latex就是一个方便编辑数学公式的一个库，简单汇总一下他的用法。
Donald为了更好的在他的著作里编写数学公式，发明了Tex这种宏语言，专门排版数学公式。Leslie封装了Tex，并自定义了更多的宏指令，成为了LaTeX。现在是国际通用的排版系统，很多论文收稿的排版要求都是LaTex，word是不接收的。
妈咪说公布了一个在线的公式编辑的网站，生成latex更方便了，非常好用：www.latexlive.com
LaTex Math的语法多且杂，我们是没法完全记住这些语法的。查询手册在手，天下我有，这里比较推荐名校莱斯Rice大学的一个语法手册，莱斯大学LaTex Math在线PDF手册 。
$c = \sqrt{a^{2}+b_{xy}^{2}+e^{x}}$
c = a 2 + b x y 2 + e x c = \sqrt{a^{2}+b_{xy}^{2}+e^{x}} c = a 2 + b x y 2 + e x
$$ c = \sqrt{a^{2}+b_{xy}^{2} +e^{x}} $$
c = a 2 + b x y 2 + e x c = \sqrt{a^{2}+b_{xy}^{2} +e^{x}}
c = a 2 + b x y 2 + e x
以下几个字符：# $ % & ~ _ ^ \ { }有特殊意义，需要表示这些字符时，需要转义，即在每个字符前加上\。
\boxed命令给公式加一个方框。
希腊字母有大写和小写之分，这个大小写是由LaTex的首字母是否大小写来控制的。
希腊字母
LaTeX形式
α A \alpha \Alpha α A \alpha \Alpha
β B \beta \Beta β B \beta \Beta
γ Γ \gamma \Gamma γ Γ \gamma \Gamma
δ Δ \delta \Delta δ Δ \delta \Delta
ϵ ε E \epsilon \varepsilon \Epsilon ϵ ε E \epsilon \varepsilon \Epsilon
ζ Z \zeta \Zeta ζ Z \zeta \Zeta
η H \eta \Eta η H \eta \Eta
θ ϑ Θ \theta \vartheta \Theta θ ϑ Θ \theta \vartheta \Theta
ι I \iota \Iota ι I \iota \Iota
κ K \kappa \Kappa κ K \kappa \Kappa
λ Λ \lambda \Lambda λ Λ \lambda \Lambda
μ M \mu M μ M \mu \Mu
ξ Ξ \xi \Xi ξ Ξ \xi \Xi
o O
o O
π Π \pi \Pi π Π \pi \Pi
ρ ϱ P \rho \varrho \Rho ρ ϱ P \rho \varrho \Rho
σ Σ \sigma \Sigma σ Σ \sigma \Sigma
τ τ \tau \tau τ τ \tau \Tau
υ Υ \upsilon \Upsilon υ Υ \upsilon \Upsilon
ϕ φ Φ \phi \varphi \Phi ϕ φ Φ \phi \varphi \Phi
χ X \chi \Chi χ X \chi \Chi
ψ Ψ \psi \Psi ψ Ψ \psi \Psi
ω Ω \omega \Omega ω Ω \omega \Omega
符号
LaTex形式
x 1 x^1 x 1 x^1
x 1 x_1 x 1 x_1
形式：\sqrt[开方次数，默认为2]{开方公式}
符号
LaTex形式
x i j 2 x x 3 x_{ij}^2\quad \sqrt{x}\quad \sqrt[3]{x} x i j 2 x 3 x x_{ij}^2\quad \sqrt{x}\quad \sqrt[3]{x}
\frac表示分数，字号工具环境设置：
\dfrac命令把字号设置为独立公式中的大小；\tfrac则把字号设置为行间公式中的大小。
符号
LaTex形式
1 2 1 2 \frac{1}{2} \dfrac{1}{2} 2 1 2 1 \frac{1}{2} \dfrac{1}{2}
1 2 1 2 \frac{1}{2} \tfrac{1}{2} 2 1 2 1 \frac{1}{2} \tfrac{1}{2}
符号
LaTex形式
tan \tan tan \tan
sin \sin sin \sin
cos \cos cos \cos
lg \lg lg \lg
arcsin \arcsin arcsin \arcsin
arctan \arctan arctan \arctan
min \min min \min
max \max max \max
exp \exp exp \exp
log \log log \log
符号
LaTex形式
1 ± 1 1\pm1 1 ± 1 \pm
× \times × \times
÷ \div ÷ \div
⋅ \cdot ⋅ \cdot
∩ \cap ∩ \cap
∪ \cup ∪ \cup
≥ \geq ≥ \geq
≤ \leq ≤ \leq
≠ \neq = \neq
≈ \approx ≈ \approx
≡ \equiv ≡ \equiv
∈ \in ∈ \in
∉ \notin ∈ / \notin
∋ \ni ∋ \ni
⊂ \subset ⊂ \subset
和、积、极限、积分等运算符，这些公式在行内公式被压缩，以适应行高，可以通过\limits和\nolimits命令显示制动是否压缩
符号
LaTex形式
∑ \sum ∑ \sum
∏ \prod ∏ \prod
lim \lim lim \lim
∫ \int ∫ \int
符号
LaTex形式
∫ \int ∫ \int
∬ \iint ∬ \iint
∫ ∫ \int \int ∫ ∫ \int \int
∭ \iiint ∭ \iiint
∫ ∫ ∫ \int \int \int ∫ ∫ ∫ \int \int \int
\iiiint
\iiiint
∫ ∫ ∫ ∫ \int \int \int \int ∫ ∫ ∫ ∫ \int \int \int \int
\idotsint
\idotsint
∫ ⋯ ∫ \int \dots \int ∫ ⋯ ∫ \int \dots \int
符号
LaTex形式
← \leftarrow ← \leftarrow
→ \rightarrow → \rightarrow
↔ \leftrightarrow ↔ \leftrightarrow
⟵ \longleftarrow ⟵ \longleftarrow
⟷ \longleftrightarrow ⟷ \longleftrightarrow
⟹ \Longrightarrow ⟹ \Longrightarrow
\xleftarrow和\xrightarrow可根据内容自动调整
符号
LaTex形式
x ˉ \bar{x} x ˉ \bar{x}
x ˊ \acute{x} x ˊ \acute{x}
x ˚ \mathring{x} x ˚ \mathring{x}
x ⃗ \vec{x} x \vec{x}
x ˋ \grave{x} x ˋ \grave{x}
x ˙ \dot{x} x ˙ \dot{x}
x ^ \hat{x} x ^ \hat{x}
x ~ \tilde{x} x ~ \tilde{x}
x ¨ \ddot{x} x ¨ \ddot{x}
x ˇ \check{x} x ˇ \check{x}
x ˘ \breve{x} x ˘ \breve{x}
\dddot{x}
\dddot{x}
用() [] {} \lange \rangle表示 () [] {} ⟨⟩
符号
LaTex形式
x x x ‾ \overline{xxx} x x x \overline{xxx}
x x x ↔ \overleftrightarrow{xxx} x x x \overleftrightarrow{xxx}
x x x ‾ \underline{xxx} x x x \underline{xxx}
x x x ↔ \underleftrightarrow{xxx} x x x \underleftrightarrow{xxx}
x x x ← \overleftarrow{xxx} x x x \overleftarrow{xxx}
x x x ⏞ \overbrace{xxx} x x x \overbrace{xxx}
x x x ← \underleftarrow{xxx} x x x \underleftarrow{xxx}
x x x ⏟ \underbrace{xxx} x x x \underbrace{xxx}
x x x → \overrightarrow{xxx} x x x \overrightarrow{xxx}
x x x ^ \widehat{xxx} x x x \widehat{xxx}
x x x → \underrightarrow{xxx} x x x \underrightarrow{xxx}
x x x ~ \widetilde{xxx} x x x \widetilde{xxx}
( ( ( ( ( x ) ) ) ) ) \Bigg( \bigg( \Big( \big((x) \big) \Big) \bigg) \Bigg) ( ( ( ( ( x ) ) ) ) ) $\Bigg( \bigg( \Big( \big((x) \big) \Big) \bigg) \Bigg)$
[ [ [ [ [ x ] ] ] ] ] \Bigg[ \bigg[ \Big[ \big[[x] \big] \Big] \bigg] \Bigg] [ [ [ [ [ x ] ] ] ] ] $\Bigg[ \bigg[ \Big[ \big[[x] \big] \Big] \bigg] \Bigg]$
{ { { { { x } } } } } \Bigg\{ \bigg\{ \Big\{ \big\{\{x\} \big\} \Big\} \bigg\} \Bigg\} { { { { { x } } } } } $\Bigg\{ \bigg\{ \Big\{ \big\{\{x\} \big\} \Big\} \bigg\} \Bigg\}$
⟨ ⟨ ⟨ ⟨ ⟨ x ⟩ ⟩ ⟩ ⟩ ⟩ \Bigg\langle \bigg\langle \Big\langle \big\langle\langle x \rangle \big\rangle \Big\rangle \bigg\rangle \Bigg\rangle ⟨ ⟨ ⟨ ⟨ ⟨ x ⟩ ⟩ ⟩ ⟩ ⟩ $\Bigg\langle \bigg\langle \Big\langle \big\langle\langle x \rangle \big\rangle \Big\rangle \bigg\rangle \Bigg\rangle$
∣ ∣ ∣ ∣ ∣ x ∣ ∣ ∣ ∣ ∣ \Bigg\lvert \bigg\lvert \Big\lvert \big\lvert\lvert x \rvert \big\rvert \Big\rvert \bigg\rvert \Bigg\rvert ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ x ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ $\Bigg\lvert \bigg\lvert \Big\lvert \big\lvert\lvert x \rvert \big\rvert \Big\rvert \bigg\rvert \Bigg\rvert$
∥ ∥ ∥ ∥ ∥ x ∥ ∥ ∥ ∥ ∥ \Bigg\lVert\bigg\lVert\Big\lVert\big\lVert\lVert x \rVert \big\rVert\Big\rVert\bigg\rVert \Bigg\rVert ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ x ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ $\Bigg\lVert\big\lVert \Big\lVert \big\lVert \lVert x \rVert \big\rVert \Big\rVert \bigg\rVert \Bigg\rVert$
省略号用 \dots \cdots \vdots \ddots表示 ，\dots和\cdots的纵向位置不同，前者一般用于有下标的序列
符号
LaTex形式
… \dots … \dots
⋯ \cdots ⋯ \cdots
⋮ \vdots ⋮ \vdots
⋱ \ddots ⋱ \ddots
x 1 , x 2 , … , x n 1 , 2 , ⋯ , n ⋮ ⋱ x_1, x_2, \dots, x_n\quad 1,2,\cdots,n\quad \vdots\quad \ddots
x 1 , x 2 , … , x n 1 , 2 , ⋯ , n ⋮ ⋱
语法
格式
实例
显示
quad空格
a \quad ba b a \quad b a b
两个quad空格
a \qquad ba b a \qquad b a b 两个m的宽度
大空格
a \: ba b a \: b a b 1/3m宽度
中等空格
a \; ba b a \; b a b 2/7m宽度
小空格
a \, ba b a \, b a b 1/6m宽度
没有空格
aba b ab a b 没有空格
缩进空格
a \! ba b a \! b a b 缩进1/6m宽度
分段函数是非常复杂的，这时候会用到LaTex的cases语法，用\begin{cases}和\end{cases}围住即可，中间则用\\来分段
x 1 x 2 … x 3 x 4 … ⋮ ⋮ ⋱ \begin{array}{ccc}
x_1 & x_2 &\dots\\
x_3 & x_4 &\dots\\
\vdots&\vdots&\ddots
\end{array}
x 1 x 3 ⋮ x 2 x 4 ⋮ … … ⋱
$$ \begin{array}{ccc} x_1 & x_2 &\dots\\ x_3 & x_4 &\dots\\ \vdots&\vdots&\ddots \end{array} $$
( a b c d ) [ a b c d ] { a b c d } ∣ a b c d ∣ ∥ a b c d ∥ \begin{pmatrix}
a & b\\
c & d \\
\end{pmatrix}
\quad
\begin{bmatrix}
a & b \\
c & d \\
\end{bmatrix}
\quad
\begin{Bmatrix}
a & b \\
c & d \\
\end{Bmatrix}
\quad
\begin{vmatrix}
a & b \\
c & d \\
\end{vmatrix}
\quad
\begin{Vmatrix}
a & b \\
c & d \\
\end{Vmatrix}
( a c b d ) [ a c b d ] { a c b d } ∣ ∣ ∣ ∣ a c b d ∣ ∣ ∣ ∣ ∥ ∥ ∥ ∥ a c b d ∥ ∥ ∥ ∥
$$ \begin{pmatrix} a & b\\ c & d \\ \end{pmatrix} \quad \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} \quad \begin{Bmatrix} a & b \\ c & d \\ \end{Bmatrix} \quad \begin{vmatrix} a & b \\ c & d \\ \end{vmatrix} \quad \begin{Vmatrix} a & b \\ c & d \\ \end{Vmatrix} $$
(
\begin{smallmatrix}
a & b \\
c & d
\end{smallmatrix}
)
$$ ( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} ) $$
无需对齐可使用multline；需要对齐使用split；用\\来分行；用&设置对齐的位置
\begin{multline}
x = a+b+c+{} \\
d+e+f+g
\end{multline}
$$ \begin{multline} x = a+b+c+{} \\ d+e+f+g \end{multline} $$
\begin{split}
x = {} & a + b + c +{}\\
& d + e + f + g
\end{split}
$$ \begin{split} x = {} & a + b + c +{}\\ & d + e + f + g \end{split} $$
不需要对齐的公式组用gather；需要对齐使用align:
\begin{gather}
a = b+c+d\\
x = y+z\\
5 = 4+1\\
\end{gather}
$$ \begin{gather} a = b+c+d\\ x = y+z\\ 5 = 4+1\\ \end{gather} $$
\begin{align}
a &=b+c+d \\
x &=y+z\\
5 &= 4+1
\end{align}
$$ \begin{align} a &=b+c+d \\ x &=y+z\\ 5 &= 4+1 \end{align} $$
y = { − x , x ≤ 0 x , x > 0 y=\begin{cases}
-x,\quad x\leq 0\\
x, \quad x>0
\end{cases}
y = { − x , x ≤ 0 x , x > 0
$$ y=\begin{cases} -x,\quad x\leq 0\\ x, \quad x>0 \end{cases} $$
\newtheorem{thm}{\bf Theorem}[section]
\begin{thm}\label{thm1}
Suppose system (\ref{l1}) satisfies Assumption (\ref{mim1}), the closed-loop system consisting of
system (\ref{l1}), the disturbance observer (\ref{g1}) and the proposed controller (\ref{n3}) is semi-globally ISS.
\end{thm}
$$ \newtheorem{thm}{\bf Theorem}[section] \begin{thm}\label{thm1} Suppose system (\ref{l1}) satisfies Assumption (\ref{mim1}), the closed-loop system consisting of system (\ref{l1}), the disturbance observer (\ref{g1}) and the proposed controller (\ref{n3}) is semi-globally ISS. \end{thm} $$
\newtheorem{lemma}{Lemma}[section]
\begin{lemma} \label{lemma1}
\end{lemma}
$$ \newtheorem{lemma}{Lemma}[section] \begin{lemma} \label{lemma1} \end{lemma} $$
\begin{proof}
***
\end{proof}
$$ \begin{proof} *** \end{proof} $$
\newtheorem{assumption}{Assumption}[section]
\begin{assumption}
***
\end{assumption}
$$ \newtheorem{assumption}{Assumption}[section] \begin{assumption} *** \end{assumption} $$
tag
latex
descript
+ + + +加
− - − -减
× \times × \times乘
÷ {\div} ÷ {\div}除
± \pm ± \pm加减
∓ \mp ∓ \mp减加
◃ \triangleleft ◃ \triangleleft正规子群
▹ \triangleright ▹ \triangleright属于正规子群
⋅ \cdot ⋅ \cdot点
∖ \setminus ∖ \setminus减号集
⋆ \star ⋆ \star星
∗ \ast ∗ \ast星号
∪ \cup ∪ \cup并集
∩ \cap ∩ \cap交集
⊔ \sqcup ⊔ \sqcup-
⊓ \sqcap ⊓ \sqcap-
∨ \vee ∨ \vee-
∧ \wedge ∧ \wedge-
∘ \circ ∘ \circ-
∙ \bullet ∙ \bullet-
⊕ \oplus ⊕ \oplus-
⊖ \ominus ⊖ \ominus-
⊙ \odot ⊙ \odot-
⊘ \oslash ⊘ \oslash-
⊗ \otimes ⊗ \otimes-
◯ \bigcirc ◯ \bigcirc-
⋄ \diamond ⋄ \diamond-
⊎ \uplus ⊎ \uplus-
△ \bigtriangleup △ \bigtriangleup-
▽ \bigtriangledown ▽ \bigtriangledown-
⊲ \lhd ⊲ \lhd-
⊳ \rhd ⊳ \rhd-
⊴ \unlhd ⊴ \unlhd-
⊵ \unrhd ⊵ \unrhd-
⨿ \amalg ⨿ \amalg-
≀ \wr ≀ \wr-
† \dagger † \dagger-
‡ \ddagger ‡ \ddagger-
tag
latex
descript
< < < <-
> > > >-
= = = =-
≤ \le ≤ \le-
≥ \ge ≥ \ge-
≡ \equiv ≡ \equiv-
≪ \ll ≪ \ll-
≫ \gg ≫ \gg-
≐ \doteq ≐ \doteq-
≜ \triangleq ≜ \triangleq-
≺ \prec ≺ \prec-
≻ \succ ≻ \succ-
∼ \sim ∼ \sim-
⪯ \preceq ⪯ \preceq-
⪰ \succeq ⪰ \succeq-
≃ \simeq ≃ \simeq-
≈ \approx ≈ \approx-
⊂ \subset ⊂ \subset-
⊃ \supset ⊃ \supset-
⊆ \subseteq ⊆ \subseteq-
⊇ \supseteq ⊇ \supseteq-
⊏ \sqsubset ⊏ \sqsubset-
⊐ \sqsupset ⊐ \sqsupset-
⊑ \sqsubseteq ⊑ \sqsubseteq-
⊒ \sqsupseteq ⊒ \sqsupseteq-
≅ \cong ≅ \cong-
⋈ \Join ⋈ \Join-
⋈ \bowtie ⋈ \bowtie-
∝ \propto ∝ \propto-
∈ \in ∈ \in-
∋ \ni ∋ \ni-
⊢ \vdash ⊢ \vdash-
⊣ \dashv ⊣ \dashv-
⊨ \models ⊨ \models-
∣ \mid ∣ \mid-
∥ \parallel ∥ \parallel-
⊥ \perp ⊥ \perp-
⌣ \smile ⌣ \smile-
⌢ \frown ⌢ \frown-
≍ \asymp ≍ \asymp-
: : : :-
∉ \notin ∈ / \notin-
≠ \ne = \ne-
tag
latex
descript
← \gets ← \gets-
→ \to → \to-
⟵ \longleftarrow ⟵ \longleftarrow-
⟶ \longrightarrow ⟶ \longrightarrow-
↑ \uparrow ↑ \uparrow-
↓ \downarrow ↓ \downarrow-
↕ \updownarrow ↕ \updownarrow-
↔ \leftrightarrow ↔ \leftrightarrow-
⇑ \Uparrow ⇑ \Uparrow-
⇓ \Downarrow ⇓ \Downarrow-
⇕ \Updownarrow ⇕ \Updownarrow-
⟷ \longleftrightarrow ⟷ \longleftrightarrow-
⇐ \Leftarrow ⇐ \Leftarrow-
⟸ \Longleftarrow ⟸ \Longleftarrow-
⇒ \Rightarrow ⇒ \Rightarrow-
⟹ \Longrightarrow ⟹ \Longrightarrow-
⇔ \Leftrightarrow ⇔ \Leftrightarrow-
⟺ \Longleftrightarrow ⟺ \Longleftrightarrow-
↦ \mapsto ↦ \mapsto-
⟼ \longmapsto ⟼ \longmapsto-
↗ \nearrow ↗ \nearrow-
↘ \searrow ↘ \searrow-
↙ \swarrow ↙ \swarrow-
↖ \nwarrow ↖ \nwarrow-
↩ \hookleftarrow ↩ \hookleftarrow-
↪ \hookrightarrow ↪ \hookrightarrow-
⇌ \rightleftharpoons ⇌ \rightleftharpoons-
⟺ \iff ⟺ \iff-
↼ \leftharpoonup ↼ \leftharpoonup-
⇀ \rightharpoonup ⇀ \rightharpoonup-
↽ \leftharpoondown ↽ \leftharpoondown-
⇁ \rightharpoondown ⇁ \rightharpoondown-
tag
latex
descript
∵ \because ∵ \because-
∴ \therefore ∴ \therefore-
… \dots … \dots-
⋯ \cdots ⋯ \cdots-
⋮ \vdots ⋮ \vdots-
⋱ \ddots ⋱ \ddots-
∀ \forall ∀ \forall-
∃ \exists ∃ \exists-
∄ \nexists ∄ \nexists-
Ⅎ \Finv Ⅎ \Finv-
¬ \neg ¬ \neg-
′ \prime ′ \prime-
∅ \emptyset ∅ \emptyset-
∞ \infty ∞ \infty-
∇ \nabla ∇ \nabla-
△ \triangle △ \triangle-
□ \Box □ \Box-
◊ \Diamond ◊ \Diamond-
⊥ \bot ⊥ \bot-
⊤ \top ⊤ \top-
∠ \angle ∠ \angle-
∡ \measuredangle ∡ \measuredangle-
∢ \sphericalangle ∢ \sphericalangle-
√ \surd √ \surd-
♢ \diamondsuit ♢ \diamondsuit-
♡ \heartsuit ♡ \heartsuit-
♣ \clubsuit ♣ \clubsuit-
♠ \spadesuit ♠ \spadesuit-
♭ \flat ♭ \flat-
♮ \natural ♮ \natural-
♯ \sharp ♯ \sharp-
tag
latex
descript
α \alpha α \alphaalpha
β \beta β \betabeta
γ \gamma γ \gammagamma
δ \delta δ \deltadelta
ϵ \epsilon ϵ \epsilonepsilon
ε \varepsilon ε \varepsilonepsilon
ζ \zeta ζ \zetazeta
η \eta η \etaeta
θ \theta θ \thetatheta
ϑ \vartheta ϑ \varthetatheta
ι \iota ι \iotaiota
κ \kappa κ \kappakappa
λ \lambda λ \lambdalambda
μ \mu μ \mumu
ν \nu ν \nunu
ξ \xi ξ \xixi
o o o oomicron
π \pi π \pipi
ϖ \varpi ϖ \varpipi
ρ \rho ρ \rhorho
ϱ \varrho ϱ \varrhorho
σ \sigma σ \sigmasigma
ς \varsigma ς \varsigmasigma
τ \tau τ \tautau
υ \upsilon υ \upsilonupsilon
ϕ \phi ϕ \phiphi
φ \varphi φ \varphiphi
χ \chi χ \chichi
ψ \psi ψ \psipsi
ω \omega ω \omegaomega
Γ \Gamma Γ \GammaGamma
Δ \Delta Δ \DeltaDelta
Θ \Theta Θ \ThetaTheta
Λ \Lambda Λ \LambdaLambda
Ξ \Xi Ξ \XiXi
Π \Pi Π \PiPi
Σ \Sigma Σ \SigmaSigma
Υ \Upsilon Υ \UpsilonUpsilon
Φ \Phi Φ \PhiPhi
Ψ \Psi Ψ \PsiPsi
Ω \Omega Ω \OmegaOmega
tag
latex
descript
ℏ \hbar ℏ \hbarh bar
ı \imath ı \imathimath
ȷ \jmath ȷ \jmathjmath
ℓ \ell ℓ \elllmath
ℜ \Re ℜ \ReReal Numbers
ℑ \Im ℑ \ImPure Imaginary Numbers
ℵ \aleph ℵ \alephaleph
ℶ \beth ℶ \bethbeth
ℷ \gimel ℷ \gimelgimel
ℸ \daleth ℸ \dalethdaleth
℘ \wp ℘ \wp-
℧ \mho ℧ \mho-
∍ \backepsilon ∍ \backepsilonbackepsilon
∂ \partial ∂ \partial-
ð \eth ð \eth-
k \Bbbk k \Bbbk-
∁ \complement ∁ \complementcomplement
Ⓢ \circledS Ⓢ \circledScircled S
\S
\Ssections
A B C \mathbb{ABC} A B C \mathbb{ABC}Blackboard bold/scripts
A B C \mathfrak{ABC} A B C \mathfrak{ABC}Fraktur typeface
A B C \mathcal{ABC} A B C \mathcal{ABC}Calligraphy/script
A B C \mathrm {ABC} A B C \mathrm {ABC}Roman typeface
d e f \mathrm{def} d e f \mathrm{def}def
tag
latex
descript
A B C A B C \frac{ABC}{ABC} A B C A B C \frac{ABC}{ABC}分数
A B C A B C \tfrac{ABC}{ABC} A B C A B C \tfrac{ABC}{ABC}小分数
d t \mathrm{d}t d t \mathrm{d}t微分
d y d x \frac{\mathrm{d} y}{\mathrm{d} x} d x d y \frac{\mathrm{d} y}{\mathrm{d} x}微分
∂ t \partial t ∂ t \partial t偏微分
∂ y ∂ x \frac{\partial y}{\partial x} ∂ x ∂ y \frac{\partial y}{\partial x}偏微分
∇ ψ \nabla\psi ∇ ψ \nabla\psiNabla算子
∂ 2 ∂ x 1 ∂ x 2 y \frac{\partial^2}{\partial x_1\partial x_2}y ∂ x 1 ∂ x 2 ∂ 2 y \frac{\partial^2}{\partial x_1\partial x_2}y偏微分
1 a + 7 b + 2 9 = c \cfrac{1}{a + \cfrac{7}{b + \cfrac{2}{9}}} =c a + b + 9 2 7 1 = c \cfrac{1}{a + \cfrac{7}{b + \cfrac{2}{9}}} =c连分数
\begin{equation} x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{a_4} } } } \end{equation}
\begin{equation} x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{a_4} } } } \end{equation}连分数
tag
latex
descript
A B C ˙ \dot{ABC} A B C ˙ \dot{ABC}一阶导数
A B C ¨ \ddot{ABC} A B C ¨ \ddot{ABC}二阶导数
A B C ′ {ABC}' A B C ′ {ABC}'一阶导数
A B C ′ ′ {ABC}'' A B C ′ ′ {ABC}''二阶导数
A B C ( n ) {ABC}^{(n)} A B C ( n ) {ABC}^{(n)}n阶导数
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a m o d b a \bmod b a m o d b a \bmod b模除
a ≡ b ( m o d m ) a \equiv b \pmod{m} a ≡ b ( m o d m ) a \equiv b \pmod{m}同余
gcd ( m , n ) \gcd(m, n) g cd( m , n ) \gcd(m, n)最大公约数
lcm ( m , n ) \operatorname{lcm}(m, n) l c m ( m , n ) \operatorname{lcm}(m, n)最小公倍数
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A B C \sqrt{ABC} A B C \sqrt{ABC}开平方
A B C \sqrt[]{ABC} A B C \sqrt[]{ABC}开方
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A B C ^{ABC} A B C ^{ABC}上标
A B C _{ABC} A B C _{ABC}下标
A B C A B C _{ABC}^{ABC} A B C A B C _{ABC}^{ABC}混合上下标
A B C A B C _{ABC}^{ABC} A B C A B C _{ABC}^{ABC}左侧混合上下标
\sideset{_1^2}{_3^4}X_a^b
\sideset{_1^2}{_3^4}X_a^b混合
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A B C ^ \hat{ABC} A B C ^ \hat{ABC}-
A B C ˇ \check{ABC} A B C ˇ \check{ABC}-
A B C ˋ \grave{ABC} A B C ˋ \grave{ABC}-
A B C ˊ \acute{ABC} A B C ˊ \acute{ABC}-
A B C ~ \tilde{ABC} A B C ~ \tilde{ABC}-
A B C ˘ \breve{ABC} A B C ˘ \breve{ABC}-
A B C ˉ \bar{ABC} A B C ˉ \bar{ABC}-
A B C ⃗ \vec{ABC} A B C \vec{ABC}-
̸ A B C \not{ABC} A B C \not{ABC}-
∘ ^{\circ} ∘ ^{\circ}-
A B C ~ \widetilde{ABC} A B C \widetilde{ABC}-
A B C ^ \widehat{ABC} A B C \widehat{ABC}-
A B C ← \overleftarrow{ABC} A B C \overleftarrow{ABC}-
A B C → \overrightarrow{ABC} A B C \overrightarrow{ABC}-
A B C ‾ \overline{ABC} A B C \overline{ABC}-
A B C ‾ \underline{ABC} A B C \underline{ABC}-
A B C ⏞ \overbrace{ABC} A B C \overbrace{ABC}-
A B C ⏟ \underbrace{ABC} A B C \underbrace{ABC}-
A B C A B C \overset{ABC}{ABC} A BC A B C \overset{ABC}{ABC}-
A B C A B C \underset{ABC}{ABC} A B C A BC \underset{ABC}{ABC}-
A B C ⌢ \stackrel\frown{ABC} A BC ⌢ \stackrel\frown{ABC}-
A B C ‾ \overline{ABC} A B C \overline{ABC}-
A B C ↔ \overleftrightarrow{ABC} A B C \overleftrightarrow{ABC}-
← A B C \overset{ABC}{\leftarrow} ← A B C \overset{ABC}{\leftarrow}-
→ A B C \overset{ABC}{\rightarrow} → A B C \overset{ABC}{\rightarrow}-
← A B C \xleftarrow[]{ABC} A B C \xleftarrow[]{ABC}-
→ A B C \xrightarrow[]{ABC} A B C \xrightarrow[]{ABC}-
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lim \lim lim \lim极限
lim x → 0 \lim_{x \to 0} lim x → 0 \lim_{x \to 0}极限
lim x → ∞ \lim_{x \to \infty} lim x → ∞ \lim_{x \to \infty}极限
lim x → 0 \textstyle \lim_{x \to 0} lim x → 0 \textstyle \lim_{x \to 0}极限
max A B C \max_{ABC} max A B C \max_{ABC}极大
min A B C \min_{ABC} min A B C \min_{ABC}极小
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log A B C A B C \log_{ABC}{ABC} log A B C A B C \log_{ABC}{ABC}对数
lg A B C A B C \lg_{ABC}{ABC} lg A B C A B C \lg_{ABC}{ABC}常用对数
ln A B C A B C \ln_{ABC}{ABC} ln A B C A B C \ln_{ABC}{ABC}自然对数
exp \exp exp \exp指数
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min x \min x min x \min x最小
max y \max y max y \max y最大
sup t \sup t sup t \sup t最小上界（上确界）
inf s \inf s inf s \inf s最大下界（下确界）
lim u \lim u lim u \lim u极限
lim sup w \limsup w l i m s u p w \limsup w上极限
lim inf v \liminf v l i m i n f v \liminf v下极限
dim p \dim p dim p \dim p维数
ker ϕ \ker\phi ker ϕ \ker\phi零空间（核）
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sin \sin sin \sin正弦
cos \cos cos \cos余弦
tan \tan tan \tan正切
cot \cot cot \cot余切
sec \sec sec \sec正割
csc \csc csc \csc余割
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sin − 1 \sin^{-1} sin − 1 \sin^{-1}反正弦
cos − 1 \cos^{-1} cos − 1 \cos^{-1}反余弦
tan − 1 \tan^{-1} tan − 1 \tan^{-1}反正切
cot − 1 \cot^{-1} cot − 1 \cot^{-1}反余切
sec − 1 \sec^{-1} sec − 1 \sec^{-1}反正割
csc − 1 \csc^{-1} csc − 1 \csc^{-1}反余割
arcsin \arcsin arcsin \arcsin反正弦
arccos \arccos arccos \arccos反余弦
arctan \arctan arctan \arctan反正切
arccot \operatorname{arccot} a r c c o t \operatorname{arccot}反余切
arcsec \operatorname{arcsec} a r c s e c \operatorname{arcsec}反正割
arccos \operatorname{arccos} a r c c o s \operatorname{arccos}反余割
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sinh \sinh sinh \sinh双曲正弦
cosh \cosh cosh \cosh双曲余弦
tanh \tanh tanh \tanh双曲正切
coth \coth coth \coth双曲余切
sech \operatorname{sech} s e c h \operatorname{sech}双曲正割
csch \operatorname{csch} c s c h \operatorname{csch}双曲余割
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sinh − 1 \sinh^{-1} sinh − 1 \sinh^{-1}反双曲正弦
cosh − 1 \cosh^{-1} cosh − 1 \cosh^{-1}反双曲余弦
tanh − 1 \tanh^{-1} tanh − 1 \tanh^{-1}反双曲正切
coth − 1 \coth^{-1} coth − 1 \coth^{-1}反双曲余切
sech − 1 \operatorname{sech}^{-1} s e c h − 1 \operatorname{sech}^{-1}反双曲正割
csch − 1 \operatorname{csch}^{-1} c s c h − 1 \operatorname{csch}^{-1}反双曲余割
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∫ \int ∫ \int积分
∫ A B C A B C \int_{ABC}^{ABC} ∫ A B C A B C \int_{ABC}^{ABC}积分
∫ A B C A B C \int\limits_{ABC}^{ABC} A B C ∫ A B C \int\limits_{ABC}^{ABC}积分
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∬ \iint ∬ \iint双重积分
∬ A B C A B C \iint_{ABC}^{ABC} ∬ A B C A B C \iint_{ABC}^{ABC}双重积分
∬ A B C A B C \iint\limits_{ABC}^{ABC} A B C ∬ A B C \iint\limits_{ABC}^{ABC}双重积分
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∭ \iiint ∭ \iiint三重积分
∭ A B C A B C \iiint_{ABC}^{ABC} ∭ A B C A B C \iiint_{ABC}^{ABC}三重积分
∭ A B C A B C \iiint\limits_{ABC}^{ABC} A B C ∭ A B C \iiint\limits_{ABC}^{ABC}三重积分
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∮ \oint ∮ \oint曲线积分
∮ A B C A B C \oint_{ABC}^{ABC} ∮ A B C A B C \oint_{ABC}^{ABC}曲线积分
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∑ \sum ∑ \sum求和
∑ A B C A B C \sum_{ABC}^{ABC} ∑ A B C A B C \sum_{ABC}^{ABC}求和
∑ A B C A B C {\textstyle \sum_{ABC}^{ABC}} ∑ A B C A B C {\textstyle \sum_{ABC}^{ABC}}求和
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∏ \prod ∏ \prod连乘积
∏ A B C A B C \prod_{ABC}^{ABC} ∏ A B C A B C \prod_{ABC}^{ABC}连乘积
∏ A B C A B C {\textstyle \prod_{ABC}^{ABC}} ∏ A B C A B C {\textstyle \prod_{ABC}^{ABC}}连乘积
∐ \coprod ∐ \coprod余积
∐ A B C A B C \coprod_{ABC}^{ABC} ∐ A B C A B C \coprod_{ABC}^{ABC}余积
∐ A B C A B C {\textstyle \coprod_{ABC}^{ABC}} ∐ A B C A B C {\textstyle \coprod_{ABC}^{ABC}}余积
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⋃ \bigcup ⋃ \bigcup并集
⋃ A B C A B C \bigcup_{ABC}^{ABC} ⋃ A B C A B C \bigcup_{ABC}^{ABC}并集
⋃ A B C A B C {\textstyle \bigcup_{ABC}^{ABC}} ⋃ A B C A B C {\textstyle \bigcup_{ABC}^{ABC}}并集
⋂ \bigcap ⋂ \bigcap交集
⋂ A B C A B C \bigcap_{ABC}^{ABC} ⋂ A B C A B C \bigcap_{ABC}^{ABC}交集
⋂ A B C A B C {\textstyle \bigcap_{ABC}^{ABC}} ⋂ A B C A B C {\textstyle \bigcap_{ABC}^{ABC}}交集
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⋁ \bigvee ⋁ \bigvee析取
⋁ A B C A B C \bigvee_{ABC}^{ABC} ⋁ A B C A B C \bigvee_{ABC}^{ABC}析取
⋁ A B C A B C {\textstyle \bigvee_{ABC}^{ABC}} ⋁ A B C A B C {\textstyle \bigvee_{ABC}^{ABC}}析取
⋀ \bigwedge ⋀ \bigwedge合取
⋀ A B C A B C \bigwedge_{ABC}^{ABC} ⋀ A B C A B C \bigwedge_{ABC}^{ABC}合取
⋀ A B C A B C {\textstyle \bigwedge_{ABC}^{ABC}} ⋀ A B C A B C {\textstyle \bigwedge_{ABC}^{ABC}}合取
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( ) \left ( \right ) ( ) \left ( \right )圆括号
[ ] \left [ \right ] [ ] \left [ \right ]方括号
⟨ ⟩ \left \langle \right \rangle ⟨ ⟩ \left \langle \right \rangle角括号
{ } \left \{ \right \} { } \left \{ \right \}花括号
$\left
\right
$
∥ ∥ \left \| \right \| ∥ ∥ \left \| \right \|双竖线，范
⌊ ⌋ \left \lfloor \right \rfloor ⌊ ⌋ \left \lfloor \right \rfloor取整函数
⌈ ⌉ \left \lceil \right \rceil ⌈ ⌉ \left \lceil \right \rceil取顶函数
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( A B C A B C ) \binom{ABC}{ABC} ( A B C A B C ) \binom{ABC}{ABC}二项式系数
[ 0 , 1 ) \left [ 0,1 \right ) [ 0 , 1 ) \left [ 0,1 \right )开闭区间
$\left\langle\psi\right
$
\left\langle\psi\right|
$\left
\psi \right \rangle$
\left | \psi \right \rangle
$\left \langle \psi
\psi \right \rangle$
\left \langle \psi | \psi \right \rangle
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… … \begin{matrix}…&…\end{matrix} … … \begin{matrix}…&…\end{matrix}矩阵
[ … … ] \begin{bmatrix}…&…\end{bmatrix} [ … … ] \begin{bmatrix}…&…\end{bmatrix}方括号矩阵
( … … ) \begin{pmatrix}…&…\end{pmatrix} ( … … ) \begin{pmatrix}…&…\end{pmatrix}圆括号矩阵
∣ … … ∣ \begin{vmatrix}…&…\end{vmatrix} ∣ ∣ … … ∣ ∣ \begin{vmatrix}…&…\end{vmatrix}单竖线矩阵
∥ … … ∥ \begin{Vmatrix}…&…\end{Vmatrix} ∥ ∥ … … ∥ ∥ \begin{Vmatrix}…&…\end{Vmatrix}双竖线矩阵
{ … … } \begin{Bmatrix}…&…\end{Bmatrix} { … … } \begin{Bmatrix}…&…\end{Bmatrix}花括号矩阵
{ … … \left\{\begin{matrix}…&…\end{matrix}\right. { … … \left\{\begin{matrix}…&…\end{matrix}\right.左单括号矩阵
… … } \left.\begin{matrix}…&…\end{matrix}\right\} … … } \left.\begin{matrix}…&…\end{matrix}\right\}右单括号矩阵
{ … if x = … \begin{cases}…& \text{ if } x=…\end{cases} { … if x = … \begin{cases}…& \text{ if } x=…\end{cases}条件等式
\begin{align*}…&…\end{align*}
\begin{align*}…&…\end{align*}多行对齐等式
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$$\left(x-1\right)\left(x+3\right)$$
\left(x-1\right)\left(x+3\right)algebra_1
$$\sqrt{a2+b 2}$$
\sqrt{a^2+b^2}algebra_2
$$\left ( \frac{a}{b}\right )^{n}= \frac{a{n}}{b {n}}$$
\left ( \frac{a}{b}\right )^{n}= \frac{a^{n}}{b^{n}}algebra_3
$$\frac{a}{b}\pm \frac{c}{d}= \frac{ad \pm bc}{bd}$$
\frac{a}{b}\pm \frac{c}{d}= \frac{ad \pm bc}{bd}algebra_4
$$\frac{x{2}}{a {2}}-\frac{y{2}}{b {2}}=1$$
\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1algebra_5
$$\frac{1}{\sqrt{a}}=\frac{\sqrt{a}}{a},a\ge 0\frac{1}{\sqrt{a}}=\frac{\sqrt{a}}{a},a\ge 0$$
\frac{1}{\sqrt{a}}=\frac{\sqrt{a}}{a},a\ge 0\frac{1}{\sqrt{a}}=\frac{\sqrt{a}}{a},a\ge 0algebra_6
$$\sqrt[n]{a^{n}}=\left ( \sqrt[n]{a}\right )^{n}$$
\sqrt[n]{a^{n}}=\left ( \sqrt[n]{a}\right )^{n}algebra_7
$$x ={-b \pm \sqrt{b^2-4ac}\over 2a}$$
x ={-b \pm \sqrt{b^2-4ac}\over 2a}algebra_8
$$y-y_{1}=k \left( x-x_{1}\right)$$
y-y_{1}=k \left( x-x_{1}\right)algebra_9
$$\left{\begin{matrix} x=a + r\text{cos}\theta \ y=b + r\text{sin}\theta \end{matrix}\right.$$
\left\{\begin{matrix} x=a + r\text{cos}\theta \\ y=b + r\text{sin}\theta \end{matrix}\right.algebra_10
$$\begin{array}{l} \text{对于方程形如：}x^{3}-1=0 \ \text{设}\text{:}\omega =\frac{-1+\sqrt{3}i}{2} \ x_{1}=1,x_{2}= \omega =\frac{-1+\sqrt{3}i}{2} \ x_{3}= \omega ^{2}=\frac{-1-\sqrt{3}i}{2} \end{array}$$
\begin{array}{l} \text{对于方程形如：}x^{3}-1=0 \\ \text{设}\text{:}\omega =\frac{-1+\sqrt{3}i}{2} \\ x_{1}=1,x_{2}= \omega =\frac{-1+\sqrt{3}i}{2} \\ x_{3}= \omega ^{2}=\frac{-1-\sqrt{3}i}{2} \end{array}algebra_11
$$\begin{array}{l} a\mathop\nolimits^2+bx+c=0 \ \Delta =\mathop\nolimits^2-4ac \ \left{\begin{matrix} \Delta \gt 0\text{方程有两个不相等的实根} \ \Delta = 0\text{方程有两个不相等的实根} \ \Delta \lt 0\text{方程有两个不相等的实根} \end{matrix}\right. \end{array}$$
\begin{array}{l} a\mathop{{x}}\nolimits^{{2}}+bx+c=0 \\ \Delta =\mathop{{b}}\nolimits^{{2}}-4ac \\ \left\{\begin{matrix} \Delta \gt 0\text{方程有两个不相等的实根} \\ \Delta = 0\text{方程有两个不相等的实根} \\ \Delta \lt 0\text{方程有两个不相等的实根} \end{matrix}\right. \end{array}algebra_12
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$$\Delta A B C$$
\Delta A B Cgeometry_1
$$a \parallel c,b \parallel c \Rightarrow a \parallel b$$
a \parallel c,b \parallel c \Rightarrow a \parallel bgeometry_2
$$l \perp \beta ,l \subset \alpha \Rightarrow \alpha \perp \beta$$
l \perp \beta ,l \subset \alpha \Rightarrow \alpha \perp \betageometry_3
$$\left.\begin{matrix} a \perp \alpha \ b \perp \alpha \end{matrix}\right}\Rightarrow a \parallel b$$
\left.\begin{matrix} a \perp \alpha \\ b \perp \alpha \end{matrix}\right\}\Rightarrow a \parallel bgeometry_4
$$P \in \alpha ,P \in \beta , \alpha \cap \beta =l \Rightarrow P \in l$$
P \in \alpha ,P \in \beta , \alpha \cap \beta =l \Rightarrow P \in lgeometry_5
$$\alpha \perp \beta , \alpha \cap \beta =l,a \subset \alpha ,a \perp l \Rightarrow a \perp \beta$$
\alpha \perp \beta , \alpha \cap \beta =l,a \subset \alpha ,a \perp l \Rightarrow a \perp \betageometry_6
$$\left.\begin{matrix} a \subset \beta ,b \subset \beta ,a \cap b=P \ a \parallel \partial ,b \parallel \partial \end{matrix}\right}\Rightarrow \beta \parallel \alpha$$
\left.\begin{matrix} a \subset \beta ,b \subset \beta ,a \cap b=P \\ a \parallel \partial ,b \parallel \partial \end{matrix}\right\}\Rightarrow \beta \parallel \alphageometry_7
$$\alpha \parallel \beta , \gamma \cap \alpha =a, \gamma \cap \beta =b \Rightarrow a \parallel b$$
\alpha \parallel \beta , \gamma \cap \alpha =a, \gamma \cap \beta =b \Rightarrow a \parallel bgeometry_8
$$A \in l,B \in l,A \in \alpha ,B \in \alpha \Rightarrow l \subset \alpha$$
A \in l,B \in l,A \in \alpha ,B \in \alpha \Rightarrow l \subset \alphageometry_9
$$\left.\begin{matrix} m \subset \alpha ,n \subset \alpha ,m \cap n=P \ a \perp m,a \perp n \end{matrix}\right}\Rightarrow a \perp \alpha$$
\left.\begin{matrix} m \subset \alpha ,n \subset \alpha ,m \cap n=P \\ a \perp m,a \perp n \end{matrix}\right\}\Rightarrow a \perp \alphageometry_10
$$\begin{array}{c} \text{直角三角形中,直角边长a,b,斜边边长c} \ a{2}+b {2}=c^{2} \end{array}$$
\begin{array}{c} \text{直角三角形中,直角边长a,b,斜边边长c} \\ a^{2}+b^{2}=c^{2} \end{array}geometry_11
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$$a > b,b > c \Rightarrow a > c$$
a > b,b > c \Rightarrow a > cinequality_1
$$a > b,c > d \Rightarrow a+c > b+d$$
a > b,c > d \Rightarrow a+c > b+dinequality_2
$$a > b > 0,c > d > 0 \Rightarrow ac bd$$
a > b > 0,c > d > 0 \Rightarrow ac bdinequality_3
$$\begin{array}{c} a \gt b,c \gt 0 \Rightarrow ac \gt bc \ a \gt b,c \lt 0 \Rightarrow ac \lt bc \end{array}$$
\begin{array}{c} a \gt b,c \gt 0 \Rightarrow ac \gt bc \\ a \gt b,c \lt 0 \Rightarrow ac \lt bc \end{array}inequality_4
$$\left
a-b \right
\geqslant \left
$$-\left
a \right
\leq a\leqslant \left
$$\left
a \right
\leqslant b \Rightarrow -b \leqslant a \leqslant \left
$$\left
a+b \right
\leqslant \left
$$\begin{array}{c} a \gt b \gt 0,n \in N^{\ast},n \gt 1 \ \Rightarrow a^{n}\gt b^{n}, \sqrt[n]{a}\gt \sqrt[n]{b} \end{array}$$
\begin{array}{c} a \gt b \gt 0,n \in N^{\ast},n \gt 1 \\ \Rightarrow a^{n}\gt b^{n}, \sqrt[n]{a}\gt \sqrt[n]{b} \end{array}inequality_9
$$\left( \sum_{k=1}^n a_k b_k \right)^{!!2}\leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right)$$
\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)inequality_10
$$\begin{array}{c} a,b \in R^{+} \ \Rightarrow \frac{a+b}2\ge \sqrt{ab} \ \left( \text{当且仅当}a=b\text{时取“}=\text{”号}\right) \end{array}$$
\begin{array}{c} a,b \in R^{+} \\ \Rightarrow \frac{a+b}{{2}}\ge \sqrt{ab} \\ \left( \text{当且仅当}a=b\text{时取“}=\text{”号}\right) \end{array}inequality_11
$$\begin{array}{c} a,b \in R \ \Rightarrow a{2}+b {2}\gt 2ab \ \left( \text{当且仅当}a=b\text{时取“}=\text{”号}\right) \end{array}$$
\begin{array}{c} a,b \in R \\ \Rightarrow a^{2}+b^{2}\gt 2ab \\ \left( \text{当且仅当}a=b\text{时取“}=\text{”号}\right) \end{array}inequality_12
$$\begin{array}{c} H_{n}=\frac{n}{\sum \limits_{i=1}^{n}\frac{1}{x_{i}}}= \frac{n}{\frac{1}{x_{1}}+ \frac{1}{x_{2}}+ \cdots + \frac{1}{x_{n}}} \ G_{n}=\sqrt[n]{\prod \limits_{i=1}^{n}x_{i}}= \sqrt[n]{x_{1}x_{2}\cdots x_{n}} \ A_{n}=\frac{1}{n}\sum \limits_{i=1}^{n}x_{i}=\frac{x_{1}+ x_{2}+ \cdots + x_{n}}{n} \ Q_{n}=\sqrt{\sum \limits_{i=1}{n}x_{i} {2}}= \sqrt{\frac{x_{1}^{2}+ x_{2}^{2}+ \cdots + x_{n}^{2}}{n}} \ H_{n}\leq G_{n}\leq A_{n}\leq Q_{n} \end{array}$$
\begin{array}{c} H_{n}=\frac{n}{\sum \limits_{i=1}^{n}\frac{1}{x_{i}}}= \frac{n}{\frac{1}{x_{1}}+ \frac{1}{x_{2}}+ \cdots + \frac{1}{x_{n}}} \\ G_{n}=\sqrt[n]{\prod \limits_{i=1}^{n}x_{i}}= \sqrt[n]{x_{1}x_{2}\cdots x_{n}} \\ A_{n}=\frac{1}{n}\sum \limits_{i=1}^{n}x_{i}=\frac{x_{1}+ x_{2}+ \cdots + x_{n}}{n} \\ Q_{n}=\sqrt{\sum \limits_{i=1}^{n}x_{i}^{2}}= \sqrt{\frac{x_{1}^{2}+ x_{2}^{2}+ \cdots + x_{n}^{2}}{n}} \\ H_{n}\leq G_{n}\leq A_{n}\leq Q_{n} \end{array}inequality_13
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$$\frac{\mathrm{d}}{\mathrm{d}x}xn=nx {n-1}$$
\frac{\mathrm{d}}{\mathrm{d}x}x^n=nx^{n-1}calculous_1
$$\frac{\mathrm{d}}{\mathrm{d}x}e{ax}=a,e {ax}$$
\frac{\mathrm{d}}{\mathrm{d}x}e^{ax}=a\,e^{ax}calculous_2
$$\frac{\mathrm{d}}{\mathrm{d}x}\ln(x)=\frac{1}{x}$$
\frac{\mathrm{d}}{\mathrm{d}x}\ln(x)=\frac{1}{x}calculous_3
$$\frac{\mathrm{d}}{\mathrm{d}x}\sin x=\cos x$$
\frac{\mathrm{d}}{\mathrm{d}x}\sin x=\cos xcalculous_4
$$\frac{\mathrm{d}}{\mathrm{d}x}\cos x=-\sin x$$
\frac{\mathrm{d}}{\mathrm{d}x}\cos x=-\sin xcalculous_5
$$\int k\mathrm{d}x = kx+C$$
\int k\mathrm{d}x = kx+Ccalculous_6
$$\frac{\mathrm{d}}{\mathrm{d}x}\tan x=\sec^2 x$$
\frac{\mathrm{d}}{\mathrm{d}x}\tan x=\sec^2 xcalculous_7
$$\frac{\mathrm{d}}{\mathrm{d}x}\cot x=-\csc^2 x$$
\frac{\mathrm{d}}{\mathrm{d}x}\cot x=-\csc^2 xcalculous_8
$$\int \frac{1}{x}\mathrm{d}x= \ln \left
x \right
+C$$
$$\int \frac{1}{\sqrt{1-x^{2}}}\mathrm{d}x= \arcsin x +C$$
\int \frac{1}{\sqrt{1-x^{2}}}\mathrm{d}x= \arcsin x +Ccalculous_10
$$\int \frac{1}{1+x^{2}}\mathrm{d}x= \arctan x +C$$
\int \frac{1}{1+x^{2}}\mathrm{d}x= \arctan x +Ccalculous_11
$$\int u \frac{\mathrm{d}v}{\mathrm{d}x},\mathrm{d}x=uv-\int \frac{\mathrm{d}u}{\mathrm{d}x}v,\mathrm{d}x$$
\int u \frac{\mathrm{d}v}{\mathrm{d}x}\,\mathrm{d}x=uv-\int \frac{\mathrm{d}u}{\mathrm{d}x}v\,\mathrm{d}xcalculous_12
$$f(x) = \int_{-\infty}^\infty \hat f(x)\xi,e^{2 \pi i \xi x} ,\mathrm{d}\xi$$
f(x) = \int_{-\infty}^\infty \hat f(x)\xi\,e^{2 \pi i \xi x} \,\mathrm{d}\xicalculous_13
$$\int x{\mu}\mathrm{d}x=\frac{x {\mu +1}}{\mu +1}+C, \left({\mu \neq -1}\right)$$
\int x^{\mu}\mathrm{d}x=\frac{x^{\mu +1}}{\mu +1}+C, \left({\mu \neq -1}\right)calculous_14
tag
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$$\begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}$$
\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}array_1
$$\begin{pmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{pmatrix}$$
\begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}array_2
$$\begin{pmatrix} a_{11} & \cdots & a_{1n} \ \vdots & \ddots & \vdots \ a_{m1} & \cdots & a_{mn} \end{pmatrix}$$
\begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & a_{mn} \end{pmatrix}array_3
$$\begin{array}{c} A=A^{T} \ A=-A^{T} \end{array}$$
\begin{array}{c} A=A^{T} \\ A=-A^{T} \end{array}array_4
$$O = \begin{bmatrix} 0 & 0 & \cdots & 0 \ 0 & 0 & \cdots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \cdots & 0 \end{bmatrix}$$
O = \begin{bmatrix} 0 & 0 & \cdots & 0 \\ 0 & 0 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 0 \end{bmatrix}array_5
$$A_{m\times n}= \begin{bmatrix} a_{11}& a_{12}& \cdots & a_{1n} \ a_{21}& a_{22}& \cdots & a_{2n} \ \vdots & \vdots & \ddots & \vdots \ a_{m1}& a_{m2}& \cdots & a_{mn} \end{bmatrix} =\left [ a_{ij}\right ]$$
A_{m\times n}= \begin{bmatrix} a_{11}& a_{12}& \cdots & a_{1n} \\ a_{21}& a_{22}& \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1}& a_{m2}& \cdots & a_{mn} \end{bmatrix} =\left [ a_{ij}\right ]array_6
$$\begin{array}{c} A={\left[ a_{ij}\right]{m \times n}},B={\left[ b {ij}\right]{n \times s}} \ c {ij}= \sum \limits_{k=1}^a_{ik}b_{kj} \ C=AB=\left[ c_{ij}\right]{m \times s} = \left[ \sum \limits {k=1}^{n}a_{ik}b_{kj}\right]_{m \times s} \end{array}$$
\begin{array}{c} A={\left[ a_{ij}\right]_{m \times n}},B={\left[ b_{ij}\right]_{n \times s}} \\ c_{ij}= \sum \limits_{k=1}^{{n}}a_{ik}b_{kj} \\ C=AB=\left[ c_{ij}\right]_{m \times s} = \left[ \sum \limits_{k=1}^{n}a_{ik}b_{kj}\right]_{m \times s} \end{array}array_7
$$\mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix} \mathbf{i}& \mathbf{j}& \mathbf{k} \ \frac{\partial X}{\partial u}& \frac{\partial Y}{\partial u}& 0 \ \frac{\partial X}{\partial v}& \frac{\partial Y}{\partial v}& 0 \ \end{vmatrix}$$
\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}array_8
tag
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$$e^{i \theta}$$
e^{i \theta}trigonometry_1
$$\left(\frac{\pi}{2}-\theta \right )$$
\left(\frac{\pi}{2}-\theta \right )trigonometry_2
$$\text{sin}^{2}\frac{\alpha}{2}=\frac{1- \text{cos}\alpha}{2}$$
\text{sin}^{2}\frac{\alpha}{2}=\frac{1- \text{cos}\alpha}{2}trigonometry_3
$$\text{cos}^{2}\frac{\alpha}{2}=\frac{1+ \text{cos}\alpha}{2}$$
\text{cos}^{2}\frac{\alpha}{2}=\frac{1+ \text{cos}\alpha}{2}trigonometry_4
$$\text{tan}\frac{\alpha}{2}=\frac{\text{sin}\alpha}{1+ \text{cos}\alpha}$$
\text{tan}\frac{\alpha}{2}=\frac{\text{sin}\alpha}{1+ \text{cos}\alpha}trigonometry_5
$$\sin \alpha + \sin \beta =2 \sin \frac{\alpha + \beta}{2}\cos \frac{\alpha - \beta}{2}$$
\sin \alpha + \sin \beta =2 \sin \frac{\alpha + \beta}{2}\cos \frac{\alpha - \beta}{2}trigonometry_6
$$\sin \alpha - \sin \beta =2 \cos \frac{\alpha + \beta}{2}\sin \frac{\alpha - \beta}{2}$$
\sin \alpha - \sin \beta =2 \cos \frac{\alpha + \beta}{2}\sin \frac{\alpha - \beta}{2}trigonometry_7
$$\cos \alpha + \cos \beta =2 \cos \frac{\alpha + \beta}{2}\cos \frac{\alpha - \beta}{2}$$
\cos \alpha + \cos \beta =2 \cos \frac{\alpha + \beta}{2}\cos \frac{\alpha - \beta}{2}trigonometry_8
$$\cos \alpha - \cos \beta =-2\sin \frac{\alpha + \beta}{2}\sin \frac{\alpha - \beta}{2}$$
\cos \alpha - \cos \beta =-2\sin \frac{\alpha + \beta}{2}\sin \frac{\alpha - \beta}{2}trigonometry_9
$$a{2}=b {2}+c^{2}-2bc\cos A$$
a^{2}=b^{2}+c^{2}-2bc\cos Atrigonometry_10
$$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}=\frac{1}{2R}$$
\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}=\frac{1}{2R}trigonometry_11
$$\sin \left ( \frac{\pi}{2}-\alpha \right ) = \cos \alpha$$
\sin \left ( \frac{\pi}{2}-\alpha \right ) = \cos \alphatrigonometry_12
$$\sin \left ( \frac{\pi}{2}+\alpha \right ) = \cos \alpha$$
\sin \left ( \frac{\pi}{2}+\alpha \right ) = \cos \alphatrigonometry_13
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$$C_{r}^{n}$$
C_{r}^{n}statistics_1
$$\frac{n!}{r!(n-r)!}$$
\frac{n!}{r!(n-r)!}statistics_2
$$\sum_{i=1}^{n}{X_i}$$
\sum_{i=1}^{n}{X_i}statistics_3
$$\sum_{i=1}{n}{X_i 2}$$
\sum_{i=1}^{n}{X_i^2}statistics_4
$$X_1, \cdots,X_n$$
X_1, \cdots,X_nstatistics_5
$$\frac{x-\mu}{\sigma}$$
\frac{x-\mu}{\sigma}statistics_6
$$\sum_{i=1}^{n}{(X_i - \overline{X})^2}$$
\sum_{i=1}^{n}{(X_i - \overline{X})^2}statistics_7
$$\begin{array}{c} \text{若}P \left( AB \right) =P \left( A \right) P \left( B \right) \ \text{则}P \left( A \left
B\right. \right) =P \left({B}\right) \end{array}$$
\begin{array}{c} \text{若}P \left( AB \right) =P \left( A \right) P \left( B \right) \\ \text{则}P \left( A \left| B\right. \right) =P \left({B}\right) \end{array}
$$P(E) ={n \choose k}p^k (1-p)^{n-k}$$
P(E) ={n \choose k}p^k (1-p)^{n-k}statistics_9
$$P \left( A \right) = \lim \limits_{n \to \infty}f_{n}\left ( A \right )$$
P \left( A \right) = \lim \limits_{n \to \infty}f_{n}\left ( A \right )statistics_10
$$P \left( \bigcup \limits_{i=1}^{+ \infty}A_{i}\right) = \prod \limits_{i=1}^{+ \infty}P{\left( A_{i}\right)}$$
P \left( \bigcup \limits_{i=1}^{+ \infty}A_{i}\right) = \prod \limits_{i=1}^{+ \infty}P{\left( A_{i}\right)}statistics_11
$$\begin{array}{c} P \left( \emptyset \right) =0 \ P \left( S \right) =1 \end{array}$$
\begin{array}{c} P \left( \emptyset \right) =0 \\ P \left( S \right) =1 \end{array}statistics_12
$$\begin{array}{c} \forall A \in S \ P \left( A \right) \ge 0 \end{array}$$
\begin{array}{c} \forall A \in S \\ P \left( A \right) \ge 0 \end{array}statistics_13
$$P \left( \bigcup \limits_{i=1}^{n}A_{i}\right) = \prod \limits_{i=1}^{n}P \left( A_{i}\right)$$
P \left( \bigcup \limits_{i=1}^{n}A_{i}\right) = \prod \limits_{i=1}^{n}P \left( A_{i}\right)statistics_14
$$\begin{array}{c} S= \binom{N}{n},A_{k}=\binom{M}{k}\cdot \binom{N-M}{n-k} \ P\left ( A_{k}\right ) = \frac{\binom{M}{k}\cdot \binom{N-M}{n-k}}{\binom{N}{n}} \end{array}$$
\begin{array}{c} S= \binom{N}{n},A_{k}=\binom{M}{k}\cdot \binom{N-M}{n-k} \\ P\left ( A_{k}\right ) = \frac{\binom{M}{k}\cdot \binom{N-M}{n-k}}{\binom{N}{n}} \end{array}statistics_15
$$\begin{array}{c} P_{n}=n! \ A_{n}^{k}=\frac{n!}{\left( n-k \left) !\right. \right.} \end{array}$$
\begin{array}{c} P_{n}=n! \\ A_{n}^{k}=\frac{n!}{\left( n-k \left) !\right. \right.} \end{array}statistics_16
tag
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$$a_{n}=a_{1}q^{n-1}$$
a_{n}=a_{1}q^{n-1}sequence_1
$$a_{n}=a_{1}+ \left( n-1 \left) d\right. \right.$$
a_{n}=a_{1}+ \left( n-1 \left) d\right. \right.sequence_2
$$S_{n}=na_{1}+\frac{n \left( n-1 \right)}2d$$
S_{n}=na_{1}+\frac{n \left( n-1 \right)}{{2}}dsequence_3
$$S_{n}=\frac{n \left( a_{1}+a_{n}\right)}{2}$$
S_{n}=\frac{n \left( a_{1}+a_{n}\right)}{2}sequence_4
$$\frac{1}{n \left( n+k \right)}= \frac{1}{k}\left( \frac{1}{n}-\frac{1}{n+k}\right)$$
\frac{1}{n \left( n+k \right)}= \frac{1}{k}\left( \frac{1}{n}-\frac{1}{n+k}\right)sequence_5
$$\frac{1}{n^{2}-1}= \frac{1}{2}\left( \frac{1}{n-1}-\frac{1}{n+1}\right)$$
\frac{1}{n^{2}-1}= \frac{1}{2}\left( \frac{1}{n-1}-\frac{1}{n+1}\right)sequence_6
$$\frac{1}{4n^{2}-1}=\frac{1}{2}\left( \frac{1}{2n-1}-\frac{1}{2n+1}\right)$$
\frac{1}{4n^{2}-1}=\frac{1}{2}\left( \frac{1}{2n-1}-\frac{1}{2n+1}\right)sequence_7
$$\frac{n+1}{n \left( n-1 \left) \cdot 2^{n}\right. \right.}= \frac{1}{\left( n-1 \left) \cdot 2^{n-1}\right. \right.}-\frac{1}{n \cdot 2^{n}}$$
\frac{n+1}{n \left( n-1 \left) \cdot 2^{n}\right. \right.}= \frac{1}{\left( n-1 \left) \cdot 2^{n-1}\right. \right.}-\frac{1}{n \cdot 2^{n}}sequence_8
$$\begin{array}{c} \text{若}\left {a_{n}\right }、\left {b_{n}\right }\text{为等差数列}, \ \text{则}\left {a_{n}+ b_{n}\right }\text{为等差数列} \end{array}$$
\begin{array}{c} \text{若}\left \{a_{n}\right \}、\left \{b_{n}\right \}\text{为等差数列}, \\ \text{则}\left \{a_{n}+ b_{n}\right \}\text{为等差数列} \end{array}sequence_9
$$(1+x)^{n} =1 + \frac{nx}{1!} + \frac{n(n-1)x^{2}}{2!} + \cdots$$
(1+x)^{n} =1 + \frac{nx}{1!} + \frac{n(n-1)x^{2}}{2!} + \cdotssequence_10
tag
latex
descript
$${E_p} = -\frac{r}$$
{E_p} = -\frac{{GMm}}{r}physics_4
$$\oint_L { \mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup}} \over E} } \cdot { \rm{d}} \mathord{ \buildrel{ \lower3pt \hbox{ \scriptscriptstyle \rightharpoonup}} \over l} = 0$
\oint_L { \mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over E} } \cdot { \rm{d}} \mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over l} = 0physics_6
$$d \vec{F}= Id \vec{l} \times \vec{B}$$
d \vec{F}= Id \vec{l} \times \vec{B}physics_8
$$Q = I ^ { 2 } R \mathrm { t }$$
Q = I ^ { 2 } R \mathrm { t }physics_13
$${E_k} = hv - {W_0}$$
{E_k} = hv - {W_0}physics_15
$${y_0} = A \cos ( \omega {t} + { \varphi _0})$$
{y_0} = A \cos ( \omega {t} + { \varphi _0})physics_19
tag
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$$\ce{SO4^2- + Ba^2+ -> BaSO4 v}$$
%此公式需要在【设置】中开启mhchem扩展支持 具体用法请参考【帮助】2.11.2 \ce{SO4^2- + Ba^2+ -> BaSO4 v}需要Mhchem扩展支持
$$\ce{A v B (v) -> B ^ B (^)}$$
%此公式需要在【设置】中开启mhchem扩展支持 具体用法请参考【帮助】2.11.2 \ce{A v B (v) -> B ^ B (^)}需要Mhchem扩展支持
$$\ce{Hg^2+ ->[I-] \underset{\mathrm{red}}{\ce{HgI2}} ->[I-] \underset{\mathrm{red}}{\ce{[Hg^{II}I4]^2-}}}$$
%此公式需要在【设置】中开启mhchem扩展支持 具体用法请参考【帮助】2.11.2 \ce{Hg^2+ ->[I-] $\underset{\mathrm{red}}{\ce{HgI2}}$ ->[I-] $\underset{\mathrm{red}}{\ce{[Hg^{II}I4]^2-}}$}需要Mhchem扩展支持
$$\ce{Zn^2+ <=>[+ 2OH-][+ 2H+] \underset{\text{amphoteres Hydroxid}}{\ce{Zn(OH)2 v}} <=>[+ 2OH-][+ 2H+] \underset{\text{Hydroxozikat}}{\ce{[Zn(OH)4]^2-}}}$$
%此公式需要在【设置】中开启mhchem扩展支持 具体用法请参考【帮助】2.11.2 \ce{Zn^2+ <=>[+ 2OH-][+ 2H+] $\underset{\text{amphoteres Hydroxid}}{\ce{Zn(OH)2 v}}$ <=>[+ 2OH-][+ 2H+] $\underset{\text{Hydroxozikat}}{\ce{[Zn(OH)4]^2-}}$}需要Mhchem扩展支持
