LaTeX
==================================
LaTeX is a high-quality typesetting system used to create professional-looking documents, such as academic papers, books, and presentations. LaTeX source code is a markup language similar to programming languages that you can use to indicate the layout, fonts, graphics, mathematical symbols, and more in your document.
Here are some useful links related to LaTeX:
LaTeX website
----------------------
#. `LaTeX Project: official website for the LaTeX Project `_
#. `Overleaf: an online LaTeX editor `_
#. `LaTeX Wikibook: a comprehensive guide to LaTeX `_
#. `The Not So Short Introduction to LATEX 2ε `_
#. `The Not So Short Introduction to LaTeX `_
#. `List of LaTeX symbols `_
#. `latex中花体字母编写汇总 `_
Online LaTeX editors
----------------------
#. `LaTeXLive `_
#. `Codecogs `_
Common symbols
----------------------
Binary operations
`````````````````````
Input:
::
+ - \times {\div} \pm \mp \triangleleft \triangleright \cdot \setminus \star \ast \cup \cap \sqcup
Output:
.. math::
+ - \times {\div} \pm \mp \triangleleft \triangleright \cdot \setminus \star \ast \cup \cap \sqcup
Input:
::
\sqcap \vee \wedge \circ \bullet \oplus \ominus \odot \oslash \otimes \bigcirc \diamond \uplus \bigtriangleup \bigtriangledown
Output:
.. math::
\sqcap \vee \wedge \circ \bullet \oplus \ominus \odot \oslash \otimes \bigcirc \diamond \uplus \bigtriangleup \bigtriangledown
Input:
::
\lhd \rhd \unlhd \unrhd \amalg \wr \dagger \ddagger
Output:
.. math::
\lhd \rhd \unlhd \unrhd \amalg \wr \dagger \ddagger
Binary relations
`````````````````````
Input:
::
< > = \le \ge \equiv \ll \gg \doteq \prec \succ \sim \preceq \succeq \simeq
Output:
.. math::
< > = \le \ge \equiv \ll \gg \doteq \prec \succ \sim \preceq \succeq \simeq
Input:
::
\approx \subset \supset \subseteq \supseteq \sqsubset \sqsupset \sqsubseteq \sqsupseteq \cong \Join \bowtie \propto \in \ni
Output:
.. math::
\approx \subset \supset \subseteq \supseteq \sqsubset \sqsupset \sqsubseteq \sqsupseteq \cong \Join \bowtie \propto \in \ni
Input:
::
\vdash \dashv \models \mid \parallel \perp \smile \frown \asymp : \notin \ne
Output:
.. math::
\vdash \dashv \models \mid \parallel \perp \smile \frown \asymp : \notin \ne
Arrows
`````````````````````
Input:
::
\gets \to \longleftarrow \longrightarrow \uparrow \downarrow \updownarrow \leftrightarrow
\Uparrow \Downarrow \Updownarrow \longleftrightarrow \Leftarrow \Longleftarrow \Rightarrow
Output:
.. math::
\gets \to \longleftarrow \longrightarrow \uparrow \downarrow \updownarrow \leftrightarrow
\Uparrow \Downarrow \Updownarrow \longleftrightarrow \Leftarrow \Longleftarrow \Rightarrow
Input:
::
\Longrightarrow \Leftrightarrow \Longleftrightarrow \mapsto \longmapsto \nearrow \searrow
\swarrow \nwarrow \hookleftarrow \hookrightarrow \rightleftharpoons \iff
Output:
.. math::
\Longrightarrow \Leftrightarrow \Longleftrightarrow \mapsto \longmapsto \nearrow \searrow
\swarrow \nwarrow \hookleftarrow \hookrightarrow \rightleftharpoons \iff
Input:
::
\leftharpoonup \rightharpoonup \leftharpoondown \rightharpoondown
Output:
.. math::
\leftharpoonup \rightharpoonup \leftharpoondown \rightharpoondown
Others
`````````````````````
Input:
::
\because \therefore \dots \cdots \vdots \ddots \forall \exists \nexists
\Finv \neg \prime \emptyset \infty \nabla
Output:
.. math::
\because \therefore \dots \cdots \vdots \ddots \forall \exists \nexists
\Finv \neg \prime \emptyset \infty \nabla
Input:
::
\triangle \Box \Diamond \bot \top \angle \measuredangle \sphericalangle \surd \diamondsuit
\heartsuit \clubsuit \spadesuit \flat \natural \sharp
Output:
.. math::
\triangle \Box \Diamond \bot \top \angle \measuredangle \sphericalangle \surd \diamondsuit
\heartsuit \clubsuit \spadesuit \flat \natural \sharp
Greek alphabet
----------------------
Lowercase
`````````````````````
Input:
::
\alpha \beta \gamma \delta \epsilon \varepsilon \zeta \eta \theta \vartheta \iota \kappa \lambda \mu
Output:
.. math::
\alpha \beta \gamma \delta \epsilon \varepsilon \zeta \eta \theta \vartheta \iota \kappa \lambda \mu
Input:
::
\nu \xi o \pi \varpi \rho \varrho \sigma \varsigma \tau \upsilon \phi \varphi \chi \psi \omega
Output:
.. math::
\nu \xi o \pi \varpi \rho \varrho \sigma \varsigma \tau \upsilon \phi \varphi \chi \psi \omega
Uppercase
`````````````````````
Input:
::
\Gamma \Delta \Theta \Lambda \Xi \Pi \Sigma \Upsilon \Phi \Psi \Omega
Output:
.. math::
\Gamma \Delta \Theta \Lambda \Xi \Pi \Sigma \Upsilon \Phi \Psi \Omega
Others
`````````````````````
Input:
::
\hbar \imath \jmath \ell \Re \Im \aleph \beth \gimel \daleth \wp \mho \backepsilon \partial
Output:
.. math::
\hbar \imath \jmath \ell \Re \Im \aleph \beth \gimel \daleth \wp \mho \backepsilon \partial
Input:
::
\eth \Bbbk \complement \circledS \S \mathbb{a} \mathfrak{a} \mathcal{a} \mathrm {a} \mathrm{def}
Output:
.. math::
\eth \Bbbk \complement \circledS \S \mathbb{a} \mathfrak{a} \mathcal{a} \mathrm {a} \mathrm{def}
Fractions & Derivative
-----------------------
Fractions
`````````````````````
Input:
::
\frac{a}{b} \tfrac{a}{b} \mathrm{d}t \frac{\mathrm{d} y}{\mathrm{d} x} \partial t
\frac{\partial y}{\partial x} \nabla\psi
\frac{\partial^2}{\partial x_1\partial x_2}y
Output:
.. math::
\frac{a}{b} \tfrac{a}{b} \mathrm{d}t \frac{\mathrm{d} y}{\mathrm{d} x} \partial t
\frac{\partial y}{\partial x} \nabla\psi
\frac{\partial^2}{\partial x_1\partial x_2}y
Input:
::
\cfrac{1}{a + \cfrac{7}{b + \cfrac{2}{9}}} =c
Output:
.. math::
\cfrac{1}{a + \cfrac{7}{b + \cfrac{2}{9}}} =c
Input:
::
\begin{equation}
x = a_0 + \cfrac{1}{a_1
+ \cfrac{1}{a_2
+ \cfrac{1}{a_3 + \cfrac{1}{a_4} } } }
\end{equation}
Output:
.. math::
\begin{equation}
x = a_0 + \cfrac{1}{a_1
+ \cfrac{1}{a_2
+ \cfrac{1}{a_3 + \cfrac{1}{a_4} } } }
\end{equation}
Derivative
`````````````````````
Input:
::
\dot{a} \ddot{a} {f}' {f}'' {f}^{(n)}
Output:
.. math::
\dot{a} \ddot{a} {f}' {f}'' {f}^{(n)}
Modular arithmetic
`````````````````````
Input:
::
a \bmod b a \equiv b \pmod{m} \gcd(m, n) \operatorname{lcm}(m, n)
Output:
.. math::
a \bmod b a \equiv b \pmod{m} \gcd(m, n) \operatorname{lcm}(m, n)
Radicals
`````````````````````
Input:
::
\sqrt{x} \sqrt[n]{x}
Output:
.. math::
\sqrt{x} \sqrt[n]{x}
Superscript and Subscript
```````````````````````````
Input:
::
x^{a} \ x_{a} \ x_{a}^{b} \ {_{a}^{b}x} \ \sideset{_1^2}{_3^4}X_a^b
Output:
.. math::
x^{a} \ x_{a} \ x_{a}^{b} \ {_{a}^{b}x} \ \sideset{_1^2}{_3^4}X_a^b
Accents and Others
```````````````````````````
Input:
::
\hat{a} \check{a} \grave{a} \acute{a} \tilde{a} \breve{a} \bar{a} \vec{a} \not{a}
Output:
.. math::
\hat{a} \check{a} \grave{a} \acute{a} \tilde{a} \breve{a} \bar{a} \vec{a} \not{a}
Input:
::
37^{\circ} \ \widetilde{abc} \ \widehat{abc} \ \overleftarrow{abc} \ \overrightarrow{abc}
Output:
.. math::
37^{\circ} \ \widetilde{abc} \ \widehat{abc} \ \overleftarrow{abc} \ \overrightarrow{abc}
Input:
::
\overline{abc} \ \underline{abc} \ \overbrace{abc} \ \underbrace{abc}
Output:
.. math::
\overline{abc} \ \underline{abc} \ \overbrace{abc} \ \underbrace{abc}
Input:
::
\overset{x}{abc} \ \underset{x}{abc} \ \stackrel\frown{AB} \ \overline{AB} \ \overleftrightarrow{AB}
Output:
.. math::
\overset{x}{abc} \ \underset{x}{abc} \ \stackrel\frown{AB} \ \overline{AB} \ \overleftrightarrow{AB}
Input:
::
\overset{a}{\leftarrow} \ \overset{a}{\rightarrow} \ \xleftarrow[abc]{x} \ \xrightarrow[abc]{x}
Output:
.. math::
\overset{a}{\leftarrow} \ \overset{a}{\rightarrow} \ \xleftarrow[abc]{x} \ \xrightarrow[abc]{x}
Limits class
------------------------------
Limits
```````````````````````````
Input:
::
\lim{a} \ \lim_{x \to 0} \ \lim_{x \to \infty} \textstyle \ \lim_{x \to 0} \max_x{y} \min_x{y}
Output:
.. math::
\lim{a} \ \lim_{x \to 0} \ \lim_{x \to \infty} \textstyle \ \lim_{x \to 0} \max_x{y} \min_x{y}
Logarithms and exponentials
```````````````````````````
Input:
::
\log_{a}{b} \ \lg_{a}{b} \ \ln_{a}{b} \ \exp a
Output:
.. math::
\log_{a}{b} \ \lg_{a}{b} \ \ln_{a}{b} \ \exp a
Bounds
```````````````````````````
::
\min x \max y \sup t \inf s \lim u \limsup w \liminf v \dim p \ker\phi
Output:
.. math::
\min x \max y \sup t \inf s \lim u \limsup w \liminf v \dim p \ker\phi
Trigonometry class
------------------------------
Trigonometric functions
``````````````````````````````
Input:
::
\sin x \cos x \tan x \cot x \sec x \csc x
Output:
.. math::
\sin x \cos x \tan x \cot x \sec x \csc x
Inverse trigonometric functions
`````````````````````````````````
Input:
::
\sin^{-1} x \cos^{-1} x \tan^{-1} x \cot^{-1} x \sec^{-1} x \arcsin x \arccos x
Output:
.. math::
\sin^{-1} x \cos^{-1} x \tan^{-1} x \cot^{-1} x \sec^{-1} x \arcsin x \arccos x
Input:
::
\arctan x \operatorname{arccot} x \operatorname{arcsec} x \operatorname{arccos} x
Output:
.. math::
\arctan x \operatorname{arccot} x \operatorname{arcsec} x \operatorname{arccos} x
Hyperbolic functions
`````````````````````````````````
Input:
::
\sinh x \cosh x \tanh x \coth x \operatorname{sech} x \operatorname{csch} x
Output:
.. math::
\sinh x \cosh x \tanh x \coth x \operatorname{sech} x \operatorname{csch} x
Inverse hyperbolic functions
`````````````````````````````````
Input:
::
\sinh^{-1} x \cosh^{-1} x \tanh^{-1} x \coth^{-1} x
\operatorname{sech}^{-1} x \operatorname{csch}^{-1}x
Output:
.. math::
\sinh^{-1} x \cosh^{-1} x \tanh^{-1} x \coth^{-1} x
\operatorname{sech}^{-1} x \operatorname{csch}^{-1}x
Integral operation
----------------------------
Integral
`````````````````````````````````
Input:
::
\int x \int_{a}^{b} x \int\limits_{a}^{b} x
Output:
.. math::
\int x \int_{a}^{b} x \int\limits_{a}^{b} x
Double integral
`````````````````````````````````
Input:
::
\iint x \iint_{a}^{b} x \iint\limits_{a}^{b} x
Output:
.. math::
\iint x \iint_{a}^{b} x \iint\limits_{a}^{b} x
Triple integral
`````````````````````````````````
Input:
::
\iiint x \iiint_{a}^{b} x \iiint\limits_{a}^{b} x
Output:
.. math::
\iiint x \iiint_{a}^{b} x \iiint\limits_{a}^{b} x
Closed line or path integral
`````````````````````````````````
Input:
::
\oint x \oint_{a}^{b} x
Output:
.. math::
\oint x \oint_{a}^{b} x
Summation
`````````````````````````````````
Input:
::
\sum s \sum_{a}^{b} s {\textstyle \sum_{a}^{b}} s
Output:
.. math::
\sum s \sum_{a}^{b} s {\textstyle \sum_{a}^{b}} s
Product and coproduct
`````````````````````````````````
Input:
::
\prod \prod_{a}^{b} {\textstyle \prod_{a}^{b}} \coprod \coprod_{a}^{b} {\textstyle \coprod_{a}^{b}}
Output:
.. math::
\prod \prod_{a}^{b} {\textstyle \prod_{a}^{b}} \coprod \coprod_{a}^{b} {\textstyle \coprod_{a}^{b}}
Union and intersection
`````````````````````````````````
Input:
::
\bigcup \bigcup_{a}^{b} {\textstyle \bigcup_{a}^{b}}
\bigcap \bigcap_{a}^{b} {\textstyle \bigcap_{a}^{b}}
Output:
.. math::
\bigcup \bigcup_{a}^{b} {\textstyle \bigcup_{a}^{b}}
\bigcap \bigcap_{a}^{b} {\textstyle \bigcap_{a}^{b}}
Disjunction and cojunction
`````````````````````````````````
Input:
::
\bigvee \bigvee_{a}^{b} {\textstyle \bigvee_{a}^{b}}
\bigwedge \bigwedge_{a}^{b} {\textstyle \bigwedge_{a}^{b}}
Output:
.. math::
\bigvee \bigvee_{a}^{b} {\textstyle \bigvee_{a}^{b}}
\bigwedge \bigwedge_{a}^{b} {\textstyle \bigwedge_{a}^{b}}
Brackets
`````````````````````````````````
Input:
::
\left ( a \right )
\left [ a \right ]
\left \langle a \right \rangle
\left \{ a \right \}
\left | a \right |
\left \| a \right \|
\left \lfloor a \right \rfloor
\left \lceil a \right \rceil
Output:
.. math::
\left ( a \right )
\left [ a \right ]
\left \langle a \right \rangle
\left \{ a \right \}
\left | a \right |
\left \| a \right \|
\left \lfloor a \right \rfloor
\left \lceil a \right \rceil
Input:
::
\binom{n}{r}
\left [ 0,1 \right )
\left \langle \psi \right |
\left | \psi \right \rangle
\left \langle \psi | \psi \right \rangle
Output:
.. math::
\binom{n}{r}
\left [ 0,1 \right )
\left \langle \psi \right |
\left | \psi \right \rangle
\left \langle \psi | \psi \right \rangle
Matrix
`````````````````````````````````
Input:
::
\begin{matrix}
1&2 &3 \\
4&5 &6
\end{matrix}
Output:
.. math::
\begin{matrix}
1&2 &3 \\
4&5 &6
\end{matrix}
Input:
::
\begin{bmatrix}
1&2 &3 \\
4&5 &6
\end{bmatrix}
Output:
.. math::
\begin{bmatrix}
1&2 &3 \\
4&5 &6
\end{bmatrix}
Input:
::
\begin{pmatrix}
1&2 &3 \\
4&5 &6
\end{pmatrix}
Output:
.. math::
\begin{pmatrix}
1&2 &3 \\
4&5 &6
\end{pmatrix}
Input:
::
\begin{vmatrix}
1&2 &3 \\
4&5 &6
\end{vmatrix}
Output:
.. math::
\begin{vmatrix}
1&2 &3 \\
4&5 &6
\end{vmatrix}
Input:
::
\begin{Vmatrix}
1&2 &3 \\
4&5 &6
\end{Vmatrix}
Output:
.. math::
\begin{Vmatrix}
1&2 &3 \\
4&5 &6
\end{Vmatrix}
Input:
::
\begin{Bmatrix}
1&2 &3 \\
4&5 &6
\end{Bmatrix}
Output:
.. math::
\begin{Bmatrix}
1&2 &3 \\
4&5 &6
\end{Bmatrix}
Input:
::
\left\{\begin{matrix}
1&2 &3 \\
4&5 &6
\end{matrix}\right.
Output:
.. math::
\left\{\begin{matrix}
1&2 &3 \\
4&5 &6
\end{matrix}\right.
Input:
::
\left.\begin{matrix}
1&2 &3 \\
4&5 &6
\end{matrix}\right\}
Output:
.. math::
\left.\begin{matrix}
1&2 &3 \\
4&5 &6
\end{matrix}\right\}
Input:
::
\begin{cases}
1& \text{ if } x= 2\\
3& \text{ if } x=4
\end{cases}
Output:
.. math::
\begin{cases}
1& \text{ if } x= 2\\
3& \text{ if } x=4
\end{cases}
Input:
::
\begin{align*}
y&=1 \\
x&=2
\end{align*}
Output:
.. math::
\begin{align*}
y&=1 \\
x&=2
\end{align*}
Formula Template
----------------------
Algebra
`````````````````````````````````
Input:
::
\left(x-1\right)\left(x+3\right)
Output:
.. math::
\left(x-1\right)\left(x+3\right)
Input:
::
\sqrt{a^2+b^2}
Output:
.. math::
\sqrt{a^2+b^2}
Input:
::
\frac{a}{b}\pm \frac{c}{d}= \frac{ad \pm bc}{bd}
Output:
.. math::
\frac{a}{b}\pm \frac{c}{d}= \frac{ad \pm bc}{bd}
Input:
::
\frac{1}{\sqrt{a}}=\frac{\sqrt{a}}{a},a\ge 0\frac{1}{\sqrt{a}}=\frac{\sqrt{a}}{a},a\ge 0
Output:
.. math::
\frac{1}{\sqrt{a}}=\frac{\sqrt{a}}{a},a\ge 0\frac{1}{\sqrt{a}}=\frac{\sqrt{a}}{a},a\ge 0
Input:
::
x ={-b \pm \sqrt{b^2-4ac}\over 2a}
Output:
.. math::
x ={-b \pm \sqrt{b^2-4ac}\over 2a}
Input:
::
\left\{\begin{matrix}
x=a + r\text{cos}\theta \\
y=b + r\text{sin}\theta
\end{matrix}\right.
Output:
.. math::
\left\{\begin{matrix}
x=a + r\text{cos}\theta \\
y=b + r\text{sin}\theta
\end{matrix}\right.
Input:
::
\left ( \frac{a}{b}\right )^{n}= \frac{a^{n}}{b^{n}}
Output:
.. math::
\left ( \frac{a}{b}\right )^{n}= \frac{a^{n}}{b^{n}}
Input:
::
\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1
Output:
.. math::
\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1
Input:
::
\sqrt[n]{a^{n}}=\left ( \sqrt[n]{a}\right )^{n}
Output:
.. math::
\sqrt[n]{a^{n}}=\left ( \sqrt[n]{a}\right )^{n}
Input:
::
y-y_{1}=k \left( x-x_{1}\right)
Output:
.. math::
y-y_{1}=k \left( x-x_{1}\right)
Input:
::
\begin{array}{l}
\text{For equations of the form: }x^{3}-1=0 \\
\text{let}\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}
Output:
.. math::
\begin{array}{l}
\text{For equations of the form: }x^{3}-1=0 \\
\text{let}\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}
Input:
::
\begin{array}{l}
a\mathop{{x}}\nolimits^{{2}}+bx+c=0 \\
\Delta =\mathop{{b}}\nolimits^{{2}}-4ac \\
\mathop{{x}}\nolimits_{{1,2}}=\frac{{-b \pm
\sqrt{{\mathop{{b}}\nolimits^{{2}}-4ac}}}}{{2a}} \\
\mathop{{x}}\nolimits_{{1}}+\mathop{{x}}\nolimits_{{2}}=-\frac{{b}}{{a}} \\
\mathop{{x}}\nolimits_{{1}}\mathop{{x}}\nolimits_{{2}}=\frac{{c}}{{a}}
\end{array}
Output:
.. math::
\begin{array}{l}
a\mathop{{x}}\nolimits^{{2}}+bx+c=0 \\
\Delta =\mathop{{b}}\nolimits^{{2}}-4ac \\
\mathop{{x}}\nolimits_{{1,2}}=\frac{{-b \pm
\sqrt{{\mathop{{b}}\nolimits^{{2}}-4ac}}}}{{2a}} \\
\mathop{{x}}\nolimits_{{1}}+\mathop{{x}}\nolimits_{{2}}=-\frac{{b}}{{a}} \\
\mathop{{x}}\nolimits_{{1}}\mathop{{x}}\nolimits_{{2}}=\frac{{c}}{{a}}
\end{array}
Input:
::
\begin{array}{l}
a\mathop{{x}}\nolimits^{{2}}+bx+c=0 \\
\Delta =\mathop{{b}}\nolimits^{{2}}-4ac \\
\left\{\begin{matrix}
\Delta \gt 0\text{ The equation has two distinct real roots} \\
\Delta = 0\text{ The equation has two equal real roots} \\
\Delta \lt 0\text{ The equation has two complex roots}
\end{matrix}\right.
\end{array}
Output:
.. math::
\begin{array}{l}
a\mathop{{x}}\nolimits^{{2}}+bx+c=0 \\
\Delta =\mathop{{b}}\nolimits^{{2}}-4ac \\
\left\{\begin{matrix}
\Delta \gt 0\text{ The equation has two distinct real roots} \\
\Delta = 0 \text{ The equation has two equal real roots }\quad \\
\Delta \lt 0\text{ The equation has two complex roots}\qquad
\end{matrix}\right.
\end{array}
Space
`````````````````````````````````
Input:
::
\begin{array}{l}
a\quad b \\
a\qquad b \\
a\enspace b \\
a\;b \\
a\:b \\
a\,b \\
a\!b
\end{array}
Output:
.. math::
\begin{array}{l}
a\quad b \\
a\qquad b \\
a\enspace b \\
a\;b \\
a\:b \\
a\,b \\
a\!b
\end{array}
Geometry
`````````````````````````````````
Input:
::
\begin{array}{l}
\Delta A B C \\
l \perp \beta ,l \subset \alpha \Rightarrow \alpha \perp \beta \\
a \parallel c,b \parallel c \Rightarrow a \parallel b \\
P \in \alpha ,P \in \beta , \alpha \cap \beta =l \Rightarrow P \in l \\
A \in l,B \in l,A \in \alpha ,B \in \alpha \Rightarrow l \subset \alpha
\end{array}
Output:
.. math::
\begin{array}{l}
\Delta A B C \\
l \perp \beta ,l \subset \alpha \Rightarrow \alpha \perp \beta \\
a \parallel c,b \parallel c \Rightarrow a \parallel b \\
P \in \alpha ,P \in \beta , \alpha \cap \beta =l \Rightarrow P \in l \\
A \in l,B \in l,A \in \alpha ,B \in \alpha \Rightarrow l \subset \alpha
\end{array}
Input:
::
\left.\begin{matrix}
a \perp \alpha \\
b \perp \alpha
\end{matrix}\right\}\Rightarrow a \parallel b
Output:
.. math::
\left.\begin{matrix}
a \perp \alpha \\
b \perp \alpha
\end{matrix}\right\}\Rightarrow a \parallel b
Input:
::
\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
Output:
.. math::
\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
Input:
::
\begin{array}{c}
\alpha \perp \beta , \alpha \cap \beta =l,a \subset \alpha ,a \perp l \Rightarrow a \perp \beta \\
\alpha \parallel \beta , \gamma \cap \alpha =a, \gamma \cap \beta =b \Rightarrow a \parallel b \\
a^{2}+b^{2}=c^{2}
\end{array}
Output:
.. math::
\begin{array}{c}
\alpha \perp \beta , \alpha \cap \beta =l,a \subset \alpha ,a \perp l \Rightarrow a \perp \beta \\
\alpha \parallel \beta , \gamma \cap \alpha =a, \gamma \cap \beta =b \Rightarrow a \parallel b \\
a^{2}+b^{2}=c^{2}
\end{array}
Inequality
`````````````````````````````````
Input:
::
\begin{array}{c}
a > b,b > c \Rightarrow a > c \\
a > b > 0,c > d > 0 \Rightarrow ac > bd \\
a > b,c > d \Rightarrow a+c > b+d \\
\left | a-b \right | \geqslant \left | a \right | -\left | b \right | \\
\left | a \right |\leqslant b \Rightarrow -b \leqslant a \leqslant \left | b \right | \\
-\left | a \right |\leq a\leqslant \left | a \right | \\
\left | a+b \right | \leqslant \left | a \right | + \left | b \right |
\end{array}
Output:
.. math::
\begin{array}{c}
a > b,b > c \Rightarrow a > c \\
a > b > 0,c > d > 0 \Rightarrow ac > bd \\
a > b,c > d \Rightarrow a+c > b+d \\
\left | a-b \right | \geqslant \left | a \right | -\left | b \right | \\
\left | a \right |\leqslant b \Rightarrow -b \leqslant a \leqslant \left | b \right | \\
-\left | a \right |\leq a\leqslant \left | a \right | \\
\left | a+b \right | \leqslant \left | a \right | + \left | b \right |
\end{array}
Input:
::
\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}
Output:
.. math::
\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}
Input:
::
\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}
Output:
.. math::
\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}
Input:
::
\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)
Output:
.. math::
\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)
Input:
::
\begin{array}{c}
a,b \in R^{+} \\
\Rightarrow \frac{a+b}{{2}}\ge \sqrt{ab} \\
\left( \text{Equality holds if and only if }a=b\right)
\end{array}
Output:
.. math::
\begin{array}{c}
a,b \in R^{+} \\
\Rightarrow \frac{a+b}{{2}}\ge \sqrt{ab} \\
\left( \text{Equality holds if and only if }a=b\right)
\end{array}
Input:
::
\begin{array}{c}
a,b \in R \\
\Rightarrow a^{2}+b^{2}\ge 2ab \\
\left( \text{Equality holds if and only if }a=b\right)
\end{array}
Output:
.. math::
\begin{array}{c}
a,b \in R \\
\Rightarrow a^{2}+b^{2}\ge 2ab \\
\left( \text{Equality holds if and only if }a=b\right)
\end{array}
Input:
::
\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}
Output:
.. math::
\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}
Integral
`````````````````````````````````
Input:
::
\begin{array}{l}
\frac{\mathrm{d}}{\mathrm{d}x}x^n=nx^{n-1} \\
\frac{\mathrm{d}}{\mathrm{d}x}\ln(x)=\frac{1}{x} \\
\frac{\mathrm{d}}{\mathrm{d}x}\cos x=-\sin x \\
\frac{\mathrm{d}}{\mathrm{d}x}\tan x=\sec^2 x \\
\int \frac{1}{x}\mathrm{d}x= \ln \left| x \right| +C \\
\int \frac{1}{1+x^{2}}\mathrm{d}x= \arctan x +C \\
f(x) = \int_{-\infty}^\infty \hat f(x)\xi\,e^{2 \pi i \xi x} \,\mathrm{d}\xi
\end{array}
Output:
.. math::
\begin{array}{l}
\frac{\mathrm{d}}{\mathrm{d}x}x^n=nx^{n-1} \\
\frac{\mathrm{d}}{\mathrm{d}x}\ln(x)=\frac{1}{x} \\
\frac{\mathrm{d}}{\mathrm{d}x}\cos x=-\sin x \\
\frac{\mathrm{d}}{\mathrm{d}x}\tan x=\sec^2 x \\
\int \frac{1}{x}\mathrm{d}x= \ln \left| x \right| +C \\
\int \frac{1}{1+x^{2}}\mathrm{d}x= \arctan x +C \\
f(x) = \int_{-\infty}^\infty \hat f(x)\xi\,e^{2 \pi i \xi x} \,\mathrm{d}\xi
\end{array}
Input:
::
\begin{array}{l}
\frac{\mathrm{d}}{\mathrm{d}x}e^{ax}=a\,e^{ax} \\
\frac{\mathrm{d}}{\mathrm{d}x}\sin x=\cos x \\
\int k\mathrm{d}x = kx+C \\
\frac{\mathrm{d}}{\mathrm{d}x}\cot x=-\csc^2 x \\
\int \frac{1}{\sqrt{1-x^{2}}}\mathrm{d}x= \arcsin x +C \\
\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 x^{\mu}\mathrm{d}x=\frac{x^{\mu +1}}{\mu +1}+C, \left({\mu \neq -1}\right)
\end{array}
Output:
.. math::
\begin{array}{l}
\frac{\mathrm{d}}{\mathrm{d}x}e^{ax}=a\,e^{ax} \\
\frac{\mathrm{d}}{\mathrm{d}x}\sin x=\cos x \\
\int k\mathrm{d}x = kx+C \\
\frac{\mathrm{d}}{\mathrm{d}x}\cot x=-\csc^2 x \\
\int \frac{1}{\sqrt{1-x^{2}}}\mathrm{d}x= \arcsin x +C \\
\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 x^{\mu}\mathrm{d}x=\frac{x^{\mu +1}}{\mu +1}+C, \left({\mu \neq -1}\right)
\end{array}
Matrix
`````````````````````````````````
Input:
::
\begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}
Output:
.. math::
\begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}
Input:
::
\begin{pmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{pmatrix}
Output:
.. math::
\begin{pmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{pmatrix}
Input:
::
\begin{pmatrix}
a_{11} & \cdots & a_{1n} \\
\vdots & \ddots & \vdots \\
a_{m1} & \cdots & a_{mn}
\end{pmatrix}
Output:
.. math::
\begin{pmatrix}
a_{11} & \cdots & a_{1n} \\
\vdots & \ddots & \vdots \\
a_{m1} & \cdots & a_{mn}
\end{pmatrix}
Input:
::
O = \begin{bmatrix}
0 & 0 & \cdots & 0 \\
0 & 0 & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & 0
\end{bmatrix}
Output:
.. math::
O = \begin{bmatrix}
0 & 0 & \cdots & 0 \\
0 & 0 & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & 0
\end{bmatrix}
Input:
::
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 ]
Output:
.. math::
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 ]
Input:
::
\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}
Output:
.. math::
\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}
Input:
::
\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}
Output:
.. math::
\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}
Input:
::
\begin{array}{c}
A=A^{T} \\
A=-A^{T}
\end{array}
Output:
.. math::
\begin{array}{c}
A=A^{T} \\
A=-A^{T}
\end{array}
Triangle
`````````````````````````````````
Input:
::
\begin{array}{l}
e^{i \theta} \\
\text{sin}^{2}\frac{\alpha}{2}=\frac{1- \text{cos}\alpha}{2} \\
\text{tan}\frac{\alpha}{2}=\frac{\text{sin}\alpha}{1+ \text{cos}\alpha} \\
\sin \alpha - \sin \beta =2 \cos \frac{\alpha + \beta}{2}\sin \frac{\alpha - \beta}{2} \\
\cos \alpha - \cos \beta =-2\sin \frac{\alpha + \beta}{2}\sin \frac{\alpha - \beta}{2} \\
\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}=\frac{1}{2R} \\
\sin \left ( \frac{\pi}{2}+\alpha \right ) = \cos \alpha
\end{array}
Output:
.. math::
\begin{array}{l}
e^{i \theta} \\
\text{sin}^{2}\frac{\alpha}{2}=\frac{1- \text{cos}\alpha}{2} \\
\text{tan}\frac{\alpha}{2}=\frac{\text{sin}\alpha}{1+ \text{cos}\alpha} \\
\sin \alpha - \sin \beta =2 \cos \frac{\alpha + \beta}{2}\sin \frac{\alpha - \beta}{2} \\
\cos \alpha - \cos \beta =-2\sin \frac{\alpha + \beta}{2}\sin \frac{\alpha - \beta}{2} \\
\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}=\frac{1}{2R} \\
\sin \left ( \frac{\pi}{2}+\alpha \right ) = \cos \alpha
\end{array}
Input:
::
\begin{array}{l}
\left(\frac{\pi}{2}-\theta \right ) \\
\text{cos}^{2}\frac{\alpha}{2}=\frac{1+ \text{cos}\alpha}{2} \\
\sin \alpha + \sin \beta =2 \sin \frac{\alpha + \beta}{2}\cos \frac{\alpha - \beta}{2} \\
\cos \alpha + \cos \beta =2 \cos \frac{\alpha + \beta}{2}\cos \frac{\alpha - \beta}{2} \\
a^{2}=b^{2}+c^{2}-2bc\cos A \\
\sin \left ( \frac{\pi}{2}-\alpha \right ) = \cos \alpha
\end{array}
Output:
.. math::
\begin{array}{l}
\left(\frac{\pi}{2}-\theta \right ) \\
\text{cos}^{2}\frac{\alpha}{2}=\frac{1+ \text{cos}\alpha}{2} \\
\sin \alpha + \sin \beta =2 \sin \frac{\alpha + \beta}{2}\cos \frac{\alpha - \beta}{2} \\
\cos \alpha + \cos \beta =2 \cos \frac{\alpha + \beta}{2}\cos \frac{\alpha - \beta}{2} \\
a^{2}=b^{2}+c^{2}-2bc\cos A \\
\sin \left ( \frac{\pi}{2}-\alpha \right ) = \cos \alpha
\end{array}
Statistics
`````````````````````````````````
Input:
::
\begin{array}{l}
C_{r}^{n} \\
\sum_{i=1}^{n}{X_i} \\
X_1, \cdots,X_n \\
\sum_{i=1}^{n}{(X_i - \overline{X})^2} \\
P(E) ={n \choose k}p^k (1-p)^{n-k} \\
\end{array}
Output:
.. math::
\begin{array}{l}
C_{r}^{n} \\
\sum_{i=1}^{n}{X_i} \\
X_1, \cdots,X_n \\
\sum_{i=1}^{n}{(X_i - \overline{X})^2} \\
P(E) ={n \choose k}p^k (1-p)^{n-k} \\
\end{array}
Input:
::
\begin{array}{l}
\frac{n!}{r!(n-r)!} \\
\sum_{i=1}^{n}{X_i^2} \\
\frac{x-\mu}{\sigma} \\
P \left( A \right) = \lim \limits_{n \to \infty}f_{n}\left ( A \right ) \\
\end{array}
Output:
.. math::
\begin{array}{l}
\frac{n!}{r!(n-r)!} \\
\sum_{i=1}^{n}{X_i^2} \\
\frac{x-\mu}{\sigma} \\
P \left( A \right) = \lim \limits_{n \to \infty}f_{n}\left ( A \right ) \\
\end{array}
Input:
::
P \left( \bigcup \limits_{i=1}^{+ \infty}A_{i}\right) =
\prod \limits_{i=1}^{+ \infty}P{\left( A_{i}\right)}
Output:
.. math::
P \left( \bigcup \limits_{i=1}^{+ \infty}A_{i}\right) =
\prod \limits_{i=1}^{+ \infty}P{\left( A_{i}\right)}
Input:
::
P \left( \bigcup \limits_{i=1}^{n}A_{i}\right) =
\prod \limits_{i=1}^{n}P \left( A_{i}\right)
Output:
.. math::
P \left( \bigcup \limits_{i=1}^{n}A_{i}\right) =
\prod \limits_{i=1}^{n}P \left( A_{i}\right)
Input:
::
\begin{array}{c}
\forall A \in S \\
P \left( A \right) \ge 0
\end{array}
Output:
.. math::
\begin{array}{c}
\forall A \in S \\
P \left( A \right) \ge 0
\end{array}
Input:
::
\begin{array}{c}
P \left( \emptyset \right) =0 \\
P \left( S \right) =1
\end{array}
Output:
.. math::
\begin{array}{c}
P \left( \emptyset \right) =0 \\
P \left( S \right) =1
\end{array}
Input:
::
\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}
Output:
.. math::
\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}
Input:
::
\begin{array}{c}
P_{n}=n! \\
A_{n}^{k}=\frac{n!}{\left( n-k \left) !\right. \right.}
\end{array}
Output:
.. math::
\begin{array}{c}
P_{n}=n! \\
A_{n}^{k}=\frac{n!}{\left( n-k \left) !\right. \right.}
\end{array}
Input:
::
\begin{array}{c}
\text{If } P(AB) = P(A)P(B) \\
\text{then } P(A|B) = \dfrac{P(B)}{1-P(\overline{B})}
\end{array}
Output:
.. math::
\begin{array}{c}
\text{If } P(AB) = P(A)P(B) \\
\text{then } P(A|B) = \dfrac{P(B)}{1-P(\overline{B})}
\end{array}
Sequence
`````````````````````````````````
Input:
::
\begin{array}{l}
a_{n}=a_{1}q^{n-1} \\
S_{n}=na_{1}+\frac{n \left( n-1 \right)}{{2}}d \\
\frac{1}{n \left( n+k \right)}= \frac{1}{k}\left( \frac{1}{n}-\frac{1}{n+k}\right) \\
\frac{1}{4n^{2}-1}=\frac{1}{2}\left( \frac{1}{2n-1}-\frac{1}{2n+1}\right) \\
\end{array}
Output:
.. math::
\begin{array}{l}
a_{n}=a_{1}q^{n-1} \\
S_{n}=na_{1}+\frac{n \left( n-1 \right)}{{2}}d \\
\frac{1}{n \left( n+k \right)}= \frac{1}{k}\left( \frac{1}{n}-\frac{1}{n+k}\right) \\
\frac{1}{4n^{2}-1}=\frac{1}{2}\left( \frac{1}{2n-1}-\frac{1}{2n+1}\right) \\
\end{array}
Input:
::
\begin{array}{l}
a_{n}=a_{1}+ \left( n-1 \left) d\right. \right. \\
S_{n}=\frac{n \left( a_{1}+a_{n}\right)}{2} \\
\frac{1}{n^{2}-1}= \frac{1}{2}\left( \frac{1}{n-1}-\frac{1}{n+1}\right) \\
\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}} \\
(1+x)^{n} =1 + \frac{nx}{1!} + \frac{n(n-1)x^{2}}{2!} + \cdots
\end{array}
Output:
.. math::
\begin{array}{l}
a_{n}=a_{1}+ \left( n-1 \left) d\right. \right. \\
S_{n}=\frac{n \left( a_{1}+a_{n}\right)}{2} \\
\frac{1}{n^{2}-1}= \frac{1}{2}\left( \frac{1}{n-1}-\frac{1}{n+1}\right) \\
\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}} \\
(1+x)^{n} =1 + \frac{nx}{1!} + \frac{n(n-1)x^{2}}{2!} + \cdots
\end{array}
Input:
::
\begin{array}{c}
\text{If}\left \{a_{n}\right \},\left \{b_{n}\right \}\text{are arithmetic progressions}, \\
\text{then}\left \{a_{n}+ b_{n}\right \}\text{is an arithmetic progression.}
\end{array}
Output:
.. math::
\begin{array}{c}
\text{If}\left \{a_{n}\right \},\left \{b_{n}\right \}\text{are arithmetic progressions}, \\
\text{then}\left \{a_{n}+ b_{n}\right \}\text{is an arithmetic progression.}
\end{array}
Physics
`````````````````````````````````
Input:
::
\sum {{{ \mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over F} }_i}} =
\frac{{d \mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over v} }}{{dt}} = 0
Output:
.. math::
\sum {{{ \mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over F} }_i}} =
\frac{{d \mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over v} }}{{dt}} = 0
Input:
::
{{ \mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over F} }_{12}} =
- {{ \mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over F} }_{21}}
Output:
.. math::
{{ \mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over F} }_{12}} =
- {{ \mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over F} }_{21}}
Input:
::
\mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over F} =
k \frac{{Qq}}{{{r^2}}} \hat{r}
Output:
.. math::
\mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over F} =
k \frac{{Qq}}{{{r^2}}} \hat{r}
Input:
::
d \mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over B} =
\frac{{{ \mu _0}}}{{4 \pi }} \frac{{Idl \times \mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over r} }}{{{r^3}}} =
\frac{{{ \mu _0}}}{{4 \pi }} \frac{{Idl \sin \theta }}{{{r^2}}}
Output:
.. math::
d \mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over B} =
\frac{{{ \mu _0}}}{{4 \pi }} \frac{{Idl \times \mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over r} }}{{{r^3}}} =
\frac{{{ \mu _0}}}{{4 \pi }} \frac{{Idl \sin \theta }}{{{r^2}}}
Input:
::
E = n{{ \Delta \Phi } \over {\Delta {t} }}
Output:
.. math::
E = n{{ \Delta \Phi } \over {\Delta {t} }}
Input:
::
\oint { \mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over E} \cdot {d\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over l}} =
- {{d{\varphi _B}} \over {dt}}}
Output:
.. math::
\oint { \mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over E} \cdot {d\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over l}} =
- {{d{\varphi _B}} \over {dt}}}
Input:
::
Q = I ^ { 2 } R \mathrm { t }
Output:
.. math::
Q = I ^ { 2 } R \mathrm { t }
Input:
::
{E_k} = hv - {W_0}
Output:
.. math::
{E_k} = hv - {W_0}
Input:
::
\Delta {x} \Delta {p} \ge \frac{h}{{4 \pi }}
Output:
.. math::
\Delta {x} \Delta {p} \ge \frac{h}{{4 \pi }}
Input:
::
{y_0} = A \cos ( \omega {t} + { \varphi _0})
Output:
.. math::
{y_0} = A \cos ( \omega {t} + { \varphi _0})
Input:
::
\mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over F} =
m \mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over a} =
m \frac{{{d^2} \mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over r} }}{{d{t^2}}}
Output:
.. math::
\mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over F} =
m \mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over a} =
m \frac{{{d^2} \mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over r} }}{{d{t^2}}}
Input:
::
{E_p} = -\frac{{GMm}}{r}
Output:
.. math::
{E_p} = -\frac{{GMm}}{r}
Input:
::
\oint_L { \mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over E} }
\cdot { \rm{d}} \mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over l} = 0
Output:
.. math::
\oint_L { \mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over E} }
\cdot { \rm{d}} \mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over l} = 0
Input:
::
d \vec{F}= Id \vec{l} \times \vec{B}
Output:
.. math::
d \vec{F}= Id \vec{l} \times \vec{B}
Input:
::
\mathop \Phi \nolimits_e =
\oint { \mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over E} \cdot {d \mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over S}} =
{1 \over {{\varepsilon _0}}}\sum {q} }
Output:
.. math::
\mathop \Phi \nolimits_e =
\oint { \mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over E} \cdot {d \mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over S}} =
{1 \over {{\varepsilon _0}}}\sum {q} }
Input:
::
\oint { \mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over B}
\cdot {d \mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over l}} =
{ \mu _0}} I + { \mu _0}{I_d}
Output:
.. math::
\oint { \mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over B}
\cdot {d \mathord{ \buildrel{ \lower3pt \hbox{$ \scriptscriptstyle \rightharpoonup$}} \over l}} =
{ \mu _0}} I + { \mu _0}{I_d}
Input:
::
F = G{{Mm} \over {{r^2}}}
Output:
.. math::
F = G{{Mm} \over {{r^2}}}
Input:
::
\lambda = \frac{{ \frac{{{c^2}}}{v}}}{{ \frac{{m{c^2}}}{h}}} = \frac{h}{{mv}} = \frac{h}{p}
Output:
.. math::
\lambda = \frac{{ \frac{{{c^2}}}{v}}}{{ \frac{{m{c^2}}}{h}}} = \frac{h}{{mv}} = \frac{h}{p}
Input:
::
l = {l_0} \sqrt {1 - {{( \frac{v}{c})}^2}}
Output:
.. math::
l = {l_0} \sqrt {1 - {{( \frac{v}{c})}^2}}
Input:
::
y(t) = A \cos ( \frac{{2 \pi {x}}}{ \lambda } + \varphi )
Output:
.. math::
y(t) = A \cos ( \frac{{2 \pi {x}}}{ \lambda } + \varphi )
Input:
::
\begin{array}{l}
\nabla \cdot \mathbf{E} =\cfrac{\rho}{\varepsilon _0} \\
\nabla \cdot \mathbf{B} = 0 \\
\nabla \times \mathbf{E} = -\cfrac{\partial \mathbf{B}}{\partial t } \\
\nabla \times \mathbf{B} = \mu _0\mathbf{J} + \mu _0\varepsilon_0 \cfrac{\partial \mathbf{E}}{\partial t }
\end{array}
Output:
.. math::
\begin{array}{l}
\nabla \cdot \mathbf{E} =\cfrac{\rho}{\varepsilon _0} \\
\nabla \cdot \mathbf{B} = 0 \\
\nabla \times \mathbf{E} = -\cfrac{\partial \mathbf{B}}{\partial t } \\
\nabla \times \mathbf{B} = \mu _0\mathbf{J} + \mu _0\varepsilon_0 \cfrac{\partial \mathbf{E}}{\partial t }
\end{array}
Input:
::
%Unicode extension support needs to be enabled in settings for this formula.
\begin{array}{l}
{\huge \unicode{8751}}_\mathbb{S} \mathbf{E} \cdot\mathrm{d}s= \cfrac{Q}{\varepsilon_0} \\
{\huge \unicode{8751}}_\mathbb{S} \mathbf{B} \cdot\mathrm{d}s= 0 \\
{\huge \oint}_{\mathbb{L}}^{} \mathbf{E} \cdot \mathrm{d}l=-\cfrac{\mathrm{d}\Phi _{\mathbf{B}}}{\mathrm{d}t } \\
{\huge \oint}_{\mathbb{L}}^{} \mathbf{B} \cdot \mathrm{d}l=\mu_0I+ \mu_0 \varepsilon_0\cfrac{\mathrm{d}\Phi _{\mathbf{E}}}{\mathrm{d}t }
\end{array}
Output:
.. math::
%Unicode extension support needs to be enabled in settings for this formula.
\begin{array}{l}
{\huge \unicode{8751}}_\mathbb{S} \mathbf{E} \cdot\mathrm{d}s= \cfrac{Q}{\varepsilon_0} \\
{\huge \unicode{8751}}_\mathbb{S} \mathbf{B} \cdot\mathrm{d}s= 0 \\
{\huge \oint}_{\mathbb{L}}^{} \mathbf{E} \cdot \mathrm{d}l=-\cfrac{\mathrm{d}\Phi _{\mathbf{B}}}{\mathrm{d}t } \\
{\huge \oint}_{\mathbb{L}}^{} \mathbf{B} \cdot \mathrm{d}l=\mu_0I+ \mu_0 \varepsilon_0\cfrac{\mathrm{d}\Phi _{\mathbf{E}}}{\mathrm{d}t }
\end{array}
Input:
::
\begin{array}{l}
\nabla \cdot \mathbf{D} =\rho _f \\
\nabla \cdot \mathbf{B} = 0 \\
\nabla \times \mathbf{E} = -\cfrac{\partial \mathbf{B}}{\partial t } \\
\nabla \times \mathbf{H} = \mathbf{J}_f + \cfrac{\partial \mathbf{D}}{\partial t }
\end{array}
Output:
.. math::
\begin{array}{l}
\nabla \cdot \mathbf{D} =\rho _f \\
\nabla \cdot \mathbf{B} = 0 \\
\nabla \times \mathbf{E} = -\cfrac{\partial \mathbf{B}}{\partial t } \\
\nabla \times \mathbf{H} = \mathbf{J}_f + \cfrac{\partial \mathbf{D}}{\partial t }
\end{array}
Input:
::
%Unicode extension support needs to be enabled in settings for this formula.
\begin{array}{l}
{\huge \unicode{8751}}_\mathbb{S} \mathbf{D} \cdot\mathrm{d}s= Q_f \\
{\huge \unicode{8751}}_\mathbb{S} \mathbf{B} \cdot\mathrm{d}s= 0 \\
{\huge \oint}_{\mathbb{L}}^{} \mathbf{E} \cdot \mathrm{d}l=-\cfrac{\mathrm{d}\Phi _{\mathbf{B}}}{\mathrm{d}t } \\
{\huge \oint}_{\mathbb{L}}^{} \mathbf{H} \cdot \mathrm{d}l=I_f+\cfrac{\mathrm{d}\Phi _{\mathbf{D}}}{\mathrm{d}t }
\end{array}
Output:
.. math::
%Unicode extension support needs to be enabled in settings for this formula.
\begin{array}{l}
{\huge \unicode{8751}}_\mathbb{S} \mathbf{D} \cdot\mathrm{d}s= Q_f \\
{\huge \unicode{8751}}_\mathbb{S} \mathbf{B} \cdot\mathrm{d}s= 0 \\
{\huge \oint}_{\mathbb{L}}^{} \mathbf{E} \cdot \mathrm{d}l=-\cfrac{\mathrm{d}\Phi _{\mathbf{B}}}{\mathrm{d}t } \\
{\huge \oint}_{\mathbb{L}}^{} \mathbf{H} \cdot \mathrm{d}l=I_f+\cfrac{\mathrm{d}\Phi _{\mathbf{D}}}{\mathrm{d}t }
\end{array}
Chemical
`````````````````````````````````
Input:
::
%This formula requires enabling the mhchem extension support in the 【Settings】.
\ce{SO4^2- + Ba^2+ -> BaSO4 v}
Output:
.. math::
%This formula requires enabling the mhchem extension support in the 【Settings】.
\ce{SO4^2- + Ba^2+ -> BaSO4 v}
Input:
::
\ce{A v B (v) -> B ^ B (^)}
Output:
.. math::
\ce{A v B (v) -> B ^ B (^)}
Input:
::
\ce{Hg^2+ ->[I-] $\underset{\mathrm{red}}{\ce{HgI2}}$ ->[I-] $\underset{\mathrm{red}}{\ce{[Hg^{II}I4]^2-}}$}
Output:
.. math::
\ce{Hg^2+ ->[I-] $\underset{\mathrm{red}}{\ce{HgI2}}$ ->[I-] $\underset{\mathrm{red}}{\ce{[Hg^{II}I4]^2-}}$}
Input:
::
\ce{Zn^2+ <=>[+ 2OH-][+ 2H+] $\underset{\text{amphoteres Hydroxid}}{\ce{Zn(OH)2 v}}$
<=>[+ 2OH-][+ 2H+] $\underset{\text{Hydroxozikat}}{\ce{[Zn(OH)4]^2-}}$}
Output:
.. math::
\ce{Zn^2+ <=>[+ 2OH-][+ 2H+] $\underset{\text{amphoteres Hydroxid}}{\ce{Zn(OH)2 v}}$
<=>[+ 2OH-][+ 2H+] $\underset{\text{Hydroxozikat}}{\ce{[Zn(OH)4]^2-}}$}
Big
---------------------
Input:
::
( \big( \Big( \bigg( \Bigg(
Output:
.. math::
( \big( \Big( \bigg( \Bigg(