iiint$ | `\iiint` |
+| Contour integral | $\oint$ | `\oint` |
+| Del / Nabla | $\nabla$ | `\nabla` |
+{: .latex_table}
+
+### Simple geometry
+
+| Name | Symbol | $\LaTeX$ code |
+|:---|:---:|---:|
+| Parallel | $\parallel$ | `\parallel` |
+| Not parallel | $\nparallel$ | `\nparallel` |
+| Perpendicular | $\perp$ | `\perp` |
+| Angle | $\angle$ | `\angle` |
+| Triangle | $\triangle$ | `\triangle` |
+| Square | $\square$ | `\square` |
+| Vector | $\overrightarrow{AB}$ | `\overrightarrow{AB}` |
+| Line segment | $\overline{AB}$ | `\overline{AB}` |
+{: .latex_table}
+
+### Other useful commands
+
+| $\LaTeX$ spacing command | Description |
+|:---|:---|
+| `\quad` | A space equal to the current font size |
+| `\qquad` | Twice the length of `\quad` |
+| `\,` | 3/18ths of `\quad` |
+| `\:` | 4/18ths of `\quad` |
+| `\;` | 5/18ths of `\quad` |
+| `\!` | -3/18ths of `\quad` |
+| `\ ` (backslash followed by a space) | A space equivalent to that in normal text |
+
+
+
+## Further resources
+Below is a list of further learning resources and reference material for $\LaTeX$ alongside guides on how you can get up and running with $\LaTeX$ quickly and easily.
+
+- [#latex-help](https://discord.com/channels/268882317391429632/840667252793802752){: .mention target="_blank"} - The server channel for $\LaTeX$ help.
+- [Overleaf Learn](https://www.overleaf.com/learn){:target="_blank"} - A high-quality and comprehensive set of guides and instructions using $\LaTeX$.
+- [DeTeXify](https://detexify.kirelabs.org/classify.html){:target="_blank"} - A useful utility for finding symbols you don't know the command for. Uses OCR handwriting recognition to recommend a symbol.
+- [Overleaf](https://www.overleaf.com){:target="_blank"} - An online cloud-based $\LaTeX$ editor. Free for personal use with additional paid features for collaboration and quality-of-life.
+- [TeX Live](https://www.tug.org/texlive/){:target="_blank"} - A multiplatform $\LaTeX$ compiler.
+- [CTAN](https://ctan.org/){:target="_blank"} - The Comprehensive $\TeX$ Archive Network, the canonical repository for $\TeX$ and $\LaTeX$ packages.
+
+[^1]: Note that braces are not always *required* to pass arguments to commands. For example, both `\frac 1 2` and `\frac{1}{2}`, as well as `\mathbb N` and `\mathbb{N}`, produce the same output; nevertheless, in each case the latter style is customarily preferred.