Giới thiệu latex

 

latex là 1 chương trình dùng để format document. Nó là: ” document markup language and document preparation system for the TeX typesetting program. Within the typesetting system, its name is styled as ” (-wikipedia).

latex là một hệ thống soạn thảo rất phù hợp với việc tạo ra các tài liệu khoa học và toán học với chất lượng bản in rất cao.
Nó rất mạnh trong việc hiển thị các biểu thức toán học với những ký tự khó như Sigma, Alpha, Square, exponent...
Nó ra đời khoảng chục năm nay, nhưng mau chóng trở thành lạc hậu vì sự tiến triển không người của các Word Editor trên Linux hay Unix như StarOffice hay OpenOffice.

Hàm, biểu tượng, ký tự đặc biệt 

Accents/diacritics

\acute{a} \grave{a} \hat{a} \tilde{a} \breve{a} \acute{a} \grave{a} \hat{a} \tilde{a} \breve{a}\,\!
\check{a} \bar{a} \ddot{a} \dot{a} \check{a} \bar{a} \ddot{a} \dot{a}\!

Standard functions

\sin a \cos b \tan c \sin a \cos b \tan c\!
\sec d \csc e \cot f \sec d \csc e \cot f\,\!
\arcsin h \arccos i \arctan j \arcsin h \arccos i \arctan j\,\!
\sinh k \cosh l \tanh m \coth n\! \sinh k \cosh l \tanh m \coth n\!
\operatorname{sh}\,o\,\operatorname{ch}\,p\,\operatorname{th}\,q\! \operatorname{sh}\,o\,\operatorname{ch}\,p\,\operatorname{th}\,q\!
\operatorname{arsinh}\,r\,\operatorname{arcosh}\,s\,\operatorname{artanh}\,t \operatorname{arsinh}\,r\,\operatorname{arcosh}\,s\,\operatorname{artanh}\,t\,\!
\lim u \limsup v \liminf w \min x \max y\! \lim u \limsup v \liminf w \min x \max y\!
\inf z \sup a \exp b \ln c \lg d \log e \log_{10} f \ker g\! \inf z \sup a \exp b \ln c \lg d \log e \log_{10} f \ker g\!
\deg h \gcd i \Pr j \det k \hom l \arg m \dim n \deg h \gcd i \Pr j \det k \hom l \arg m \dim n\!

Modular arithmetic

s_k \equiv 0 \pmod{m} s_k \equiv 0 \pmod{m}\,\!
a\,\bmod\,b a\,\bmod\,b\,\!
\mid \; \nmid  \mid \; \nmid \!

Derivatives

\nabla \, \partial x \, dx \, \dot x \, \ddot y\, dy/dx\, \frac{dy}{dx}\, \frac{\partial^2 y}{\partial x_1\,\partial x_2} \nabla \, \partial x \, dx \, \dot x \, \ddot y\, dy/dx\, \frac{dy}{dx}\, \frac{\partial^2 y}{\partial x_1\,\partial x_2}

Sets

\forall \exists \nexists \empty \emptyset \varnothing \forall \exists \nexists \empty \emptyset \varnothing\,\!
\in \ni \not\in \notin \subset \subseteq \supset \supseteq \in \ni \not\in \notin \subset \subseteq \supset \supseteq\,\!
\cap \bigcap \cup \bigcup \biguplus \setminus \smallsetminus \cap \bigcap \cup \bigcup \biguplus \setminus \smallsetminus\,\!
\sqsubset \sqsubseteq \sqsupset \sqsupseteq \sqcap \sqcup \bigsqcup \sqsubset \sqsubseteq \sqsupset \sqsupseteq \sqcap \sqcup \bigsqcup\,\!

Operators

+ \oplus \bigoplus \pm \mp - + \oplus \bigoplus \pm \mp - \,\!
\times \otimes \bigotimes \cdot \circ \bullet \bigodot \times \otimes \bigotimes \cdot \circ \bullet \bigodot\,\!
\star * / \div \frac{1}{2} \star * / \div \frac{1}{2}\,\!

Logic

\land (or \and) \wedge \bigwedge \bar{q} \overline{q} \to \And \land \wedge \bigwedge \bar{q} \ \overline{q} \to \And\,\!
\lor \vee \bigvee \lnot \neg q \leftrightarrow \lor \vee \bigvee \lnot \, \neg q \leftrightarrow\,\!

Root

\sqrt{2} \sqrt[n]{x} \sqrt{2} \sqrt[n]{x}\,\!

Relations

\sim \approx \simeq \cong \dot= \overset{\underset{\mathrm{def}}{}}{=} :=\!\, \sim \approx \simeq \cong \dot=  \overset{\underset{\mathrm{def}}{}}{=} :=\!\,
\le < \ll \gg \ge > \equiv \not\equiv \ne \mbox{or} \neq \propto \le < \ll \gg \ge > \equiv \not\equiv \ne \mbox{or} \neq \propto\,\!

Geometric

\Diamond \Box \triangle \angle \perp \parallel 45^\circ \Diamond \, \Box \, \triangle \, \angle \perp \, \parallel \, 45^\circ\,\!

Arrows

\leftarrow (or \gets) \rightarrow (or \to) \nleftarrow \nrightarrow \leftrightarrow \nleftrightarrow \longleftarrow \longrightarrow \longleftrightarrow \leftarrow \rightarrow \nleftarrow \not\to \leftrightarrow \nleftrightarrow \longleftarrow \longrightarrow \longleftrightarrow \,\!
\Leftarrow \Rightarrow \nLeftarrow \nRightarrow \Leftrightarrow \nLeftrightarrow \Longleftarrow \Longrightarrow \Longleftrightarrow (or \iff) \Leftarrow \Rightarrow \nLeftarrow \nRightarrow \Leftrightarrow \nLeftrightarrow \Longleftarrow \Longrightarrow \Longleftrightarrow \!
\uparrow \downarrow \updownarrow \Uparrow \Downarrow \Updownarrow \nearrow \searrow \swarrow \nwarrow \uparrow \downarrow \updownarrow \Uparrow \Downarrow \Updownarrow  \nearrow \searrow \swarrow \nwarrow \!
\rightharpoonup \rightharpoondown \leftharpoonup \leftharpoondown \upharpoonleft \upharpoonright \downharpoonleft \downharpoonright \rightleftharpoons \leftrightharpoons \rightharpoonup \rightharpoondown \leftharpoonup \leftharpoondown \upharpoonleft \upharpoonright \downharpoonleft \downharpoonright \rightleftharpoons \leftrightharpoons \,\!
\curvearrowleft \circlearrowleft \Lsh \upuparrows \rightrightarrows \rightleftarrows \Rrightarrow \rightarrowtail \looparrowright \curvearrowleft \circlearrowleft \Lsh \upuparrows \rightrightarrows \rightleftarrows \Rrightarrow \rightarrowtail \looparrowright \,\!
\curvearrowright \circlearrowright \Rsh \downdownarrows \leftleftarrows \leftrightarrows \Lleftarrow \leftarrowtail \looparrowleft \curvearrowright \circlearrowright \Rsh \downdownarrows \leftleftarrows \leftrightarrows \Lleftarrow \leftarrowtail \looparrowleft \,\!
\mapsto \longmapsto \hookrightarrow \hookleftarrow \multimap \leftrightsquigarrow \rightsquigarrow \twoheadrightarrow \twoheadleftarrow \mapsto \longmapsto \hookrightarrow \hookleftarrow \multimap \leftrightsquigarrow \rightsquigarrow \twoheadrightarrow \twoheadleftarrow\,\!

Special

\And \eth \S \P \% \dagger \ddagger \ldots \cdots \And \eth \S \P \% \dagger \ddagger \ldots \cdots\,\!
\smile \frown \wr \triangleleft \triangleright \infty \bot \top \smile \frown \wr \triangleleft \triangleright \infty \bot \top\,\!
\vdash \vDash \Vdash \models \lVert \rVert \imath \hbar \vdash \vDash \Vdash \models \lVert \rVert \imath \hbar\,\!
\ell \mho \Finv \Re \Im \wp \complement \ell \mho \Finv \Re \Im \wp \complement\,\!
\diamondsuit \heartsuit \clubsuit \spadesuit \Game \flat \natural \sharp \diamondsuit \heartsuit \clubsuit \spadesuit \Game \flat \natural \sharp\,\!

Unsorted (new stuff)

\vartriangle \triangledown \lozenge \circledS \measuredangle \Bbbk \backprime \blacktriangle \blacktriangledown  \vartriangle \triangledown \lozenge \circledS \measuredangle \Bbbk \backprime \blacktriangle \blacktriangledown
\blacksquare \blacklozenge \bigstar \sphericalangle \diagup \diagdown \dotplus \Cap \Cup \barwedge  \blacksquare \blacklozenge \bigstar \sphericalangle \diagup \diagdown \dotplus \Cap \Cup \barwedge\!
\veebar \doublebarwedge \boxminus \boxtimes \boxdot \boxplus \divideontimes \ltimes \rtimes \leftthreetimes  \veebar \doublebarwedge \boxminus \boxtimes \boxdot \boxplus \divideontimes \ltimes \rtimes \leftthreetimes
\rightthreetimes \curlywedge \curlyvee \circleddash \circledast \circledcirc \centerdot \intercal \leqq \leqslant  \rightthreetimes \curlywedge \curlyvee \circleddash \circledast \circledcirc \centerdot \intercal \leqq \leqslant
\eqslantless \lessapprox \approxeq \lessdot \lll \lessgtr \lesseqgtr \lesseqqgtr \doteqdot \risingdotseq  \eqslantless \lessapprox \approxeq \lessdot \lll \lessgtr \lesseqgtr \lesseqqgtr \doteqdot \risingdotseq
\fallingdotseq \backsim \backsimeq \subseteqq \Subset \preccurlyeq \curlyeqprec \precsim \precapprox \vartriangleleft  \fallingdotseq \backsim \backsimeq \subseteqq \Subset \preccurlyeq \curlyeqprec \precsim \precapprox \vartriangleleft
\Vvdash \bumpeq \Bumpeq \geqq \geqslant \eqslantgtr \gtrsim \gtrapprox \eqsim \gtrdot  \Vvdash \bumpeq \Bumpeq \geqq \geqslant \eqslantgtr \gtrsim \gtrapprox \eqsim \gtrdot
\ggg \gtrless \gtreqless \gtreqqless \eqcirc \circeq \triangleq \thicksim \thickapprox \supseteqq  \ggg \gtrless \gtreqless \gtreqqless \eqcirc \circeq \triangleq \thicksim \thickapprox \supseteqq
\Supset \succcurlyeq \curlyeqsucc \succsim \succapprox \vartriangleright \shortmid \shortparallel \between \pitchfork  \Supset \succcurlyeq \curlyeqsucc \succsim \succapprox \vartriangleright \shortmid \shortparallel \between \pitchfork
\varpropto \blacktriangleleft \therefore \backepsilon \blacktriangleright \because \nleqslant \nleqq \lneq \lneqq  \varpropto \blacktriangleleft \therefore \backepsilon \blacktriangleright \because \nleqslant \nleqq \lneq \lneqq
\lvertneqq \lnsim \lnapprox \nprec \npreceq \precneqq \precnsim \precnapprox \nsim \nshortmid  \lvertneqq \lnsim \lnapprox \nprec \npreceq \precneqq \precnsim \precnapprox \nsim \nshortmid
\nvdash \nVdash \ntriangleleft \ntrianglelefteq \nsubseteq \nsubseteqq \varsubsetneq \subsetneqq \varsubsetneqq \ngtr  \nvdash \nVdash \ntriangleleft \ntrianglelefteq \nsubseteq \nsubseteqq \varsubsetneq \subsetneqq \varsubsetneqq \ngtr
\subsetneq \subsetneq
\ngeqslant \ngeqq \gneq \gneqq \gvertneqq \gnsim \gnapprox \nsucc \nsucceq \succneqq  \ngeqslant \ngeqq \gneq \gneqq \gvertneqq \gnsim \gnapprox \nsucc \nsucceq \succneqq
\succnsim \succnapprox \ncong \nshortparallel \nparallel \nvDash \nVDash \ntriangleright \ntrianglerighteq \nsupseteq  \succnsim \succnapprox \ncong \nshortparallel \nparallel \nvDash \nVDash \ntriangleright \ntrianglerighteq \nsupseteq
\nsupseteqq \varsupsetneq \supsetneqq \varsupsetneqq  \nsupseteqq \varsupsetneq \supsetneqq \varsupsetneqq
\jmath \surd \ast \uplus \diamond \bigtriangleup \bigtriangledown \ominus \jmath \surd \ast \uplus \diamond \bigtriangleup \bigtriangledown \ominus\,\!
\oslash \odot \bigcirc \amalg \prec \succ \preceq \succeq \oslash \odot \bigcirc \amalg \prec \succ \preceq \succeq\,\!
\dashv \asymp \doteq \parallel \dashv \asymp \doteq \parallel\,\!
\ulcorner \urcorner \llcorner \lrcorner \ulcorner \urcorner \llcorner \lrcorner

 

Các biểu thức phức tạp:

Subscripts, superscripts, integrals

Feature Syntax How it looks rendered
HTML PNG
Superscript a^2 a2 a^2 \,\!
Subscript a_2 a2 a_2 \,\!
Grouping a^{2+2} a2 + 2 a^{2+2}\,\!
a_{i,j} ai,j a_{i,j}\,\!
Combining sub & super without and with horizontal separation x_2^3 x_2^3 x_2^3 \,\!
{x_2}^3 {x_2}^3 {x_2}^3 \,\!
Super super 10^{10^{ \,\!{8} } 10^{10^{ \,\! 8 } }
Super super 10^{10^{ \overset{8}{} }} 10^{10^{ \overset{8}{} }}
Super super (wrong in HTML in some browsers) 10^{10^8} 10^{10^8}
Preceding and/or additional sub & super \sideset{_1^2}{_3^4}\prod_a^b \sideset{_1^2}{_3^4}\prod_a^b
{}_1^2\!\Omega_3^4 {}_1^2\!\Omega_3^4
Stacking \overset{\alpha}{\omega} \overset{\alpha}{\omega}
\underset{\alpha}{\omega} \underset{\alpha}{\omega}
\overset{\alpha}{\underset{\gamma}{\omega}} \overset{\alpha}{\underset{\gamma}{\omega}}
\stackrel{\alpha}{\omega} \stackrel{\alpha}{\omega}
Derivative (forced PNG) x', y'', f', f''\!   x', y'', f', f''\!
Derivative (f in italics may overlap primes in HTML) x', y'', f', f'' x',y'',f',f'' x', y'', f', f''\!
Derivative (wrong in HTML) x^\prime, y^{\prime\prime} x^\prime, y^{\prime\prime} x^\prime, y^{\prime\prime}\,\!
Derivative (wrong in PNG) x\prime, y\prime\prime x\prime, y\prime\prime x\prime, y\prime\prime\,\!
Derivative dots \dot{x}, \ddot{x} \dot{x}, \ddot{x}
Underlines, overlines, vectors \hat a \ \bar b \ \vec c \hat a \ \bar b \ \vec c
\overrightarrow{a b} \ \overleftarrow{c d} \ \widehat{d e f} \overrightarrow{a b} \ \overleftarrow{c d} \ \widehat{d e f}
\overline{g h i} \ \underline{j k l} \overline{g h i} \ \underline{j k l}
Arrows A \xleftarrow{n+\mu-1} B \xrightarrow[T]{n\pm i-1} C  A \xleftarrow{n+\mu-1} B \xrightarrow[T]{n\pm i-1} C
Overbraces \overbrace{ 1+2+\cdots+100 }^{5050} \overbrace{ 1+2+\cdots+100 }^{5050}
Underbraces \underbrace{ a+b+\cdots+z }_{26} \underbrace{ a+b+\cdots+z }_{26}
Sum \sum_{k=1}^N k^2 \sum_{k=1}^N k^2
Sum (force \textstyle) \textstyle \sum_{k=1}^N k^2 \textstyle \sum_{k=1}^N k^2
Sum in a fraction (default \textstyle) \frac{\sum_{k=1}^N k^2}{a} \frac{\sum_{k=1}^N k^2}{a}
Sum in a fraction (force \displaystyle) \frac{\displaystyle \sum_{k=1}^N k^2}{a} \frac{\displaystyle \sum_{k=1}^N k^2}{a}
Product \prod_{i=1}^N x_i \prod_{i=1}^N x_i
Product (force \textstyle) \textstyle \prod_{i=1}^N x_i \textstyle \prod_{i=1}^N x_i
Coproduct \coprod_{i=1}^N x_i \coprod_{i=1}^N x_i
Coproduct (force \textstyle) \textstyle \coprod_{i=1}^N x_i \textstyle \coprod_{i=1}^N x_i
Limit \lim_{n \to \infty}x_n \lim_{n \to \infty}x_n
Limit (force \textstyle) \textstyle \lim_{n \to \infty}x_n \textstyle \lim_{n \to \infty}x_n
Integral \int\limits_{1}^{3}\frac{e^3/x}{x^2}\, dx \int\limits_{1}^{3}\frac{e^3/x}{x^2}\, dx
Integral (alternate limits style) \int_{1}^{3}\frac{e^3/x}{x^2}\, dx \int_{1}^{3}\frac{e^3/x}{x^2}\, dx
Integral (force \textstyle) \textstyle \int\limits_{-N}^{N} e^x\, dx \textstyle \int\limits_{-N}^{N} e^x\, dx
Integral (force \textstyle, alternate limits style) \textstyle \int_{-N}^{N} e^x\, dx \textstyle \int_{-N}^{N} e^x\, dx
Double integral \iint\limits_D \, dx\,dy \iint\limits_D \, dx\,dy
Triple integral \iiint\limits_E \, dx\,dy\,dz \iiint\limits_E \, dx\,dy\,dz
Quadruple integral \iiiint\limits_F \, dx\,dy\,dz\,dt \iiiint\limits_F \, dx\,dy\,dz\,dt
Line or path integral \int_C x^3\, dx + 4y^2\, dy \int_C x^3\, dx + 4y^2\, dy
Closed line or path integral \oint_C x^3\, dx + 4y^2\, dy \oint_C x^3\, dx + 4y^2\, dy
Intersections \bigcap_1^n p \bigcap_1^n p
Unions \bigcup_1^k p \bigcup_1^k p

Fractions, matrices, multilines

Feature Syntax How it looks rendered
Fractions \frac{2}{4}=0.5 \frac{2}{4}=0.5
Small fractions \tfrac{2}{4} = 0.5 \tfrac{2}{4} = 0.5
Large (normal) fractions \dfrac{2}{4} = 0.5 \qquad \dfrac{2}{c + \dfrac{2}{d + \dfrac{2}{4}}} = a \dfrac{2}{4} = 0.5 \qquad \dfrac{2}{c + \dfrac{2}{d + \dfrac{2}{4}}} = a
Large (nested) fractions \cfrac{2}{c + \cfrac{2}{d + \cfrac{2}{4}}} = a \cfrac{2}{c + \cfrac{2}{d + \cfrac{2}{4}}} = a
Binomial coefficients \binom{n}{k} \binom{n}{k}
Small binomial coefficients \tbinom{n}{k} \tbinom{n}{k}
Large (normal) binomial coefficients \dbinom{n}{k} \dbinom{n}{k}
Matrices
\begin{matrix}
  x & y \\
  z & v 
\end{matrix}
\begin{matrix} x & y \\ z & v
\end{matrix}
\begin{vmatrix}
  x & y \\
  z & v 
\end{vmatrix}
\begin{vmatrix} x & y \\ z & v
\end{vmatrix}
\begin{Vmatrix}
  x & y \\
  z & v
\end{Vmatrix}
\begin{Vmatrix} x & y \\ z & v
\end{Vmatrix}
\begin{bmatrix}
  0      & \cdots & 0      \\
  \vdots & \ddots & \vdots \\ 
  0      & \cdots & 0
\end{bmatrix}
\begin{bmatrix} 0 & \cdots & 0 \\ \vdots
& \ddots & \vdots \\ 0 & \cdots &
0\end{bmatrix}
\begin{Bmatrix}
  x & y \\
  z & v
\end{Bmatrix}
\begin{Bmatrix} x & y \\ z & v
\end{Bmatrix}
\begin{pmatrix}
  x & y \\
  z & v 
\end{pmatrix}
\begin{pmatrix} x & y \\ z & v
\end{pmatrix}
\bigl( \begin{smallmatrix}
  a&b\\ c&d
\end{smallmatrix} \bigr)
\bigl( \begin{smallmatrix}
  a&b\\ c&d
\end{smallmatrix} \bigr)
Case distinctions
f(n) = 
\begin{cases} 
  n/2,  & \mbox{if }n\mbox{ is even} \\
  3n+1, & \mbox{if }n\mbox{ is odd} 
\end{cases}
f(n) = 
\begin{cases}
  n/2,  & \mbox{if }n\mbox{ is even} \\ 
  3n+1, & \mbox{if }n\mbox{ is odd} 
\end{cases}
Multiline equations
\begin{align}
 f(x) & = (a+b)^2 \\
      & = a^2+2ab+b^2 \\
\end{align}
\begin{align}
 f(x) & = (a+b)^2 \\
      & = a^2+2ab+b^2 \\
\end{align}
\begin{alignat}{2}
 f(x) & = (a-b)^2 \\
      & = a^2-2ab+b^2 \\
\end{alignat}
\begin{alignat}{2}
 f(x) & = (a-b)^2 \\
      & = a^2-2ab+b^2 \\
\end{alignat}
Multiline equations (must define number of colums used ({lcr}) (should not be used unless needed)
\begin{array}{lcl}
  z        & = & a \\
  f(x,y,z) & = & x + y + z  
\end{array}
\begin{array}{lcl}
  z        & = & a \\
  f(x,y,z) & = & x + y + z  
\end{array}
Multiline equations (more)
\begin{array}{lcr}
  z        & = & a \\
  f(x,y,z) & = & x + y + z     
\end{array}
\begin{array}{lcr}
  z        & = & a \\
  f(x,y,z) & = & x + y + z     
\end{array}
Breaking up a long expression so that it wraps when necessary, at the expense of destroying correct spacing
<math>f(x) \,\!</math>
<math>= \sum_{n=0}^\infty a_n x^n </math>
<math>= a_0+a_1x+a_2x^2+\cdots</math>

f(x) \,\!= \sum_{n=0}^\infty a_n x^n = a_0 +a_1x+a_2x^2+\cdots

Simultaneous equations
\begin{cases}
    3x + 5y +  z \\
    7x - 2y + 4z \\
   -6x + 3y + 2z 
\end{cases}
\begin{cases} 3x + 5y + z \\ 7x - 2y + 4z \\ -6x + 3y + 2z \end{cases}
Arrays
\begin{array}{|c|c||c|} a & b & S \\
\hline
0&0&1\\
0&1&1\\
1&0&1\\
1&1&0\\
\end{array}
\begin{array}{|c|c||c|} a & b & S \\
\hline
0&0&1\\
0&1&1\\
1&0&1\\
1&1&0\\
\end{array}

Parenthesizing big expressions, brackets, bars

Feature Syntax How it looks rendered
Bad ( \frac{1}{2} ) ( \frac{1}{2} )
Good \left ( \frac{1}{2} \right ) \left ( \frac{1}{2} \right )

You can use various delimiters with \left and \right:

Feature Syntax How it looks rendered
Parentheses \left ( \frac{a}{b} \right ) \left ( \frac{a}{b} \right )
Brackets \left [ \frac{a}{b} \right ] \quad \left \lbrack \frac{a}{b} \right \rbrack \left [ \frac{a}{b} \right ] \quad \left \lbrack \frac{a}{b} \right \rbrack
Braces \left \{ \frac{a}{b} \right \} \quad \left \lbrace \frac{a}{b} \right \rbrace \left \{ \frac{a}{b} \right \} \quad \left \lbrace \frac{a}{b} \right \rbrace
Angle brackets \left \langle \frac{a}{b} \right \rangle \left \langle \frac{a}{b} \right \rangle
Bars and double bars \left | \frac{a}{b} \right \vert \left \Vert \frac{c}{d} \right \| \left | \frac{a}{b} \right \vert \left \Vert \frac{c}{d} \right \|
Floor and ceiling functions: \left \lfloor \frac{a}{b} \right \rfloor \left \lceil \frac{c}{d} \right \rceil \left \lfloor \frac{a}{b} \right \rfloor \left \lceil \frac{c}{d} \right \rceil
Slashes and backslashes \left / \frac{a}{b} \right \backslash \left / \frac{a}{b} \right \backslash
Up, down and up-down arrows \left \uparrow \frac{a}{b} \right \downarrow \quad \left \Uparrow \frac{a}{b} \right \Downarrow \quad \left \updownarrow \frac{a}{b} \right \Updownarrow \left \uparrow \frac{a}{b} \right \downarrow \quad \left \Uparrow \frac{a}{b} \right \Downarrow \quad \left \updownarrow \frac{a}{b} \right \Updownarrow
Delimiters can be mixed,
as long as \left and \right match
\left [ 0,1 \right )</code> <br/> <code>\left \langle \psi \right | \left [ 0,1 \right )
\left \langle \psi \right |
Use \left. and \right. if you don't
want a delimiter to appear:
\left . \frac{A}{B} \right \} \to X \left . \frac{A}{B} \right \} \to X
Size of the delimiters \big( \Big( \bigg( \Bigg( \dots \Bigg] \bigg] \Big] \big]/ \big( \Big( \bigg( \Bigg( \dots \Bigg] \bigg] \Big] \big]
\big\{ \Big\{ \bigg\{ \Bigg\{ \dots \Bigg\rangle \bigg\rangle \Big\rangle \big\rangle \big\{ \Big\{ \bigg\{ \Bigg\{ \dots \Bigg\rangle \bigg\rangle \Big\rangle \big\rangle
\big\| \Big\| \bigg\| \Bigg\| \dots \Bigg| \bigg| \Big| \big| \big\| \Big\| \bigg\| \Bigg\| \dots \Bigg| \bigg| \Big| \big|
\big\lfloor \Big\lfloor \bigg\lfloor \Bigg\lfloor \dots \Bigg\rceil \bigg\rceil \Big\rceil \big\rceil \big\lfloor \Big\lfloor \bigg\lfloor \Bigg\lfloor \dots \Bigg\rceil \bigg\rceil \Big\rceil \big\rceil
\big\uparrow \Big\uparrow \bigg\uparrow \Bigg\uparrow \dots \Bigg\Downarrow \bigg\Downarrow \Big\Downarrow \big\Downarrow \big\uparrow \Big\uparrow \bigg\uparrow \Bigg\uparrow \dots \Bigg\Downarrow \bigg\Downarrow \Big\Downarrow \big\Downarrow
\big\updownarrow \Big\updownarrow \bigg\updownarrow \Bigg\updownarrow \dots \Bigg\Updownarrow \bigg\Updownarrow \Big\Updownarrow \big\Updownarrow \big\updownarrow \Big\updownarrow \bigg\updownarrow \Bigg\updownarrow \dots \Bigg\Updownarrow \bigg\Updownarrow \Big\Updownarrow \big\Updownarrow
\big / \Big / \bigg / \Bigg / \dots \Bigg\backslash \bigg\backslash \Big\backslash \big\backslash \big / \Big / \bigg / \Bigg / \dots \Bigg\backslash \bigg\backslash \Big\backslash \big\backslash

Alphabets and typefaces

Texvc cannot render arbitrary Unicode characters. Those it can handle can be entered by the expressions below. For others, such as Cyrillic, they can be entered as Unicode or HTML entities in running text, but cannot be used in displayed formulas.

Greek alphabet
\Alpha \Beta \Gamma \Delta \Epsilon \Zeta \Alpha \Beta \Gamma \Delta \Epsilon \Zeta \,\!
\Eta \Theta \Iota \Kappa \Lambda \Mu \Eta \Theta \Iota \Kappa \Lambda \Mu \,\!
\Nu \Xi \Pi \Rho \Sigma \Tau \Nu \Xi \Pi \Rho \Sigma \Tau\,\!
\Upsilon \Phi \Chi \Psi \Omega \Upsilon \Phi \Chi \Psi \Omega \,\!
\alpha \beta \gamma \delta \epsilon \zeta \alpha \beta \gamma \delta \epsilon \zeta \,\!
\eta \theta \iota \kappa \lambda \mu \eta \theta \iota \kappa \lambda \mu \,\!
\nu \xi \pi \rho \sigma \tau \nu \xi \pi \rho \sigma \tau \,\!
\upsilon \phi \chi \psi \omega \upsilon \phi \chi \psi \omega \,\!
\varepsilon \digamma \vartheta \varkappa \varepsilon \digamma \vartheta \varkappa \,\!
\varpi \varrho \varsigma \varphi \varpi \varrho \varsigma \varphi\,\!
Blackboard bold/scripts
\mathbb{A} \mathbb{B} \mathbb{C} \mathbb{D} \mathbb{E} \mathbb{F} \mathbb{G} \mathbb{A} \mathbb{B} \mathbb{C} \mathbb{D} \mathbb{E} \mathbb{F} \mathbb{G} \,\!
\mathbb{H} \mathbb{I} \mathbb{J} \mathbb{K} \mathbb{L} \mathbb{M} \mathbb{H} \mathbb{I} \mathbb{J} \mathbb{K} \mathbb{L} \mathbb{M} \,\!
\mathbb{N} \mathbb{O} \mathbb{P} \mathbb{Q} \mathbb{R} \mathbb{S} \mathbb{T} \mathbb{N} \mathbb{O} \mathbb{P} \mathbb{Q} \mathbb{R} \mathbb{S} \mathbb{T} \,\!
\mathbb{U} \mathbb{V} \mathbb{W} \mathbb{X} \mathbb{Y} \mathbb{Z} \mathbb{U} \mathbb{V} \mathbb{W} \mathbb{X} \mathbb{Y} \mathbb{Z}\,\!
Boldface (vectors)
\mathbf{A} \mathbf{B} \mathbf{C} \mathbf{D} \mathbf{E} \mathbf{F} \mathbf{G} \mathbf{A} \mathbf{B} \mathbf{C} \mathbf{D} \mathbf{E} \mathbf{F} \mathbf{G} \,\!
\mathbf{H} \mathbf{I} \mathbf{J} \mathbf{K} \mathbf{L} \mathbf{M} \mathbf{H} \mathbf{I} \mathbf{J} \mathbf{K} \mathbf{L} \mathbf{M} \,\!
\mathbf{N} \mathbf{O} \mathbf{P} \mathbf{Q} \mathbf{R} \mathbf{S} \mathbf{T} \mathbf{N} \mathbf{O} \mathbf{P} \mathbf{Q} \mathbf{R} \mathbf{S} \mathbf{T} \,\!
\mathbf{U} \mathbf{V} \mathbf{W} \mathbf{X} \mathbf{Y} \mathbf{Z} \mathbf{U} \mathbf{V} \mathbf{W} \mathbf{X} \mathbf{Y} \mathbf{Z} \,\!
\mathbf{a} \mathbf{b} \mathbf{c} \mathbf{d} \mathbf{e} \mathbf{f} \mathbf{g} \mathbf{a} \mathbf{b} \mathbf{c} \mathbf{d} \mathbf{e} \mathbf{f} \mathbf{g} \,\!
\mathbf{h} \mathbf{i} \mathbf{j} \mathbf{k} \mathbf{l} \mathbf{m} \mathbf{h} \mathbf{i} \mathbf{j} \mathbf{k} \mathbf{l} \mathbf{m} \,\!
\mathbf{n} \mathbf{o} \mathbf{p} \mathbf{q} \mathbf{r} \mathbf{s} \mathbf{t} \mathbf{n} \mathbf{o} \mathbf{p} \mathbf{q} \mathbf{r} \mathbf{s} \mathbf{t} \,\!
\mathbf{u} \mathbf{v} \mathbf{w} \mathbf{x} \mathbf{y} \mathbf{z} \mathbf{u} \mathbf{v} \mathbf{w} \mathbf{x} \mathbf{y} \mathbf{z} \,\!
\mathbf{0} \mathbf{1} \mathbf{2} \mathbf{3} \mathbf{4} \mathbf{0} \mathbf{1} \mathbf{2} \mathbf{3} \mathbf{4} \,\!
\mathbf{5} \mathbf{6} \mathbf{7} \mathbf{8} \mathbf{9} \mathbf{5} \mathbf{6} \mathbf{7} \mathbf{8} \mathbf{9}\,\!
Boldface (Greek)
\boldsymbol{\Alpha} \boldsymbol{\Beta} \boldsymbol{\Gamma} \boldsymbol{\Delta} \boldsymbol{\Epsilon} \boldsymbol{\Zeta} \boldsymbol{\Alpha} \boldsymbol{\Beta} \boldsymbol{\Gamma} \boldsymbol{\Delta} \boldsymbol{\Epsilon} \boldsymbol{\Zeta} \,\!
\boldsymbol{\Eta} \boldsymbol{\Theta} \boldsymbol{\Iota} \boldsymbol{\Kappa} \boldsymbol{\Lambda} \boldsymbol{\Mu} \boldsymbol{\Eta} \boldsymbol{\Theta} \boldsymbol{\Iota} \boldsymbol{\Kappa} \boldsymbol{\Lambda} \boldsymbol{\Mu}\,\!
\boldsymbol{\Nu} \boldsymbol{\Xi} \boldsymbol{\Pi} \boldsymbol{\Rho} \boldsymbol{\Sigma} \boldsymbol{\Tau} \boldsymbol{\Nu} \boldsymbol{\Xi} \boldsymbol{\Pi} \boldsymbol{\Rho} \boldsymbol{\Sigma} \boldsymbol{\Tau}\,\!
\boldsymbol{\Upsilon} \boldsymbol{\Phi} \boldsymbol{\Chi} \boldsymbol{\Psi} \boldsymbol{\Omega} \boldsymbol{\Upsilon} \boldsymbol{\Phi} \boldsymbol{\Chi} \boldsymbol{\Psi} \boldsymbol{\Omega}\,\!
\boldsymbol{\alpha} \boldsymbol{\beta} \boldsymbol{\gamma} \boldsymbol{\delta} \boldsymbol{\epsilon} \boldsymbol{\zeta} \boldsymbol{\alpha} \boldsymbol{\beta} \boldsymbol{\gamma} \boldsymbol{\delta} \boldsymbol{\epsilon} \boldsymbol{\zeta}\,\!
\boldsymbol{\eta} \boldsymbol{\theta} \boldsymbol{\iota} \boldsymbol{\kappa} \boldsymbol{\lambda} \boldsymbol{\mu} \boldsymbol{\eta} \boldsymbol{\theta} \boldsymbol{\iota} \boldsymbol{\kappa} \boldsymbol{\lambda} \boldsymbol{\mu}\,\!
\boldsymbol{\nu} \boldsymbol{\xi} \boldsymbol{\pi} \boldsymbol{\rho} \boldsymbol{\sigma} \boldsymbol{\tau} \boldsymbol{\nu} \boldsymbol{\xi} \boldsymbol{\pi} \boldsymbol{\rho} \boldsymbol{\sigma} \boldsymbol{\tau}\,\!
\boldsymbol{\upsilon} \boldsymbol{\phi} \boldsymbol{\chi} \boldsymbol{\psi} \boldsymbol{\omega} \boldsymbol{\upsilon} \boldsymbol{\phi} \boldsymbol{\chi} \boldsymbol{\psi} \boldsymbol{\omega}\,\!
\boldsymbol{\varepsilon} \boldsymbol{\digamma} \boldsymbol{\vartheta} \boldsymbol{\varkappa} \boldsymbol{\varepsilon} \boldsymbol{\digamma} \boldsymbol{\vartheta} \boldsymbol{\varkappa} \,\!
\boldsymbol{\varpi} \boldsymbol{\varrho} \boldsymbol{\varsigma} \boldsymbol{\varphi} \boldsymbol{\varpi} \boldsymbol{\varrho} \boldsymbol{\varsigma} \boldsymbol{\varphi}\,\!
Italics
\mathit{A} \mathit{B} \mathit{C} \mathit{D} \mathit{E} \mathit{F} \mathit{G} \mathit{A} \mathit{B} \mathit{C} \mathit{D} \mathit{E} \mathit{F} \mathit{G} \,\!
\mathit{H} \mathit{I} \mathit{J} \mathit{K} \mathit{L} \mathit{M} \mathit{H} \mathit{I} \mathit{J} \mathit{K} \mathit{L} \mathit{M} \,\!
\mathit{N} \mathit{O} \mathit{P} \mathit{Q} \mathit{R} \mathit{S} \mathit{T} \mathit{N} \mathit{O} \mathit{P} \mathit{Q} \mathit{R} \mathit{S} \mathit{T} \,\!
\mathit{U} \mathit{V} \mathit{W} \mathit{X} \mathit{Y} \mathit{Z} \mathit{U} \mathit{V} \mathit{W} \mathit{X} \mathit{Y} \mathit{Z} \,\!
\mathit{a} \mathit{b} \mathit{c} \mathit{d} \mathit{e} \mathit{f} \mathit{g} \mathit{a} \mathit{b} \mathit{c} \mathit{d} \mathit{e} \mathit{f} \mathit{g} \,\!
\mathit{h} \mathit{i} \mathit{j} \mathit{k} \mathit{l} \mathit{m} \mathit{h} \mathit{i} \mathit{j} \mathit{k} \mathit{l} \mathit{m} \,\!
\mathit{n} \mathit{o} \mathit{p} \mathit{q} \mathit{r} \mathit{s} \mathit{t} \mathit{n} \mathit{o} \mathit{p} \mathit{q} \mathit{r} \mathit{s} \mathit{t} \,\!
\mathit{u} \mathit{v} \mathit{w} \mathit{x} \mathit{y} \mathit{z} \mathit{u} \mathit{v} \mathit{w} \mathit{x} \mathit{y} \mathit{z} \,\!
\mathit{0} \mathit{1} \mathit{2} \mathit{3} \mathit{4} \mathit{0} \mathit{1} \mathit{2} \mathit{3} \mathit{4} \,\!
\mathit{5} \mathit{6} \mathit{7} \mathit{8} \mathit{9} \mathit{5} \mathit{6} \mathit{7} \mathit{8} \mathit{9}\,\!
Roman typeface
\mathrm{A} \mathrm{B} \mathrm{C} \mathrm{D} \mathrm{E} \mathrm{F} \mathrm{G} \mathrm{A} \mathrm{B} \mathrm{C} \mathrm{D} \mathrm{E} \mathrm{F} \mathrm{G} \,\!
\mathrm{H} \mathrm{I} \mathrm{J} \mathrm{K} \mathrm{L} \mathrm{M} \mathrm{H} \mathrm{I} \mathrm{J} \mathrm{K} \mathrm{L} \mathrm{M} \,\!
\mathrm{N} \mathrm{O} \mathrm{P} \mathrm{Q} \mathrm{R} \mathrm{S} \mathrm{T} \mathrm{N} \mathrm{O} \mathrm{P} \mathrm{Q} \mathrm{R} \mathrm{S} \mathrm{T} \,\!
\mathrm{U} \mathrm{V} \mathrm{W} \mathrm{X} \mathrm{Y} \mathrm{Z} \mathrm{U} \mathrm{V} \mathrm{W} \mathrm{X} \mathrm{Y} \mathrm{Z} \,\!
\mathrm{a} \mathrm{b} \mathrm{c} \mathrm{d} \mathrm{e} \mathrm{f} \mathrm{g} \mathrm{a} \mathrm{b} \mathrm{c} \mathrm{d} \mathrm{e} \mathrm{f} \mathrm{g}\,\!
\mathrm{h} \mathrm{i} \mathrm{j} \mathrm{k} \mathrm{l} \mathrm{m} \mathrm{h} \mathrm{i} \mathrm{j} \mathrm{k} \mathrm{l} \mathrm{m} \,\!
\mathrm{n} \mathrm{o} \mathrm{p} \mathrm{q} \mathrm{r} \mathrm{s} \mathrm{t} \mathrm{n} \mathrm{o} \mathrm{p} \mathrm{q} \mathrm{r} \mathrm{s} \mathrm{t} \,\!
\mathrm{u} \mathrm{v} \mathrm{w} \mathrm{x} \mathrm{y} \mathrm{z} \mathrm{u} \mathrm{v} \mathrm{w} \mathrm{x} \mathrm{y} \mathrm{z} \,\!
\mathrm{0} \mathrm{1} \mathrm{2} \mathrm{3} \mathrm{4} \mathrm{0} \mathrm{1} \mathrm{2} \mathrm{3} \mathrm{4} \,\!
\mathrm{5} \mathrm{6} \mathrm{7} \mathrm{8} \mathrm{9} \mathrm{5} \mathrm{6} \mathrm{7} \mathrm{8} \mathrm{9}\,\!
Fraktur typeface
\mathfrak{A} \mathfrak{B} \mathfrak{C} \mathfrak{D} \mathfrak{E} \mathfrak{F} \mathfrak{G} \mathfrak{A} \mathfrak{B} \mathfrak{C} \mathfrak{D} \mathfrak{E} \mathfrak{F} \mathfrak{G} \,\!
\mathfrak{H} \mathfrak{I} \mathfrak{J} \mathfrak{K} \mathfrak{L} \mathfrak{M} \mathfrak{H} \mathfrak{I} \mathfrak{J} \mathfrak{K} \mathfrak{L} \mathfrak{M} \,\!
\mathfrak{N} \mathfrak{O} \mathfrak{P} \mathfrak{Q} \mathfrak{R} \mathfrak{S} \mathfrak{T} \mathfrak{N} \mathfrak{O} \mathfrak{P} \mathfrak{Q} \mathfrak{R} \mathfrak{S} \mathfrak{T} \,\!
\mathfrak{U} \mathfrak{V} \mathfrak{W} \mathfrak{X} \mathfrak{Y} \mathfrak{Z} \mathfrak{U} \mathfrak{V} \mathfrak{W} \mathfrak{X} \mathfrak{Y} \mathfrak{Z} \,\!
\mathfrak{a} \mathfrak{b} \mathfrak{c} \mathfrak{d} \mathfrak{e} \mathfrak{f} \mathfrak{g} \mathfrak{a} \mathfrak{b} \mathfrak{c} \mathfrak{d} \mathfrak{e} \mathfrak{f} \mathfrak{g} \,\!
\mathfrak{h} \mathfrak{i} \mathfrak{j} \mathfrak{k} \mathfrak{l} \mathfrak{m} \mathfrak{h} \mathfrak{i} \mathfrak{j} \mathfrak{k} \mathfrak{l} \mathfrak{m} \,\!
\mathfrak{n} \mathfrak{o} \mathfrak{p} \mathfrak{q} \mathfrak{r} \mathfrak{s} \mathfrak{t} \mathfrak{n} \mathfrak{o} \mathfrak{p} \mathfrak{q} \mathfrak{r} \mathfrak{s} \mathfrak{t} \,\!
\mathfrak{u} \mathfrak{v} \mathfrak{w} \mathfrak{x} \mathfrak{y} \mathfrak{z} \mathfrak{u} \mathfrak{v} \mathfrak{w} \mathfrak{x} \mathfrak{y} \mathfrak{z} \,\!
\mathfrak{0} \mathfrak{1} \mathfrak{2} \mathfrak{3} \mathfrak{4} \mathfrak{0} \mathfrak{1} \mathfrak{2} \mathfrak{3} \mathfrak{4} \,\!
\mathfrak{5} \mathfrak{6} \mathfrak{7} \mathfrak{8} \mathfrak{9} \mathfrak{5} \mathfrak{6} \mathfrak{7} \mathfrak{8} \mathfrak{9}\,\!
Calligraphy/script
\mathcal{A} \mathcal{B} \mathcal{C} \mathcal{D} \mathcal{E} \mathcal{F} \mathcal{G} \mathcal{A} \mathcal{B} \mathcal{C} \mathcal{D} \mathcal{E} \mathcal{F} \mathcal{G} \,\!
\mathcal{H} \mathcal{I} \mathcal{J} \mathcal{K} \mathcal{L} \mathcal{M} \mathcal{H} \mathcal{I} \mathcal{J} \mathcal{K} \mathcal{L} \mathcal{M} \,\!
\mathcal{N} \mathcal{O} \mathcal{P} \mathcal{Q} \mathcal{R} \mathcal{S} \mathcal{T} \mathcal{N} \mathcal{O} \mathcal{P} \mathcal{Q} \mathcal{R} \mathcal{S} \mathcal{T} \,\!
\mathcal{U} \mathcal{V} \mathcal{W} \mathcal{X} \mathcal{Y} \mathcal{Z} \mathcal{U} \mathcal{V} \mathcal{W} \mathcal{X} \mathcal{Y} \mathcal{Z}\,\!
Hebrew
\aleph \beth \gimel \daleth \aleph \beth \gimel \daleth\,\!

Mixed text faces

Feature Syntax How it looks rendered
Non-italicised characters \mbox{abc} abc \mbox{abc} \,\!
Mixed italics (bad) \mbox{if} n \mbox{is even} ifnis even \mbox{if} n \mbox{is even} \,\!
Mixed italics (good) \mbox{if }n\mbox{ is even} if n is even \mbox{if }n\mbox{ is even} \,\!
Mixed italics (more legible: ~ is a non-breaking space, while "\ " forces a space) \mbox{if}~n\ \mbox{is even} \mbox{if}~n\ \mbox{is even} \mbox{if}~n\ \mbox{is even} \,\!

 

 

Examples

Quadratic polynomial

ax2 + bx + c = 0

<math>ax^2 + bx + c = 0</math>

Quadratic polynomial (force PNG rendering)

ax^2 + bx + c = 0\,\!

<math>ax^2 + bx + c = 0\,\!</math>

Quadratic formula

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

<math>x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}</math>

Tall parentheses and fractions

2 = \left( \frac{\left(3-x\right) \times 2}{3-x} \right)

<math>2 = \left(
 \frac{\left(3-x\right) \times 2}{3-x}
 \right)</math>
S_{\text{new}} = S_{\text{old}} - \frac{ \left( 5-T \right) ^2} {2}

 <math>S_{\text{new}} = S_{\text{old}} - \frac{ \left( 5-T \right) ^2} {2}</math>
 

Integrals

\int_a^x \!\!\!\int_a^s f(y)\,dy\,ds = \int_a^x f(y)(x-y)\,dy

<math>\int_a^x \!\!\!\int_a^s f(y)\,dy\,ds
 = \int_a^x f(y)(x-y)\,dy</math>

Summation

\sum_{m=1}^\infty\sum_{n=1}^\infty\frac{m^2\,n}{3^m\left(m\,3^n+n\,3^m\right)}

<math>\sum_{m=1}^\infty\sum_{n=1}^\infty\frac{m^2\,n}
 {3^m\left(m\,3^n+n\,3^m\right)}</math>

Differential equation

u'' + p(x)u' + q(x)u=f(x),\quad x>a

<math>u'' + p(x)u' + q(x)u=f(x),\quad x>a</math>

Complex numbers

|\bar{z}| = |z|, |(\bar{z})^n| = |z|^n, \arg(z^n) = n \arg(z)

<math>|\bar{z}| = |z|,
 |(\bar{z})^n| = |z|^n,
 \arg(z^n) = n \arg(z)</math>

Limits

\lim_{z\rightarrow z_0} f(z)=f(z_0)

<math>\lim_{z\rightarrow z_0} f(z)=f(z_0)</math>

Integral equation

\phi_n(\kappa)
 = \frac{1}{4\pi^2\kappa^2} \int_0^\infty \frac{\sin(\kappa R)}{\kappa R}  \frac{\partial}{\partial R}  \left[R^2\frac{\partial D_n(R)}{\partial R}\right]\,dR

<math>\phi_n(\kappa) =
 \frac{1}{4\pi^2\kappa^2} \int_0^\infty
 \frac{\sin(\kappa R)}{\kappa R}
 \frac{\partial}{\partial R}
 \left[R^2\frac{\partial D_n(R)}{\partial R}\right]\,dR</math>

Example

\phi_n(\kappa) = 0.033C_n^2\kappa^{-11/3},\quad \frac{1}{L_0}\ll\kappa\ll\frac{1}{l_0}

<math>\phi_n(\kappa) = 
 0.033C_n^2\kappa^{-11/3},\quad
 \frac{1}{L_0}\ll\kappa\ll\frac{1}{l_0}</math>

Continuation and cases

f(x) = \begin{cases}1 & -1 \le x < 0 \\
 \frac{1}{2} & x = 0 \\ 1 - x^2 & \mbox{otherwise}\end{cases}

<math>
 f(x) =
 \begin{cases}
 1 & -1 \le x < 0 \\
 \frac{1}{2} & x = 0 \\
 1 - x^2 & \mbox{otherwise}
 \end{cases}
 </math>

Prefixed subscript

{}_pF_q(a_1,\dots,a_p;c_1,\dots,c_q;z) = \sum_{n=0}^\infty \frac{(a_1)_n\cdots(a_p)_n}{(c_1)_n\cdots(c_q)_n}\frac{z^n}{n!}

 <math>{}_pF_q(a_1,\dots,a_p;c_1,\dots,c_q;z)
 = \sum_{n=0}^\infty
 \frac{(a_1)_n\cdots(a_p)_n}{(c_1)_n\cdots(c_q)_n}
 \frac{z^n}{n!}</math>

Fraction and small fraction

 \frac {a}{b} \tfrac {a}{b} 
<math> \frac {a}{b}\  \tfrac {a}{b} </math>

Area of a Quadrilateral

S=\frac{1}{2}dD\sin\alpha 
<math>S=\frac{1}{2}dD\sin\alpha </math>

Volume of a sphere-stand

V=\frac{1}{6}\pi h[3(r_1^2+r_2^2)+h^2]
<math>V=\frac{1}{6}\pi h[3(r_1^2+r_2^2)+h^2]</math>