docs: add Math Preview

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+---
+title: "Math Preview"
+author: "Murray Bourne"
+date: 2019-03-04T16:01:23+08:00
+lastmod: 2019-03-05T16:01:23+08:00
+draft: false
+tags: ["preview", "math", "tag-6"]
+categories: ["docs", "math", "index"]
+
+# mathjax: true
+
+# Use KaTeX
+# See https://github.com/KaTeX/KaTeX
+katex: true
+
+# Use Mmark
+# See https://gohugo.io/content-management/formats/#mmark
+markup: mmark
+
+menu:
+  main:
+    parent: "docs"
+    weight: 5
+---
+
+[KaTeX and MathJax Comparison Demo, currently processed as KaTex](https://www.intmath.com/cg5/katex-mathjax-comparison.php)
+
+<!--more-->
+
+## Repeating fractions
+$$
+\frac{1}{\Bigl(\sqrt{\phi \sqrt{5}}-\phi\Bigr) e^{\frac25 \pi}} \equiv 1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}} {1+\frac{e^{-8\pi}} {1+\cdots} } } }
+$$
+
+
+## Summation notation
+$$
+\left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right)
+$$
+
+
+## Sum of a Series
+I broke up the next two examples into separate lines so it behaves better on a mobile phone. That's why they include \displaystyle.
+
+$$
+\displaystyle\sum_{i=1}^{k+1}i
+$$
+
+$$
+\displaystyle= \left(\sum_{i=1}^{k}i\right) +(k+1)
+$$
+
+$$
+\displaystyle= \frac{k(k+1)}{2}+k+1
+$$
+
+$$
+\displaystyle= \frac{k(k+1)+2(k+1)}{2}
+$$
+
+$$
+\displaystyle= \frac{(k+1)(k+2)}{2}
+$$
+
+$$
+\displaystyle= \frac{(k+1)((k+1)+1)}{2}
+$$
+
+## Product notation
+$$
+\displaystyle 1 + \frac{q^2}{(1-q)}+\frac{q^6}{(1-q)(1-q^2)}+\cdots = \displaystyle \prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})}, \displaystyle\text{ for }\lvert q\rvert < 1.
+$$
+
+
+## Inline math
+And here is some in-line math: $$ k_{n+1} = n^2 + k_n^2 - k_{n-1} $$ , followed by some more text.
+
+
+## Greek Letters
+$$
+\Gamma\ \Delta\ \Theta\ \Lambda\ \Xi\ \Pi\ \Sigma\ \Upsilon\ \Phi\ \Psi\ \Omega
+\alpha\ \beta\ \gamma\ \delta\ \epsilon\ \zeta\ \eta\ \theta\ \iota\ \kappa\ \lambda\ \mu\ \nu\ \xi \ \omicron\ \pi\ \rho\ \sigma\ \tau\ \upsilon\ \phi\ \chi\ \psi\ \omega\ \varepsilon\ \vartheta\ \varpi\ \varrho\ \varsigma\ \varphi
+$$
+
+
+## Arrows
+$$
+\gets\ \to\ \leftarrow\ \rightarrow\ \uparrow\ \Uparrow\ \downarrow\ \Downarrow\ \updownarrow\ \Updownarrow
+$$
+
+$$
+\Leftarrow\ \Rightarrow\ \leftrightarrow\ \Leftrightarrow\ \mapsto\ \hookleftarrow
+\leftharpoonup\ \leftharpoondown\ \rightleftharpoons\ \longleftarrow\ \Longleftarrow\ \longrightarrow
+$$
+
+$$
+\Longrightarrow\ \longleftrightarrow\ \Longleftrightarrow\ \longmapsto\ \hookrightarrow\ \rightharpoonup
+$$
+
+$$
+\rightharpoondown\ \leadsto\ \nearrow\ \searrow\ \swarrow\ \nwarrow
+$$
+
+
+## Symbols
+$$
+\surd\ \barwedge\ \veebar\ \odot\ \oplus\ \otimes\ \oslash\ \circledcirc\ \boxdot\ \bigtriangleup
+$$
+
+$$
+\bigtriangledown\ \dagger\ \diamond\ \star\ \triangleleft\ \triangleright\ \angle\ \infty\ \prime\ \triangle
+$$
+
+
+## Calculus
+$$
+\int u \frac{dv}{dx}\,dx=uv-\int \frac{du}{dx}v\,dx
+$$
+
+$$
+f(x) = \int_{-\infty}^\infty \hat f(\xi)\,e^{2 \pi i \xi x}
+$$
+
+$$
+\oint \vec{F} \cdot d\vec{s}=0
+$$
+
+
+## Lorenz Equations
+$$
+\begin{aligned} \dot{x} & = \sigma(y-x) \\ \dot{y} & = \rho x - y - xz \\ \dot{z} & = -\beta z + xy \end{aligned}
+$$
+
+
+## Cross Product
+This works in KaTeX, but the separation of fractions in this environment is not so good.
+
+$$
+\mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\ \frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0 \end{vmatrix}
+$$
+
+Here's a workaround: make the fractions smaller with an extra class that targets the spans with "mfrac" class (makes no difference in the MathJax case):
+
+$$
+\mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\ \frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0 \end{vmatrix}
+$$
+
+
+## Accents
+$$
+\hat{x}\ \vec{x}\ \ddot{x}
+$$
+
+
+## Stretchy brackets
+$$
+\left(\frac{x^2}{y^3}\right)
+$$
+
+
+## Evaluation at limits
+$$
+\left.\frac{x^3}{3}\right|_0^1
+$$
+
+
+## Case definitions
+$$
+f(n) = \begin{cases} \frac{n}{2}, & \text{if } n\text{ is even} \\ 3n+1, & \text{if } n\text{ is odd} \end{cases}
+$$
+
+
+## Maxwell's Equations
+$$
+\begin{aligned} \nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\ \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\ \nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\ \nabla \cdot \vec{\mathbf{B}} & = 0 \end{aligned}
+$$
+
+These equations are quite cramped. We can add vertical spacing using (for example) [1em] after each line break (\\). as you can see here:
+
+$$
+\begin{aligned} \nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\[1em] \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\[0.5em] \nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\[1em] \nabla \cdot \vec{\mathbf{B}} & = 0 \end{aligned}
+$$
+
+
+## Statistics
+Definition of combination:
+
+$$
+\frac{n!}{k!(n-k)!} = {^n}C_k
+{n \choose k}
+$$
+
+## Fractions on fractions
+$$
+\frac{\frac{1}{x}+\frac{1}{y}}{y-z}
+$$
+
+
+## n-th root
+$$
+\sqrt[n]{1+x+x^2+x^3+\ldots}
+$$
+
+
+## Matrices
+$$
+\begin{pmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{pmatrix}
+\begin{bmatrix} 0 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 0 \end{bmatrix}
+$$
+
+
+## Punctuation
+$$
+f(x) = \sqrt{1+x} \quad (x \ge -1)
+f(x) \sim x^2 \quad (x\to\infty)
+$$
+
+Now with punctuation:
+
+$$
+f(x) = \sqrt{1+x}, \quad x \ge -1
+f(x) \sim x^2, \quad x\to\infty
+$$