\frac{a}{b} = \frac{1}{2}\[ \frac{a}{b} = \frac{1}{2} \]
$$
a^2 + b^2 = c^2
$$\[ a^2 + b^2 = c^2 \]
$$
1^2 + 2^2 + \ldots +(n - 1)^{n + 1}
$$\[ 1^2 + 2^2 + \ldots +(n - 1)^{n + 1}+\sqrt{x} \quad \quad \hat{y_i} = \hat{\beta_0} + \hat{\beta_1} x_i + \mu_i \]
$$
\lim \limits_{x \rightarrow \infty} \frac{f(x)}{g(x)} = \frac{\lim \limits_{x \rightarrow \infty} f(x) }{ \lim \limits_{x \rightarrow \infty} g(x) }
$$\[ \lim \limits_{x \rightarrow \infty} \frac{f(x)}{g(x)} = \frac{\lim \limits_{x \rightarrow \infty} f(x) }{ \lim \limits_{x \rightarrow \infty} g(x) } \]
$$
\sum^{\infty} _{n = 1} \frac{n!}{n^n}:
\lim \limits_{n \rightarrow \infty} \frac{a_n + 1}{a_n} =
\lim \limits_{n \rightarrow \infty} \frac{1}{(1 + \frac{1}{n})^n } =
\frac{1}{e} < 1
$$\[ \sum^{\infty} _{n = 1} \frac{n!}{n^n}: \lim \limits_{n \rightarrow \infty} \frac{a_n + 1}{a_n} = \lim \limits_{n \rightarrow \infty} \frac{1}{(1 + \frac{1}{n})^n } = \frac{1}{e} < 1 \]
$$
f'(a) = \lim \limits_{h \rightarrow 0} \frac{f(a + h)}{h}
$$\[ f'(a) = \lim \limits_{h \rightarrow 0} \frac{f(a + h)}{h} \]
$$
\lim \limits_{x \rightarrow 1} \frac{1 + x^2}{\sin(\pi x)} = \frac{2}{\pi}
$$\[ \lim \limits_{x \rightarrow 1} \frac{1 + x^2}{\sin(\pi x)} = \frac{2}{\pi} \]
$$
\nabla \quad
\partial x \quad
dx \quad
\dot x \quad
\ddot y \quad
dy/dx \quad
\frac{dy}{dx} \quad
\frac{\partial^2 z}{\partial x\,\partial y} \quad
$$\[ \nabla \quad \partial x \quad dx \quad \dot x \quad \ddot y \quad dy/dx \quad \frac{dy}{dx} \quad \frac{\partial^2 z}{\partial x\,\partial y} \quad \]
$$
y = f(s) = 2x^4 + 3x^2 + 6x +1
\Rightarrow \frac{dy}{dx} = 8x^3 + 6x + 6
$$\[ y = f(s) = 2x^4 + 3x^2 + 6x +1 \Rightarrow \frac{dy}{dx} = 8x^3 + 6x + 6 \]
$$
z = f(x(t), y(t)) \longrightarrow
\frac{dz}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}
$$\[ z = f(x(t), y(t)) \longrightarrow \frac{dz}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt} \]
$$
\overrightarrow{\mu} =
\frac{\partial f(x_0, y_0)}{\partial x} \overrightarrow{i} +
\frac{\partial f(x_0, y_0)}{\partial y} \overrightarrow{j} +
(-1) \overrightarrow{k}
$$\[ \overrightarrow{\mu} = \frac{\partial f(x_0, y_0)}{\partial x} \overrightarrow{i} + \frac{\partial f(x_0, y_0)}{\partial y} \overrightarrow{j} + (-1) \overrightarrow{k} \]
$$
grad(\phi) = \frac{\partial \phi}{\partial x} \overrightarrow{i} +
\frac{\partial \phi}{\partial y} \overrightarrow{j} +
\frac{\partial \phi}{\partial z} \overrightarrow{k}
$$\[ grad(\phi) = \frac{\partial \phi}{\partial x} \overrightarrow{i} + \frac{\partial \phi}{\partial y} \overrightarrow{j} + \frac{\partial \phi}{\partial z} \overrightarrow{k} \]
$$
\nabla_x \overrightarrow{f} = \left(
\frac{\partial}{\partial x} \overrightarrow{i} +
\frac{\partial}{\partial y} \overrightarrow{j} +
\frac{\partial}{\partial z} \overrightarrow{k} \right)
\times
\left(
f_1 \overrightarrow{i} + f_2 \overrightarrow{j} + f_3 \overrightarrow{k} \right) =
\begin{vmatrix}
\overrightarrow{i} & \overrightarrow{j} & \overrightarrow{k} \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
f_1 & f_2 & f_3
\end{vmatrix}
$$\[ \nabla_x \overrightarrow{f} = \left( \frac{\partial}{\partial x} \overrightarrow{i} + \frac{\partial}{\partial y} \overrightarrow{j} + \frac{\partial}{\partial z} \overrightarrow{k} \right) \times \left( f_1 \overrightarrow{i} + f_2 \overrightarrow{j} + f_3 \overrightarrow{k} \right) = \begin{vmatrix} \overrightarrow{i} & \overrightarrow{j} & \overrightarrow{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ f_1 & f_2 & f_3 \end{vmatrix} \]
$$
A(R) =\lim \limits_{b \rightarrow \infty} \sum^n _{i = 1} f(c_i) \Delta x
\quad
\mathrm{donde} \quad \Delta x = \frac{b - a}{n}; c_i = a + i \Delta x
$$\[ A(R) =\lim \limits_{b \rightarrow \infty} \sum^n _{i = 1} f(c_i) \Delta x \quad \mathrm{donde} \quad \Delta x = \frac{b - a}{n}; c_i = a + i \Delta x \]
$$
G(x) = \int f(x)dx = F(x) +c, \quad \forall x \in I
$$\[ G(x) = \int f(x)dx = F(x) +c, \quad \forall x \in I \]
$$
\int \frac{du}{u^2 - a^2} = \frac{1}{2a} \ln | \frac{u - a}{u + a}| + c
$$\[ \int \frac{du}{u^2 - a^2} = \frac{1}{2a} \ln | \frac{u - a}{u + a}| + c \]
$$
\int ^b _a f(x)dx = F(x)|^b _a = F(b) - F(a)
$$\[ \int ^b _a f(x)dx = F(x)|^b _a = F(b) - F(a) \]
$$
\iint \limits_{D} f(x,y)dx dy = \int ^b _a \left( \int ^{\psi(y)} _{\varphi(x)} f(x,y)dx \right) dy
$$\[ \iint \limits_{D} f(x,y)dx dy = \int ^b _a \left( \int ^{\psi(y)} _{\varphi(x)} f(x,y)dx \right) dy \]
$$
\int ^2 _1 \int ^{2x} _x \int ^{\sqrt{2xy}} _{\sqrt{1 - x^2 - y^2}} \frac{z}{x^2 + y^2 + z^2} dx dy dz = \ln \left( \frac{81 \sqrt{3} }{ 2 } - \frac{9}{4} \right)
$$\[ \int ^2 _1 \int ^{2x} _x \int ^{\sqrt{2xy}} _{\sqrt{1 - x^2 - y^2}} \frac{z}{x^2 + y^2 + z^2} dx dy dz = \ln \left( \frac{81 \sqrt{3} }{ 2 } - \frac{9}{4} \right) \]
$$
C = \oint \overrightarrow{F} d \overrightarrow{r} =
\oint_{\Gamma} \left(
P(x,y,z)dx + Q(x,y,z)dy + R(x,y,z)dz
\right)
$$\[ C = \oint \overrightarrow{F} d \overrightarrow{r} = \oint_{\Gamma} \left( P(x,y,z)dx + Q(x,y,z)dy + R(x,y,z)dz \right) \]
\begin{equation}
\nonumber \vec{a} =
\begin{pmatrix}
1 \\ 2 \\ 3 \\ 4 \\ 5
\end{pmatrix}
\end{equation}\[\begin{equation} \nonumber \vec{a} = \begin{pmatrix} 1 \\ 2 \\ 3 \\ 4 \\ 5 \end{pmatrix} \end{equation}\]
$$
A =
\begin{pmatrix}
1 & 2 \\ 2 & 3 \\ 3 & 4 \\ 4 & 5 \\ 5 & 6
\end{pmatrix}
$$\[ A = \begin{pmatrix} 1 & 2 \\ 2 & 3 \\ 3 & 4 \\ 4 & 5 \\ 5 & 6 \end{pmatrix} \]
$$
A_{3 \times 3} =
\begin{pmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{pmatrix} _{3 \times 3}
$$\[ A_{3 \times 3} = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} _{3 \times 3} \]
$$
\left [
\begin{matrix}
c_{(1,1)} & c_{(1,2)} & c_{(1,3)} \\
c_{(2,1)} & c_{(2,2)} & c_{(2,3)} \\
c_{(3,1)} & c_{(3,2)} & c_{(3,3)}
\end{matrix}
\right ]
$$\[ \left [ \begin{matrix} c_{(1,1)} & c_{(1,2)} & c_{(1,3)} \\ c_{(2,1)} & c_{(2,2)} & c_{(2,3)} \\ c_{(3,1)} & c_{(3,2)} & c_{(3,3)} \end{matrix} \right ] \]
$$
A =
\begin{pmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{pmatrix}
$$\[ A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} \]
$$
\det A =
\begin{vmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{vmatrix} = (1 \times 5 \times 9 + 4 \times 8 \times 3 + 7 \times 2 \times 6) -
(3 \times 5 \times 7 + 6 \times 8 \times 1 + 9 \times 2 \times 4) = 0
$$\[ \det A = \begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{vmatrix} = \]
$$
f(x) = \left\{
\begin{array}{lcc}
\frac{ \sqrt{x+3} - \sqrt{3x+1} }{ \sqrt{x-1} }, \quad x > 1 \\ \\
ax + b, \quad -2 \le x \le 1 \\ \\
\frac{ x^2 + 2x }{ x^2 + x -2 }, \quad x < -2
\end{array}
\right.
$$\[ f(x) = \left\{ \begin{array}{lcc} \frac{ \sqrt{x+3} - \sqrt{3x+1} }{ \sqrt{x-1} }, \quad x > 1 \\ \\ ax + b, \quad -2 \le x \le 1 \\ \\ \frac{ x^2 + 2x }{ x^2 + x -2 }, \quad x < -2 \end{array} \right. \]
$$
\begin{eqnarray}
a &=& b
a &=& b
\end{eqnarray}
$$\[ \begin{eqnarray} a &=& b \\ a &=& b \end{eqnarray} \] Otras:
$$
\begin{eqnarray[align*}
a &=& b
a &=& b
\end{align*}
$$\[ \begin{align*} a &=& b \\ a &=& b \end{align*} \]
$$
\begin{array}{rcl}
f(n) & = & (n+1)^3 \\
& = & n^3 + 3n^2 +3n + 1
\end{array}
$$\[ \begin{array}{rcl} f(n) & = & (n+1)^3 \\ & = & n^3 + 3n^2 +3n + 1 \end{array} \]
$$
\xleftarrow[abajo]{arriba} \xrightarrow[abajo]{arriba}
$$\[ \xleftarrow[abajo]{arriba} \quad \xrightarrow[abajo]{arriba} \]
$$
\overbrace{
\underbrace{ \sin(x) \cos(y) }_{T_1}
\underbrace{+ 35 \,x y }_{T_2}
\underbrace{- x^3 y^4 }_{T_3}
}^{Primer \; miembro}
=
\overbrace{
\underbrace{ \log(2x^3) e^{2y} }_{T_1}
\underbrace{- x^3 (y^2 -5) }_{T_2}
}^{Segundo \; miembro}
$$\[ \overbrace{ \underbrace{ \sin(x) \cos(y) }_{T_1} \underbrace{+ 35 \,x y }_{T_2} \underbrace{- x^3 y^4 }_{T_3} }^{Primer \; miembro} = \overbrace{ \underbrace{ \log(2x^3) e^{2y} }_{T_1} \underbrace{- x^3 (y^2 -5) }_{T_2} }^{Segundo \; miembro} \] ## Tachar o cancelar Expresión
$$
$$
$$
\big [
\Big [
\bigg [
\Bigg [
\quad
\Bigg ]
\bigg ]
\Big ]
\big ]
$$\[ \big [ \Big [ \bigg [ \Bigg [ \quad \Bigg ] \bigg ] \Big ] \big ] \]
$$
\big (
\Big (
\bigg (
\Bigg (
\quad
\Bigg )
\bigg )
\Big )
\big )
$$\[ \big ( \Big ( \bigg ( \Bigg ( \quad \Bigg ) \bigg ) \Big ) \big ) \]
Aproximado
$$
\approx
$$\[ \approx \] ## Terminar la demostración
$$
\square
\blacksquare
$$\[ \square \quad \blacksquare \]
$$
\begin{array}{llll}
alpha & \Alpha & \alpha \\
beta & \Beta & \beta \\
gamma & \Gamma & \gamma \\
delta & \Delta & \delta \\
epsilon & \Epsilon & \epsilon & \varepsilon \\
zeta & \Zeta & \zeta \\
eta & \Eta & \eta \\
theta & \Theta & \theta & \vartheta \\
iota & \Iota & \iota \\
kappa & \Kappa & \kappa & \varkappa \\
lambda & \Lambda & \lambda \\
mu & \Mu & \mu \\
nu & \Nu & \nu \\
xi & \Xi & \xi \\
omicron & \Omicron & \omicron \\
pi & \Pi & \pi & \varpi \\
rho & \Rho & \rho & \varrho \\
sigma & \Sigma & \sigma & \varsigma \\
tau & \Tau & \tau \\
upsilon & \Upsilon & \upsilon \\
phi & \Phi & \phi & \varphi \\
chi & \Chi & \chi \\
psi & \Psi & \psi \\
omega & \Omega & \omega \\
\end{array}
$$\[ \begin{array}{llll} alpha & \Alpha & \alpha \\ beta & \Beta & \beta \\ gamma & \Gamma & \gamma \\ delta & \Delta & \delta \\ epsilon & \Epsilon & \epsilon & \varepsilon \\ zeta & \Zeta & \zeta \\ eta & \Eta & \eta \\ theta & \Theta & \theta & \vartheta \\ iota & \Iota & \iota \\ kappa & \Kappa & \kappa & \varkappa \\ lambda & \Lambda & \lambda \\ mu & \Mu & \mu \\ nu & \Nu & \nu \\ xi & \Xi & \xi \\ omicron & \Omicron & \omicron \\ pi & \Pi & \pi & \varpi \\ rho & \Rho & \rho & \varrho \\ sigma & \Sigma & \sigma & \varsigma \\ tau & \Tau & \tau \\ upsilon & \Upsilon & \upsilon \\ phi & \Phi & \phi & \varphi \\ chi & \Chi & \chi \\ psi & \Psi & \psi \\ omega & \Omega & \omega \\ \end{array} \]
$$
{ \color{Blue} y} =
{ \color{Sepia} 3x^2 } -
{ \color{Red} 5x } +
{ \color{Green} 2 }
$$\[ { \color{Blue} y} = { \color{Sepia} 3x^2 } - { \color{Red} 5x } + { \color{Green} 2 } \]
$$
SO_2 + NO_2
\longrightarrow \;
NO + SO_3
$$\[ SO_2 + NO_2 \longrightarrow \; NO + SO_3 \]
$$
E + S {\stackrel {k_{1}} {\underset {k_{- 1}} {\rightleftharpoons}}} ES \xrightarrow{k_{2}} E + P
$$\[ E + S {\stackrel {k_{1}} {\underset {k_{- 1}} {\rightleftharpoons}}} ES \xrightarrow{k_{2}} E + P \]
$$
\sideset
{_\llcorner^\ulcorner}{_\lrcorner^\urcorner}
{\operatorname{\pi \simeq 3{,}14159265}}
$$\[ \sideset {_\llcorner^\ulcorner}{_\lrcorner^\urcorner} {\operatorname{ \pi \simeq 3{,}14159265} } \]
$$
\sideset
{_\llcorner^\ulcorner}{_\lrcorner^\urcorner}
{\operatorname{\pi \simeq 3{,}14159265}}
$$\[ \begin{equation} \boxed{ x^2+1} \end{equation} \]
$$
\begin{eqnarray*}
x&=&y\\
x^2&=&xy\\
x^2-y^2&=&xy-y^2\\
(x+y)(x-y)&=&y(x-y)\\
x+y&=&y\\
2y&=&y\quad \mbox{(por la primera ecuación)}\\
2&=&1
\end{eqnarray*}
$$\[ \begin{eqnarray*} x&=&y\\ x^2&=&xy\\ x^2-y^2&=&xy-y^2\\ (x+y)(x-y)&=&y(x-y)\\ x+y&=&y\\ 2y&=&y\quad \mbox{(por la primera ecuación)}\\ 2&=&1 \end{eqnarray*} \]
$$
\begin{array}{|c|c||c|}
\hline
a & b & a \lor b \\
\hline
0 & 0 & 0 \\
0 & 1 & 1 \\
1 & 0 & 1 \\
1 & 1 & 1 \\
\hline
\end{array}
\quad
\begin{array}{|c|c||c|}
\hline
a & b & a \land b \\
\hline
0 & 0 & 0 \\
0 & 1 & 0 \\
1 & 0 & 0 \\
1 & 1 & 1 \\
\hline
\end{array}
$$\[ \begin{array}{|c|c||c|} \hline a & b & a \lor b \\ \hline 0 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 1 \\ \hline \end{array} \quad \begin{array}{|c|c||c|} \hline a & b & a \land b \\ \hline 0 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \\ 1 & 1 & 1 \\ \hline \end{array} \]