Ayuda:Fórmulas Matemáticas

De Wikillerato

A continuación ofrecemos un cuadro de referencia con nociones básicas y ejemplos que sirven de ayuda para escribir fórmulas utilizando el código LaTeX.

Tabla de contenidos

1 Básicos
2 Subíndices, superíndices, integrales
3 Fracciones, matrices, multilíneas
4 Alfabetos
5 Añadiendo paréntesis a grandes expresiones
6 Espaciado
7 Ejemplos

Básicos

Acentos
`\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}`	$\check{a} \bar{a} \ddot{a} \dot{a}$
Funciones estándar
`\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`	$\sinh k \cosh l \tanh m \coth n$
`\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`	$\inf z \sup a \exp b \ln c \lg d \log e \log_{10} f \ker g$
Derivadas
`\nabla \partial x dx \dot x \ddot y`	$\nabla \partial x dx \dot x \ddot y$
Conjuntos
`\forall \exists \emptyset \varnothing`	$\forall \exists \emptyset \varnothing$
`\in \ni \notin \subset \subseteq \supset \supseteq`	$\in \ni \notin \subset \subseteq \supset \supseteq$
`\cap \bigcap \cup \bigcup \biguplus \setminus \smallsetminus`	$\cap \bigcap \cup \bigcup \biguplus \setminus \smallsetminus$
`\sqsubset \sqsubseteq \sqsupset \sqsupseteq \sqcap \sqcup \bigsqcup`	$\sqsubset \sqsubseteq \sqsupset \sqsupseteq \sqcap \sqcup \bigsqcup$
Operadores
`+ \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}`	$\star * / \div \frac{1}{2}$
Lógica
`\land \wedge \bigwedge \bar{q} \to p`	$\land \wedge \bigwedge \bar{q} \to p$
`\lor \vee \bigvee \lnot \neg q \And`	$\lor \vee \bigvee \lnot \neg q \And$
Raíces
`\sqrt{2} \sqrt[n]{x}`	$\sqrt{2} \sqrt[n]{x}$
Relaciones
`\sim \approx \simeq \cong`	$\sim \approx \simeq \cong$
`\le < \ll \gg \ge > \equiv \not\equiv \ne \mbox{or} \neq \propto`	$\le < \ll \gg \ge > \equiv \not\equiv \ne \mbox{or} \neq \propto$
Geometría
`\Diamond \Box \triangle \angle \perp \mid \nmid \\| 45^\circ`	$\Diamond \Box \triangle \angle \perp \mid \nmid \\| 45^\circ$
Flechas
`\leftarrow \gets \rightarrow \to \not\to \leftrightarrow \longleftarrow \longrightarrow`	$\leftarrow \gets \rightarrow \to \not\to \leftrightarrow \longleftarrow \longrightarrow$
`\uparrow \downarrow \updownarrow \rightharpoonup \rightharpoondown \leftharpoonup \leftharpoondown \upharpoonleft`	$\uparrow \downarrow \updownarrow \rightharpoonup \rightharpoondown \leftharpoonup \leftharpoondown \upharpoonleft$
`\upharpoonright \downharpoonleft \downharpoonright \rightleftharpoons \Leftarrow \Rightarrow \Leftrightarrow \Longleftarrow`	$\upharpoonright \downharpoonleft \downharpoonright \rightleftharpoons \Leftarrow \Rightarrow \Leftrightarrow \Longleftarrow$
`\Longrightarrow \Uparrow \Downarrow \Updownarrow`	$\Longrightarrow \Uparrow \Downarrow \Updownarrow$
`\nLeftrightarrow \longleftrightarrow`	$\nLeftrightarrow \longleftrightarrow$
Especial
`\eth \S \P \% \dagger \ddagger \ldots \cdots`	$\eth \S \P \% \dagger \ddagger \ldots \cdots$
`\smile \frown \wr \triangleleft \triangleright \infty \bot \top`	$\smile \frown \wr \triangleleft \triangleright \infty \bot \top$
`\vdash \vDash \Vdash \models \lVert \rVert \imath \hbar`	$\vdash \vDash \Vdash \models \lVert \rVert \imath \hbar$
Otros
`\vartriangle \triangledown \lozenge \circledS \measuredangle \nexists \Bbbk \backprime \blacktriangle \blacktriangledown`	$\vartriangle \triangledown \lozenge \circledS \measuredangle \nexists \Bbbk \backprime \blacktriangle \blacktriangledown$
`\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`	$\eqslantless \lessapprox \approxeq \lessdot \lll \lessgtr \lesseqgtr \lesseqqgtr \doteqdot$
`\fallingdotseq \backsim \backsimeq \subseteqq \Subset \preccurlyeq \curlyeqprec \precsim \precapprox`	$\fallingdotseq \backsim \backsimeq \subseteqq \Subset \preccurlyeq \curlyeqprec \precsim \precapprox$
`\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 \ngtr`	$\nvdash \nVdash \ntriangleleft \ntrianglelefteq \nsubseteq \nsubseteqq \ngtr$
`\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`	$\nsupseteqq \varsupsetneq \supsetneqq \varsupsetneqq$

Subíndices, superíndices, integrales

	Sintaxis	Cómo se verá
Superíndice	`a^2`	$a^2$
Subíndice	`a_2`	$a_2$
Agrupar	`a^{2+2}`	$a^{2+2}$
Agrupar	`a_{i,j}`	$a_{i,j}$
Combinar superindice y subíndice	`x_2^3`	$x_2^3$
Superíndices y subíndices, anteriores, posteriores, arriba y abajo	`\sideset {_1^2} {_3^4} \prod_a^b`	$\sideset {_1^2} {_3^4} \prod_a^b$
	`{}_1^2 \! \Omega_3^4`	${}_1^2 \! \Omega_3^4$
Apilar	`\overset { \alpha} { \omega}`	$\overset { \alpha} { \omega}$
	`\overset { \alpha} { \underset { \gamma} { \omega}}`	$\overset { \alpha} { \underset { \gamma} { \omega}}$
	`\stackrel { \alpha} { \omega}`	$\stackrel { \alpha} { \omega}$
Derivadas	`x', y, f', f`	$x', y'', f', f''$
Subrayado, línea superior, vectores	`\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}`	$\overline {g h i} \underline {j k l}$
Flechas	`A \xleftarrow {n+ \mu-1} B \xrightarrow[T] {n \pm i-1} C`	$A \xleftarrow {n+ \mu-1} B \xrightarrow[T] {n \pm i-1} C$
Llaves superiores	`\overbrace{ 1+2+ \cdots+100 } ^ {5050}`	$\overbrace{ 1+2+ \cdots+100 } ^ {5050}$
Llaves inferiores	`\underbrace { a+b+ \cdots+z }_{26}`	$\underbrace { a+b+ \cdots+z }_{26}$
Sumatorios	`\sum_{k=1}^N k^2`	$\sum_{k=1}^N k^2$
Productorio	`\prod_{i=1}^N x_i`	$\prod_{i=1}^N x_i$
Coproducto	`\coprod_{i=1}^N x_i`	$\coprod_{i=1}^N x_i$
Límite	`\lim_{n \to \infty}x_n`	$\lim_{n \to \infty}x_n$
Integral	`\int_{-N}^{N} e^x\, dx`	$\int_{-N}^{N} e^x\, dx$
Integral doble	`\iint_{D}^{W} \, dx\,dy`	$\iint_{D}^{W} \, dx\,dy$
Integral triple	`\iiint_{E}^{V} \, dx\,dy\,dz`	$\iiint_{E}^{V} \, dx\,dy\,dz$
Integral de línea	`\oint_{C} x^3\, dx + 4y^2\, dy`	$\oint_{C} x^3\, dx + 4y^2\, dy$
Intersecciones	`\bigcap_1^{n} p`	$\bigcap_1^{n} p$
Uniones	`\bigcup_1^{k} p`	$\bigcup_1^{k} p$

Fracciones, matrices, multilíneas

	Sintaxis	Cómo se verá
Fracciones	`\frac{2}{4}=0.5`	$\frac{2}{4}=0.5$
Coeficiente binomial	`\binom{n}{k}`	$\binom{n}{k}$
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)	$\bigl( \begin{smallmatrix} a&b\\ c&d \end{smallmatrix} \bigr)$
Distinción de casos	f(n) = \begin{cases} n/2, & \mbox{if }n\mbox{ is even} \\ 3n+1, & \mbox{if }n\mbox{ is odd} \end{cases}	$f(n) = \begin{cases} n/2, & \mbox{if }n\mbox{ is even} \\ 3n+1, & \mbox{if }n\mbox{ is odd} \end{cases}$
Ecuaciones multilínea (se debe definir el número de columnas con {lcl})	\begin{array}{lcl} z & = & a \\ f(x,y,z) & = & x + y + z \end{array}	$\begin{array}{lcl} z & = & a \\ f(x,y,z) & = & x + y + z \end{array}$
	\begin{array}{lclcl} z & = & a & = & \sqrt 2\\ f(x,y,z) & = & x + y + z & = & t^2\\ f(z) & = & x+y & = & 2 \pi \end{array}	$\begin{array}{lclcl} z & = & a & = & \sqrt 2\\ f(x,y,z) & = & x + y + z & = & t^2\\ f(z) & = & x+y & = & 2 \pi \end{array}$
Romper largas expresiones para hacer más legible el código	<math>f(x) \,\!</math> <math>= \sum_{n=0}^\infty a_n x^n </math> <math>= a_0+a_1x+a_2x^2+\cdots</math>	$f(x) \,\!$ $= \sum_{n=0}^ \infty a_n x^n$ $= a_0+a_1x+a_2x^2+ \cdots$
Ecuaciones simultáneas	\begin{cases} 3x + 5y + z \\ 7x - 2y + 4z \\ -6x + 3y + 2z \end{cases}	$\begin{cases} 3x + 5y + z \\ 7x - 2y + 4z \\ -6x + 3y + 2z \end{cases}$

Alfabetos

Alfabeto griego
`\Delta \Theta \Lambda`	$\Delta \Theta \Lambda$
`\Xi \Pi \Sigma`	$\Xi \Pi \Sigma$
`\Upsilon \Phi \Psi \Omega`	$\Upsilon \Phi \Psi \Omega$
`\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`	$\varpi \varrho \varsigma \varphi$

Añadiendo paréntesis a grandes expresiones

	Sintaxis	Cómo se verá
Mal	( \frac{1}{2} )	$( \frac{1}{2} )$
Bien	\left ( \frac{1}{2} \right )	$\left ( \frac{1} {2} \right )$

	Sintaxis	Cómo se verá
Paréntesis	\left ( \frac{a}{b} \right )	$\left ( \frac{a}{b} \right )$
Corchetes	\left [ \frac{a}{b} \right ] \quad \left \lbrack \frac{a}{b} \right \rbrack	$\left [ \frac{a}{b} \right ] \quad \left \lbrack \frac{a}{b} \right \rbrack$
Llaves	\left \{ \frac{a}{b} \right \} \quad \left \lbrace \frac{a}{b} \right \rbrace	$\left \{ \frac{a}{b} \right \} \quad \left \lbrace \frac{a}{b} \right \rbrace$
Barras y dobles barras	\left \| \frac{a}{b} \right \vert \left \Vert \frac{c}{d} \right \\|	$\left \| \frac{a}{b} \right \vert \left \Vert \frac{c}{d} \right \\|$
Barras invertidas	\left / \frac{a}{b} \right \backslash	$\left / \frac{a}{b} \right \backslash$
Flechas arriba y abajo	\left \uparrow \frac{a}{b} \right \downarrow \quad \left \Uparrow \frac{a}{b} \right \Downarrow \quad \left \updownarrow \frac{a}{b} \right \Updownarrow	$\left \uparrow \frac{a}{b} \right \downarrow \quad \left \Uparrow \frac{a}{b} \right \Downarrow \quad \left \updownarrow \frac{a}{b} \right \Updownarrow$
Los delimitadores pueden mezclarse	\left [ 0,1 \right )	$\left [ 0,1 \right )$
Usa \left. y \right. si no quieres que un delimitador aparezca	\left . \frac{A}{B} \right \} \to X	$\left . \frac{A}{B} \right \} \to X$
Tamaño de los delimitadores	\big( \Big( \bigg( \Bigg( ... \Bigg] \bigg] \Big] \big]	$\big( \Big( \bigg( \Bigg( ... \Bigg] \bigg] \Big] \big]$
	\big\{ \Big\{ \bigg\{ \Bigg\{ ... \Bigg\rangle \bigg\rangle \Big\rangle \big\rangle	$\big\{ \Big\{ \bigg\{ \Bigg\{ ... \Bigg\rangle \bigg\rangle \Big\rangle \big\rangle$
	\big\uparrow \Big\uparrow \bigg\uparrow \Bigg\uparrow ... \Bigg\Downarrow \bigg\Downarrow \Big\Downarrow \big\Downarrow	$\big\uparrow \Big\uparrow \bigg\uparrow \Bigg\uparrow ... \Bigg\Downarrow \bigg\Downarrow \Big\Downarrow \big\Downarrow$
	\big\updownarrow \Big\updownarrow \bigg\updownarrow \Bigg\updownarrow ... \Bigg\Updownarrow \bigg\Updownarrow \Big\Updownarrow \big\Updownarrow	$\big\updownarrow \Big\updownarrow \bigg\updownarrow \Bigg\updownarrow ... \Bigg\Updownarrow \bigg\Updownarrow \Big\Updownarrow \big\Updownarrow$
	\big / \Big / \bigg / \Bigg / ... \Bigg\backslash \bigg\backslash \Big\backslash \big\backslash	$\big / \Big / \bigg / \Bigg / ... \Bigg\backslash \bigg\backslash \Big\backslash \big\backslash$

Espaciado

Nota: TeX elimina los espacios automáticamente, pero puedes controlarlos manualmente.

	Sintaxis	Cómo se verá
Espacio en blanco	a\ b	$a\ b$

Ejemplos

Polinomio cuadrático	`ax^2 + bx + c = 0\,\!`	$ax^2 + bx + c = 0\,\!$
Fórmula cuadrática	`x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}`	$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Paréntesis altos y fracciones	`2 = \left( \frac{\left(3-x\right) \times 2}{3-x} \right)`	$2 = \left( \frac{\left(3-x\right) \times 2}{3-x} \right)$
Paréntesis altos y fracciones	`S_{new} = S_{old} + \frac{ \left( 5-T \right) ^2} {2}`	$S_{new} = S_{old} + \frac{ \left( 5-T \right) ^2} {2}$
Integrales	`\int_a^x \,dy = \int_a^x f(y)`	$\int_a^x \,dy = \int_a^x f(y)$
Sumatorios	`\sum_{n=1}^\infty\frac{m^2\,n}{3^m}`	$\sum_{n=1}^\infty\frac{m^2\,n}{3^m}$
Ecuaciones diferenciales	`u + p(x)u' = f(x),\quad x>a`	$u'' + p(x)u' = f(x),\quad x>a$
Límites	`\lim_{z\rightarrow z_0} f(z)=f(z_0)`	$\lim_{z\rightarrow z_0} f(z)=f(z_0)$
Casos	`f(x) = \begin{cases} 1 & -1 \le x < 0 \\ \frac{1}{2} & x = 0 \\ 1 - x^2 & 0 < x\le 1 \end{cases}`	$f(x) = \begin {cases} 1 & -1 \le x < 0 \\ \frac{1}{2} & x = 0 \\ 1 - x^2 & 0 < x\le 1 </p> \end {cases}$