227 lines
5.4 KiB
Markdown
227 lines
5.4 KiB
Markdown
---
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title: "数学公式渲染"
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description: "主题支持mathjs和katex两种不同插件的数学公式渲染方案。"
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keywords: "math,formula"
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date: 2022-09-11T10:16:02+08:00
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lastmod: 2022-09-11T10:16:02+08:00
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categories:
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- 示例
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tags:
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- 数学公式
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- mathjax
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- katex
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url: "post/math-formula.html"
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math: mathjax
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---
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本主题支持 `mathjax` 和 `katex` 两种不的方案支持数学公式的渲染，可根据自已的需求进行选择。
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<!--more-->
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接下的示例中，将使用 [MathJax](https://www.mathjax.org/) 方案来展示渲染效果。
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{{< note info >}}
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- 使用 `hugo new` 命令创建一篇新的文章
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- 可以全局启用数据公式渲染，请在项目配置参数 `math: katex` 或 `math: mathjax`
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- 或是将该参数配置到需要显示数学公式的页面头部（减少不必要的加载消耗）
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{{< /note >}}
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**注意：** 使用[支持的TeX功能](https://docs.mathjax.org/en/latest/input/tex/index.html)的联机参考资料。
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### 例子
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## 重复的分数
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$$
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\frac{1}{\Bigl(\sqrt{\phi \sqrt{5}}-\phi\Bigr) e^{\frac25 \pi}} \equiv 1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}} {1+\frac{e^{-8\pi}} {1+\cdots} } } }
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$$
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## 总和记号
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$$
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\left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right)
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$$
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## 几何级数之和
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我把接下来的两个例子分成了几行，这样它在手机上表现得更好。这就是为什么它们包含 `\displaystyle`。
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$$
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\displaystyle\sum_{i=1}^{k+1}i
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$$
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$$
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\displaystyle= \left(\sum_{i=1}^{k}i\right) +(k+1)
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$$
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$$
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\displaystyle= \frac{k(k+1)}{2}+k+1
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$$
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$$
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\displaystyle= \frac{k(k+1)+2(k+1)}{2}
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$$
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$$
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\displaystyle= \frac{(k+1)(k+2)}{2}
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$$
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$$
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\displaystyle= \frac{(k+1)((k+1)+1)}{2}
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$$
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## 乘记号
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$$
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\displaystyle 1 + \frac{q^2}{(1-q)}+\frac{q^6}{(1-q)(1-q^2)}+\cdots = \displaystyle \prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})}, \displaystyle\text{ for }\lvert q\rvert < 1.
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$$
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## 随文数式
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这是一些线性数学: $$ k_{n+1} = n^2 + k_n^2 - k_{n-1} $$ ， 然后是更多的文本。
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## 希腊字母
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$$
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\Gamma\ \Delta\ \Theta\ \Lambda\ \Xi\ \Pi\ \Sigma\ \Upsilon\ \Phi\ \Psi\ \Omega
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\alpha\ \beta\ \gamma\ \delta\ \epsilon\ \zeta\ \eta\ \theta\ \iota\ \kappa\ \lambda\ \mu\ \nu\ \xi \ \omicron\ \pi\ \rho\ \sigma\ \tau\ \upsilon\ \phi\ \chi\ \psi\ \omega\ \varepsilon\ \vartheta\ \varpi\ \varrho\ \varsigma\ \varphi
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$$
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## 箭头
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$$
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\gets\ \to\ \leftarrow\ \rightarrow\ \uparrow\ \Uparrow\ \downarrow\ \Downarrow\ \updownarrow\ \Updownarrow
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$$
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$$
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\Leftarrow\ \Rightarrow\ \leftrightarrow\ \Leftrightarrow\ \mapsto\ \hookleftarrow
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\leftharpoonup\ \leftharpoondown\ \rightleftharpoons\ \longleftarrow\ \Longleftarrow\ \longrightarrow
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$$
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$$
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\Longrightarrow\ \longleftrightarrow\ \Longleftrightarrow\ \longmapsto\ \hookrightarrow\ \rightharpoonup
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$$
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$$
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\rightharpoondown\ \leadsto\ \nearrow\ \searrow\ \swarrow\ \nwarrow
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$$
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## 符号
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$$
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\surd\ \barwedge\ \veebar\ \odot\ \oplus\ \otimes\ \oslash\ \circledcirc\ \boxdot\ \bigtriangleup
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$$
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$$
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\bigtriangledown\ \dagger\ \diamond\ \star\ \triangleleft\ \triangleright\ \angle\ \infty\ \prime\ \triangle
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$$
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## 微积分学
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$$
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\int u \frac{dv}{dx}\,dx=uv-\int \frac{du}{dx}v\,dx
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$$
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$$
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f(x) = \int_{-\infty}^\infty \hat f(\xi)\,e^{2 \pi i \xi x}
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$$
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$$
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\oint \vec{F} \cdot d\vec{s}=0
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$$
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## 洛伦茨方程
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$$
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\begin{aligned} \dot{x} & = \sigma(y-x) \\\\ \dot{y} & = \rho x - y - xz \\\\ \dot{z} & = -\beta z + xy \end{aligned}
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$$
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## 交叉乘积
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这在KaTeX中是可行的，但在这种环境中馏分的分离不是很好。
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$$
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\mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\\\ \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\\\ \frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0 \end{vmatrix}
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$$
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这里有一个解决方案:使用“mfrac”类(在MathJax情况下没有区别)的额外类使分数更小:
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$$
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\mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\\\ \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\\\ \frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0 \end{vmatrix}
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$$
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## 强调
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$$
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\hat{x}\ \vec{x}\ \ddot{x}
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$$
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## 有弹性的括号
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$$
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\left(\frac{x^2}{y^3}\right)
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$$
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## 评估范围
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$$
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\left.\frac{x^3}{3}\right|_0^1
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$$
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## 诊断标准
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$$
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f(n) = \begin{cases} \frac{n}{2}, & \text{if } n\text{ is even} \\\\ 3n+1, & \text{if } n\text{ is odd} \end{cases}
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$$
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## 麦克斯韦方程组
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$$
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\begin{aligned} \nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\\\ \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\\\ \nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\\\ \nabla \cdot \vec{\mathbf{B}} & = 0 \end{aligned}
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$$
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## 统计学
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固定词组：
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$$
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\frac{n!}{k!(n-k)!} = {^n}C_k
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{n \choose k}
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$$
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## 分数在分数
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$$
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\frac{\frac{1}{x}+\frac{1}{y}}{y-z}
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$$
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## ｎ次方根
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$$
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\sqrt[n]{1+x+x^2+x^3+\ldots}
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$$
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## 矩阵
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$$
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\begin{pmatrix} a_{11} & a_{12} & a_{13}\\\\ a_{21} & a_{22} & a_{23}\\\\ a_{31} & a_{32} & a_{33} \end{pmatrix}
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\begin{bmatrix} 0 & \cdots & 0 \\\\ \vdots & \ddots & \vdots \\\\ 0 & \cdots & 0 \end{bmatrix}
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$$
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## 标点符号
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$$
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f(x) = \sqrt{1+x} \quad (x \ge -1)
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f(x) \sim x^2 \quad (x\to\infty)
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$$
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现在用标点符号:
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$$
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f(x) = \sqrt{1+x}, \quad x \ge -1
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f(x) \sim x^2, \quad x\to\infty
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$$ |