mdbook_mathjax

Section 1

Section 2

Chapter 2

Section 1

Section 2

Chapter 3

Section 1

Section 2

Chapter 4

Section 1

Section 2

Chapter 5

Section 1

Section 2

Chapter 6

Section 1

Section 2

Here is an inline example, $\pi (\theta )$,

an equation,

$$\nabla f(x)\in {\mathbb{R}}^n,$$

and a regular $ symbol.

Define $f(x)$:

$$\begin{gathered} f(x)=x^2 \\ x\in \mathbb{R} \end{gathered}$$

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以下中文

• 扩展欧几里得算法（Python）: \\被转义成\,目前原因不明。

\begin{aligned} & \texttt{def ex_gcd(a, b):}\\ & \qquad \texttt{if b == 0: }\\ & \qquad \qquad \texttt{return 1, 0, a}\\ & \qquad \texttt{else: }\\ & \qquad \qquad \texttt{x, y, g = ex_gcd(b, a % b)}\\ & \qquad \qquad \texttt{return y, x - (a // b) * y, g}\\ \\ \\ & \texttt{N = 5}\\ & \texttt{for n in range(1, N + 1):}\\ & \qquad \texttt{for m in range(1, N + 1):}\\ & \qquad \qquad \texttt{print(ex_gcd(n, m))} \end{aligned}

• 范德蒙德行列式

$$\left|\begin{array}{cccc}1& 1& \dots & 1\\ x_1& x_2& \dots & x_n\\ ⋮& ⋮& \ddots & ⋮\\ x_1^{n-1}& x_2^{n-2}& \dots & x_n^{n-1}\end{array}\right|=\prod _{1\le j\le i\le n}(x_i-x_j)$$

• 碳酸氢钙溶液加热

$$\mathrm{Ca}(\mathrm{HC}{\mathrm{O}}_3{)}_2\stackrel{△}{\to }\mathrm{CaC}{\mathrm{O}}_3\downarrow +\mathrm{C}{\mathrm{O}}_2\uparrow +{\mathrm{H}}_2\mathrm{O}$$

• 薛定谔方程

$$i\hslash \frac{\partial }{\partial t}|\psi (t)⟩=\hat{H}|\psi (t)⟩$$

$$i\hslash \frac{\partial }{\partial t}\Psi (x,t)=\left[-\frac{{\hslash }^2}{2m}\frac{{\partial }^2}{\partial x^2}+V(x,t)\right]\Psi (x,t)$$

• 麦克斯韦方程组

$$\{\begin{array}{ll}\nabla \cdot \mathbf{E}=\frac{\rho }{{\epsilon }_0}& \text{Gauss’s law}\\ \nabla \cdot \mathbf{B}=0& \text{Gauss’s law for magnetism}\\ \nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}& \text{Maxwell–Faraday equation}\\ \nabla \times \mathbf{B}={\mu }_0\left(\mathbf{J}+{\epsilon }_0\frac{\partial \mathbf{E}}{\partial t}\right)& \text{Ampeˋre’s circuital law}\end{array}$$

• 异或运算的真值表

$$\begin{array}{lll}x& y& S\\ 0& 0& 0\\ 0& 1& 1\\ 1& 0& 1\\ 1& 0& 0\end{array}$$

• 狄利克雷函数

$$1_{\mathbb{Q}}(x)=\{\begin{array}{ll}1& x\in \mathbb{Q}\\ 0& x\notin \mathbb{Q}\end{array}$$

• 雅可比符号

$$\left(\frac{a}{p}\right)=\{\begin{array}{ll}0& \text{if a≡0(modp)}\\ 1& \text{if }a\not\equiv 0(\mathrm{mod}p)\wedge \exists x:x^2\equiv a(\mathrm{mod}p)\\ -1& \text{if }a\not\equiv 0(\mathrm{mod}p)\wedge \nexists x:x^2\equiv a(\mathrm{mod}p)\end{array}$$

• 斯托克斯公式

$$\begin{array}{rl}{\oint }_{\Gamma }Pdx+Qdy+Rdz=& {\iint }_{\Sigma }\left(\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}\right)dydz\\ & +\left(\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x}\right)dzdx\\ & +\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dxdy\end{array}$$

• 涛函数与黎曼函数

$$\begin{array}{rlr}\Gamma (z)=& {\int }_0^{\infty }{t}^{z-1}{\mathrm{e}}^{-t}dt& \\ \zeta (s)=& \sum _{n=1}^{\infty }\frac{1}{n^s}& \end{array}$$

• 德摩根律

$$\begin{array}{rl}& \neg (P\vee Q)⟺\neg P\wedge \neg Q\\ & \neg (P\wedge Q)⟺\neg P\vee \neg Q\\ & (A\cup B{)}^{\complement }=A^{\complement }\cap B^{\complement }\\ & (A\cap B{)}^{\complement }=A^{\complement }\cup B^{\complement }\\ & \overline{A\cup B}=\overline{A}\cap \overline{B}\\ & \overline{A\cap B}=\overline{A}\cup \overline{B}\end{array}\text{\hspace{1em}(propositional logic)(set theory)(probability theory)}$$

• 贝叶斯公式

$$P(A_i|B)=\frac{P(B|A_i)\cdot P(A_i)}{{\sum }_{j=1}^{n}P(B|A_j)\cdot P(A_j)}$$

• 麦克劳林级数

$$\begin{array}{rl}& e^x=\sum _{n=0}^{\infty }\frac{x^n}{n!}\\ & \sin x=\sum _{n=0}^{\infty }(-1{)}^n\frac{x^{2n+1}}{(2n+1)!}\\ & \cos x=\sum _{n=0}^{\infty }(-1{)}^n\frac{x^{2n}}{(2n)!}\\ & \ln (x+1)=\sum _{n=0}^{\infty }(-1{)}^n\frac{x^{n+1}}{(n+1)!}\end{array}$$

• 自然对数e的定义

$$e=\underset{x\to \infty }{\lim }{\left(1+\frac{1}{x}\right)}^x$$

• 二项式定理

$$(x+y{)}^n=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})x^k{y}^{n-k}$$