$$\sin{x}^2 + \cos{x}^2 = 1 \\ e^{\pi i} + 1 = 0$$
        % following does not work in Katex
$$\sin{x}^2 + \cos{x}^2 = 1 \\ e^{pi i} + 1 = 0$$

$\vec{v} \\ \vec{AB} \\ \bar{v} \\ \bar{p_0} \\ \overline{p_0} \\ \overline{v} \\ \overline{AB} \\ \overrightarrow{AB}$
        $\vec{v} \\ \vec{AB} \\ \bar{v} \\ \bar{p_0} \\ \overline{p_0} \\ \overline{v} \\ \overline{AB} \\ \overrightarrow{AB}$

$\cos{\alpha} = \frac{\vec{BA} \vec{BC}}{|\vec{BA}| | \vec{BC} |} \\ \cos{\alpha} = \frac{\vec{BA} \, \vec{BC}}{|\vec{BA}| | \vec{BC} |} \\ \cos{\alpha} = \frac{\vec{BA} \quad \vec{BC}}{|\vec{BA}| | \vec{BC} |} \\ \cos{\alpha} = \frac{\vec{BA} \qquad \vec{BC}}{|\vec{BA}| | \vec{BC} |}$

\cos{\alpha} = \frac{\vec{BA} \vec{BC}}{|\vec{BA}| | \vec{BC} |} \\
\cos{\alpha} = \frac{\vec{BA} \, \vec{BC}}{|\vec{BA}| | \vec{BC} |}     \\
\cos{\alpha} = \frac{\vec{BA} \quad \vec{BC}}{|\vec{BA}| | \vec{BC} |}   \\
\cos{\alpha} = \frac{\vec{BA} \qquad \vec{BC}}{|\vec{BA}| | \vec{BC} |}

$I_2 = \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}$
    I_2 = \begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}

\begin{aligned} A &= \begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{bmatrix} \\ \det{A}&=a_1 (-1)^{1 + 1} \begin{vmatrix} b_2 & b_3 \\ c_2 & c_3 \end{vmatrix} + b_1 (-1)^{2 + 1} \begin{vmatrix} a_2 & a_3 \\ c_2 & c_3 \end{vmatrix} + c_1 (-1)^{3 + 1} \begin{vmatrix} a_2 & a_3 \\ b_2 & b_3 \\ \end{vmatrix} \end{aligned}
        \begin{aligned} A &= \begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{bmatrix} \\ \det{A}&=a_1 (-1)^{1 + 1} \begin{vmatrix} b_2 & b_3 \\ c_2 & c_3 \end{vmatrix} + b_1 (-1)^{2 + 1} \begin{vmatrix} a_2 & a_3 \\ c_2 & c_3 \end{vmatrix} + c_1 (-1)^{3 + 1} \begin{vmatrix} a_2 & a_3 \\ b_2 & b_3 \\ \end{vmatrix} \end{aligned}

$I_3 = \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix}$
     I_3 = \begin{bmatrix}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1
\end{bmatrix}

$R_2 = \begin{bmatrix} \cos \beta & -\sin \beta\\ \sin \beta & \cos \beta \end{bmatrix} \\$
     R_2 = \begin{bmatrix}
\cos(\beta) & -\sin(\beta)\\
\sin(\beta) & \cos(\beta)
\end{bmatrix} \\

3x3 Rotaton by Y Matrix
$M_{y}(\alpha) =\begin{bmatrix} \cos\alpha & \sin\alpha & 0\\ 0 & 1 & 0 \\ -\sin\alpha & \cos\alpha & 0 \end{bmatrix} \\$
     M_{y}(\alpha) =\begin{bmatrix}
\cos\alpha & \sin\alpha & 0\\
0      &   1    & 0    \\
-\sin\alpha & \cos\alpha & 0
\end{bmatrix} \\

3x3 Rotaton by X Matrix
$M_{x}(\alpha) =\begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\alpha & \sin\alpha\\ 0 & -\sin\alpha& \cos\alpha \end{bmatrix} \\$
     M_{x}(\alpha) =\begin{bmatrix}
1 &   0      &     0   \\
0 & \cos\alpha & \sin\alpha\\
0 & -\sin\alpha& \cos\alpha
\end{bmatrix} \\

3x3 Rotaton by Z Matrix
$M_{z}(\alpha) =\begin{bmatrix} \cos\alpha & \sin\alpha & 0\\ -\sin\alpha & \cos\alpha & 0\\ 0 & 0 & 1 \end{bmatrix} \\$
     M_{z}(\alpha) =\begin{bmatrix}
\cos\alpha & \sin\alpha & 0\\
-\sin\alpha & \cos\alpha & 0\\
0      &   0    & 1
\end{bmatrix} \\

Translate to [x, y, z] Matrix
$S = \begin{bmatrix} 1 & 0 & 0 & x\\ 0 & 1 & 0 & y\\ 0 & 0 & 1 & z\\ 0 & 0 & 0 & 1 \end{bmatrix} \\$
     S = \begin{bmatrix}
1 & 0 & 0 & x\\
0 & 1 & 0 & y\\
0 & 0 & 1 & z\\
0 & 0 & 0 & 1
\end{bmatrix} \\

3x3 Matrix, Stretching or Squeezing
$T =\begin{bmatrix} x & 0 & 0\\ 0 & y & 0\\ 0 & 0 & z \end{bmatrix} \\$
     T =\begin{bmatrix}
x & 0 & 0\\
0 & y & 0\\
0 & 0 & z
\end{bmatrix} \\

Matrix dots and indices
$A_{m,n} = \begin{pmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m,1} & a_{m,2} & \cdots & a_{m,n} \end{pmatrix} \\$
  A_{m,n} =
\begin{pmatrix}
a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\
a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\
\vdots  & \vdots  & \ddots & \vdots  \\
a_{m,1} & a_{m,2} & \cdots & a_{m,n}
\end{pmatrix} \\

Matrix with Round Bracket
$A =\left( \begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i \end{array} \right)\\$
     A =\left( \begin{array}{ccc}
a & b & c \\
d & e & f \\
g & h & i \end{array} \right)\\

Matrix with Curly Bracket
$A =\left \{ \begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i \end{array} \right \} \\$
     A =\left \{ \begin{array}{ccc}
a & b & c \\
d & e & f \\
g & h & i \end{array} \right \} \\

Matrix with Square Bracket
$A =\left[ \begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i \end{array} \right] \\$
     A =\left[ \begin{array}{ccc}
a & b & c \\
d & e & f \\
g & h & i \end{array} \right] \\

3x3 Matrix With Vertical Bar
$A = \left| \begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i \end{array} \right| \\$
     A = \left| \begin{array}{ccc}
a & b & c \\
d & e & f \\
g & h & i \end{array} \right|  \\

Matrix With Vertical Bar [simpler form]
$\begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}$
        \begin{vmatrix}
a_1 & a_2 & a_3 \\
b_1 & b_2 & b_3 \\
c_1 & c_2 & c_3
\end{vmatrix}

Matrix With Double Vertical Bar
$\begin{Vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{Vmatrix}$
        \begin{Vmatrix}
a_1 & a_2 & a_3 \\
b_1 & b_2 & b_3 \\
c_1 & c_2 & c_3
\end{Vmatrix}

Matrix Without brackets
$\begin{matrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{matrix}$
        \begin{matrix}
a_1 & a_2 & a_3 \\
b_1 & b_2 & b_3 \\
c_1 & c_2 & c_3
\end{matrix}

$M_{1}= \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ \end{bmatrix} \\ M_{i}= \begin{bmatrix} 0 & -1 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 0 &-1\\ 0 & 0 & 1 & 0\\ \end{bmatrix} \\ M_{j}= \begin{bmatrix} 0 & 0 &-1 & 0\\ 0 & 0 & 0 & 1\\ 1 & 0 & 0 & 0\\ 0 &-1 & 0 & 0\\ \end{bmatrix} \\ M_{k}= \begin{bmatrix} 0 & 0 & 0 &-1\\ 0 & 0 &-1 & 0\\ 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0\\ \end{bmatrix} \\ M_{-1}= \begin{bmatrix} -1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1\\ \end{bmatrix}$
                        M_{1}= \begin{bmatrix}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1\\
\end{bmatrix}

M_{i}= \begin{bmatrix}
0 & -1 & 0 & 0\\
1 & 0 & 0 & 0\\
0 & 0 & 0 &-1\\
0 & 0 & 1 & 0\\
\end{bmatrix} \\

M_{j}= \begin{bmatrix}
0 & 0 &-1 & 0\\
0 & 0 & 0 & 1\\
1 & 0 & 0 & 0\\
0 &-1 & 0 & 0\\
\end{bmatrix}

M_{k}= \begin{bmatrix}
0 & 0 & 0 &-1\\
0 & 0 &-1 & 0\\
0 & 1 & 0 & 0\\
1 & 0 & 0 & 0\\
\end{bmatrix} \\

M_{-1}= \begin{bmatrix}
-1 & 0 & 0 & 0\\
0 & -1 & 0 & 0\\
0 & 0 & -1 & 0\\
0 & 0 & 0 & -1\\
\end{bmatrix}

$\chi(\lambda) = \left| \begin{array}{ccc} \lambda - a & -b & -c \\ -d & \lambda - e & -f \\ -g & -h & \lambda - i \end{array} \right| \\$
    \chi(\lambda) = \left| \begin{array}{ccc}
\lambda - a & -b & -c \\
-d & \lambda - e & -f \\
-g & -h & \lambda - i \end{array} \right| \\

Inner Product with 2D Vector
$\left< \left[ \begin{array}{cc} 1 \\ 3 \end{array} \right]^{\ast} \,, \left[ \begin{array}{cc} 2 \\ 3 \end{array} \right]^{\ast} \right>$
                \left<
\left[ \begin{array}{cc}
1 \\
3
\end{array}
\right]^{\ast}
\,,
\left[ \begin{array}{cc}
2 \\
3
\end{array}
\right]^{\ast}
\right>

3D Vector
$v =\left[ \begin{array}{cc} c_1 \\ c_2 \\ c_n \end{array} \right]$
    v =\left[ \begin{array}{cc}
c_1 \\
c_2 \\
c_n
\end{array}
\right]

Vector with Dots
$v =\left[ \begin{array}{cc} c_1 \\ c_2 \\ \vdots \\ c_n \end{array} \right] \\$
     v =\left[ \begin{array}{cc}
c_1 \\
c_2 \\
\vdots \\
c_n
\end{array}
\right] \\

$\left[ \begin{array}{c} x_1 \\ x_2 \end{array} \right] = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \times \left[ \begin{array}{c} y_1 \\ y_2 \end{array} \right] \\$
 \left[ \begin{array}{c}
x_1 \\
x_2
\end{array} \right]
= \begin{bmatrix}
A & B \\
C & D
\end{bmatrix}
\times
\left[
\begin{array}{c}
y_1 \\
y_2
\end{array}
\right] \\

$\begin{bmatrix} xz & xw \\ yz & yw \end{bmatrix} = \left[ \begin{array}{c} x \\ y \end{array} \right] \times \left[ \begin{array}{cc} z & w \end{array} \right]\\$
   \begin{bmatrix}
xz & xw \\
yz & yw
\end{bmatrix} = \left[
\begin{array}{c}
x \\
y
\end{array}
\right] \times \left[
\begin{array}{cc}
z & w
\end{array}
\right]\\

$A =\begin{Bmatrix} x & y \\ z & v \end{Bmatrix} \\$
 A =\begin{Bmatrix}
x & y \\
z & v
\end{Bmatrix} \\

$A = \begin{pmatrix} x & y \\ z & v \end{pmatrix} \\$
     A = \begin{pmatrix}
x & y \\
z & v
\end{pmatrix} \\

$v = \left[w_1, w_2, w_3 \right] \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{bmatrix} \left[ \begin{array}{cc} c_1 \\ c_2 \\ c_3 \end{array} \right] \\$
 v =
\left[w_1, w_2, w_3 \right]
\begin{bmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}  \\
\end{bmatrix}
\left[ \begin{array}{cc}
c_1 \\
c_2 \\
c_3
\end{array}
\right] \\

$v = \left[w_1, w_2, w_3 \right] \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{bmatrix} \\$
 v =
\left[w_1, w_2, w_3 \right]
\begin{bmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}  \\
\end{bmatrix} \\

$v = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{bmatrix} \left[ \begin{array}{cc} c_1 \\ c_2 \\ c_3 \end{array} \right] \\$
     v =
\begin{bmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}  \\
\end{bmatrix}
\left[ \begin{array}{cc}
c_1 \\
c_2 \\
c_3
\end{array}
\right] \\

$v = \left[w_1, w_2, \cdots, w_n \right] \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix} \left[ \begin{array}{cc} c_1 \\ c_2 \\ \vdots \\ c_n \end{array} \right] \\$
     v =
\left[w_1, w_2, \cdots, w_n \right]
\begin{bmatrix}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots  & \ddots & \vdots  \\
a_{n1} & a_{n2} & \cdots & a_{nn}
\end{bmatrix}
\left[ \begin{array}{cc}
c_1 \\
c_2 \\
\vdots \\
c_n
\end{array}
\right] \\

$\begin{eqnarray} \textbf{C}'(t) &=& 3\left[ \textbf{B}_{2,{\color{red}-1}}(t) - \textbf{B}_{2,0}(t) \right]\textbf{P}_0 + \nonumber \\ && \> 3\left[ \textbf{B}_{2,0}(t) - \textbf{B}_{2,1}(t) \right]\textbf{P}_1 + \nonumber \\ && \> 3\left[ \textbf{B}_{2,1}(t) - \textbf{B}_{2,2}(t) \right] \textbf{P}_2+ \nonumber \\ && \> 3\left[ \textbf{B}_{2,2}(t) - \textbf{B}_{2,{\color{red}3}}(t) \right] \textbf{P}_3 \nonumber \\ &=& 3 \textbf{B}_{2,0}(t)(\textbf{P}_1 - \textbf{P}_0) + \nonumber \\ && \> 3 \textbf{B}_{2,1}(t)(\textbf{P}_2 - \textbf{P}_1) + \nonumber \\ && \> 3 \textbf{B}_{2,2}(t)(\textbf{P}_3 - \textbf{P}_2) \nonumber \\ \newline \end{eqnarray}$
\begin{eqnarray}
\textbf{C}'(t) &=& 3\left[ \textbf{B}_{2,{\color{red}-1}}(t) - \textbf{B}_{2,0}(t) \right]\textbf{P}_0 + \nonumber \\
&& \> 3\left[ \textbf{B}_{2,0}(t) - \textbf{B}_{2,1}(t) \right]\textbf{P}_1 + \nonumber \\
&& \> 3\left[ \textbf{B}_{2,1}(t) - \textbf{B}_{2,2}(t) \right] \textbf{P}_2+ \nonumber \\
&& \> 3\left[ \textbf{B}_{2,2}(t) - \textbf{B}_{2,{\color{red}3}}(t) \right] \textbf{P}_3 \nonumber \\
&=&  3 \textbf{B}_{2,0}(t)(\textbf{P}_1 - \textbf{P}_0) + \nonumber \\
&& \> 3 \textbf{B}_{2,1}(t)(\textbf{P}_2 - \textbf{P}_1) + \nonumber \\
&& \> 3 \textbf{B}_{2,2}(t)(\textbf{P}_3 - \textbf{P}_2) \nonumber \\
\newline
\end{eqnarray}

$\mathcal{O}(2^n) \\ \mathcal{O}(n\log{}n)$
                    \mathcal{O}(2^n) \\
\mathcal{O}(n\log{}n)

\begin{aligned} M_{z}(\beta) & =\begin{bmatrix} \cos\beta & -\sin\beta & 0\\ \sin\beta & \cos\beta & 0\\ 0 & 0 & 1 \end{bmatrix} \\ M_{y}(\beta) & =\begin{bmatrix} \cos \beta & \sin\beta & 0\\ 0 & 1 & 0 \\ -sin\beta & \cos\beta & 0 \end{bmatrix} \\ M_{x}(\beta) & =\begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\beta & -\sin\beta\\ 0 & \sin\beta& \cos\beta \end{bmatrix} \end{aligned}
\begin{aligned} M_{z}(\beta) & =\begin{bmatrix} \cos\beta & -\sin\beta & 0\\ \sin\beta & \cos\beta & 0\\ 0 & 0 & 1 \end{bmatrix} \\ M_{y}(\beta) & =\begin{bmatrix} \cos \beta & \sin\beta & 0\\ 0 & 1 & 0 \\ -sin\beta & \cos\beta & 0 \end{bmatrix} \\ M_{x}(\beta) & =\begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\beta & -\sin\beta\\ 0 & \sin\beta& \cos\beta \end{bmatrix} \end{aligned}

$\text{Equation}$ \begin{aligned} x & = y + 1 \\ y & = z + 1 \\ z & = x + 1 \end{aligned}
                    \begin{aligned} x & = y + 1 \\ y & = z + 1 \\ z & = x + 1 \end{aligned}

Limit
$\lim_{h \rightarrow 0} \frac{f(x + h)}{h}$
\lim_{h \rightarrow 0} \frac{f(x + h)}{h}

$\text{Inner Product}$ $\langle \vec{u} \,, \vec{v} \rangle \\ \langle \cdot \,, \cdot \rangle \\ \langle \,, \rangle \\ \left\langle u\mathbf{M}^{\ast} \,, v\right\rangle = \left\langle u \,, \mathbf{M}^{\ast} v\right\rangle$
\langle \vec{u} \,, \vec{v} \rangle \\
\langle \cdot \,, \cdot \rangle \\
\langle \,, \rangle \\
\left\langle u\mathbf{M}^{\ast} \,, v\right\rangle = \left\langle u \,, \mathbf{M}^{\ast} v\right\rangle

$\text{Derivatives}$ $\frac{du}{dt} \\ \frac{d^2u}{dx^{2}} \\ \frac{\partial u}{\partial t} = h^2 \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right)$
                \frac{du}{dt} \\
\frac{d^2u}{dx^{2}} \\

\frac{\partial u}{\partial t} = h^2 \left(
\frac{\partial^2 u}{\partial x^2} +
\frac{\partial^2 u}{\partial y^2} +
\frac{\partial^2 u}{\partial z^2}
\right)

$\text{Norm}$ $\|\vec{v} \|$
                \|\vec{v} \|

Series
\begin{aligned} \sin x & = \sum_{k=0}^{\infty} (-1)^k \frac{x^{2k+1}}{(2k+1)!} \\ \sin x & = \frac{x^1}{1!} - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \\ \cos x & = \sum_{k=0}^{\infty} (-1)^k \frac{x^{2k}}{2k!} \\ \cos x & = \frac{x^0}{0!} - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \end{aligned}
\begin{aligned} \sin x & = \sum_{k=0}^{\infty} (-1)^k \frac{x^{2k+1}}{(2k+1)!} \\ \sin x & = \frac{x^1}{1!} - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \\ \cos x & = \sum_{k=0}^{\infty} (-1)^k \frac{x^{2k}}{2k!} \\ \cos x & = \frac{x^0}{0!} - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \end{aligned}

Derivative evaluated at
\begin{aligned} M_{z}(\theta) & =\begin{bmatrix} \cos\theta & -\sin\theta & 0\\ \sin\theta & \cos\theta & 0\\ 0 & 0 & 1 \end{bmatrix} \\ & \Rightarrow \left. {\frac{d M_z}{d\theta}} \right|_{\theta=0} = \begin{bmatrix} 0 & -1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 0 \end{bmatrix} \end{aligned}
\begin{aligned} M_{z}(\theta) & =\begin{bmatrix} \cos\theta & -\sin\theta & 0\\ \sin\theta & \cos\theta & 0\\ 0 & 0 & 1 \end{bmatrix} & \Rightarrow \left. {\frac{d M_z}{d\theta}} \right|_{\theta=0} & = \begin{bmatrix} 0 & -1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 0 \end{bmatrix} \end{aligned}

Lie Algebra Symbol
$\mathfrak{so}(3) \tau \mathcal{T} \mathcal{M} \mathcal{U} \mathcal{X}$
                \mathfrak{so}(3)
\tau
\mathcal{T}
\mathcal{M}
\mathcal{U}
\mathcal{X}

Double newlines
$\mbox{\\\~\\\ or \medskip or \bigskip}$
                \\~\\
\medskip
\bigskip

$\mid \\ \nmid \\ \vert \\ \lvert \\ \big|_{\theta = 0}^{\theta = 2\pi} \\ \bigg|_{\theta = 0}^{\theta = 2\pi} \\ \left\langle \right\rangle \\ \left< \right> \\ \left( \right) \\$
                \mid \\
\nmid \\
\vert \\
\lvert \\
\big|_{\theta = 0}^{\theta = 2\pi} \\
\bigg|_{\theta = 0}^{\theta = 2\pi} \\
\left\langle \right\rangle \\
\left< \right> \\
\left( \right) \\

                \begin{scope}[xshift=4cm]
\node[main node] (1) {$1$};
\node[main node] (2) [right = 2cm  of 1] {$2$};
\node[main node] (3) [below = 2cm  of 1] {$3$};
\node[main node] (4) [right = 2cm  of 3] {$4$};
\path[draw,thick]
(1) edge[->] node {} (2)
(2) edge[->] node {} (4)
(4) edge[->] node {} (3)
(3) edge[->] node {} (1)
(4) edge[->] node {} (1);
\end{scope}

\begin{align*} \begin{split} g(x) &= x^2 + bx + c \\ f(x) &= x^4 + 3x^3 + 2x^2 + 3x + 10 \\ &\quad + x^4 + 3x^3 + 2x^2 + 3x + 10 \end{split} \end{align*}
                \begin{align*}
\begin{split}
g(x) &= x^2 + bx + c \\
f(x) &= x^4 + 3x^3 + 2x^2 + 3x + 10 \\
&\quad + x^4 + 3x^3 + 2x^2 + 3x + 10
\end{split}
\end{align*}

split long equation with aligned
\begin{aligned} F ={} & \{F_{x} \in F_{c} : (|S| > |C|) \\ & \cap (\mathrm{minPixels} < |S| < \mathrm{maxPixels}) \\ & \cap (|S_{\mathrm{conected}}|> |S| - \epsilon)\} \end{aligned}
                \begin{aligned} F ={} & \{F_{x} \in F_{c} : (|S| > |C|) \\ & \cap (\mathrm{minPixels} < |S| < \mathrm{maxPixels}) \\ & \cap (|S_{\mathrm{conected}}| > |S| - \epsilon)\} \end{aligned}

% generate png file: pdflatex -shell-escape file.tex
$\documentclass[border=4pt,convert={density=600,size=300x300,outext=.png}]{standalone}$
$\usepackage{tikz}$
$\usetikzlibrary{automata,positioning}$
$\begin{document}$
$\begin{tikzpicture}[shorten >=1pt,node distance=2cm,on grid,auto]$
$\node[state,initial] (q_0) {$q_0$};$
$\node[state] (q_1) [above right=of q_0] {$q_1$};$
$\node[state] (q_2) [below right=of q_0] {$q_2$};$
$\node[state,accepting](q_3) [below right=of q_1] {$q_3$};$
$\path[->]$
$(q_0) edge node {0} (q_1)$
$edge node [swap] {1} (q_2)$
$(q_1) edge node {1} (q_3)$
$edge [loop above] node {0} ()$
$(q_2) edge node [swap] {0} (q_3)$
$edge [loop below] node {1} ();$
$\end{tikzpicture}$
$\end{document}$

\documentclass[border=4pt,convert={density=800,size=500x300,outext=.png}]{standalone}
\usepackage{tikz}
\usetikzlibrary{automata,positioning}
\begin{document}
\begin{tikzpicture}[shorten >=1pt,node distance=4cm,on grid,auto]
\tikzstyle{every state}=[fill=red,draw=none,text=white]
\node[state,initial] (q_0)   {$q_0$};
\node[state] (q_1) [above right=of q_0] {$q_1$};
\node[state] (q_2) [below right=of q_1] {$q_2$};
\node[state, accepting] (q_3) [below right=of q_0] {$q_3$};
\path[->]
(q_0) edge  node {[} (q_1)
(q_1) edge [bend right] node [below] {0-9} (q_2)
edge [loop above] node {[} ()
(q_2) edge  node {]} (q_3)
edge [loop below] node {0-9} ()
(q_2) edge [bend right] node [above] {[} (q_1)
(q_1) edge [bend right] node {]} (q_3)
(q_3) edge [loop below] node {]} ()
(q_3) edge  node {[} (q_1);
\end{tikzpicture}
\end{document}

\documentclass[border=4pt,convert={density=800,size=600x600,outext=.png}]{standalone}
\usepackage{tikz}
\begin{document}
% first method
\begin{tikzpicture}[shorten >=1pt, auto, node distance=3cm, ultra thick,
node_style/.style={circle,draw=blue,fill=blue!20!,font=\sffamily\Large\bfseries},
edge_style/.style={draw=black, ultra thick}]
\node[node_style] (v1) at (-2,2) {2};
\node[node_style] (v2) at (2,2) {3};
\node[node_style] (v3) at (4,0) {6};
\node[node_style] (v4) at (2,-2) {4};
\node[node_style] (v5) at (-2,-2) {5};
\node[node_style] (v6) at (-4,0) {1};
\draw[edge_style]  (v1) edge node{1} (v2);
\draw[edge_style]  (v2) edge node{2} (v3);
\draw[edge_style]  (v3) edge node{3} (v4);
\draw[edge_style]  (v4) edge node{4} (v5);
\draw[edge_style]  (v5) edge node{5} (v6);
\draw[edge_style]  (v6) edge node{6} (v1);
\draw[edge_style]  (v5) edge node{7} (v1);
\draw[edge_style]  (v5) edge node{8} (v2);
\draw[edge_style]  (v4) edge node{9} (v2);
\end{tikzpicture}
\end{document}


\begin{align*}
&\mathbb{R}  \quad \text{ is Real Number } \\
&\mathbb{C}  \quad \text{ is Complex Number} \\
&\mathbb{O}  \quad \text{ is Octonion } \\
&\mathbb{H}  \quad \text{ is Quaternion } \\
&\mathbb{N}  \quad \text{ is Integer } \\
&\mathbb{Q}  \quad \text{ is Rational Number } \\
&\mathbb{Z}  \quad \text{ is Complex Number} \\
&\mathbb{I}  \quad \text{ is Irrational Number} \\
&\mathbb{P}  \quad \text{ is Prime Number } \\
&\mathbb{W}  \quad \text{ is Whole, I never see the def.} \\
&\mathbb{A}  \quad \text{ is Algebra Number } \\
& z = a + bi \quad \text{ complex number} \\
&\Re         \quad \text{ is Real Part} \\
&\Im         \quad \text{ is Imaginary Part} \\
& \Re(z) = a \quad \text{ or } \operatorname{Re}(z) \\
& \Im(z) = b \quad \text{ or } \operatorname{Im}(z) \\
\end{align*}


\begin{equation*} \left.\begin{aligned} B’&=-\partial\times E,\\ E’&=\partial\times B - 4\pi j, \end{aligned} \right\} \qquad \text{Maxwell’s equations} \end{equation*} \begin{equation*} \left\{\begin{aligned} B’&=-\partial\times E,\\ E’&=\partial\times B - 4\pi j, \end{aligned} \right\} \qquad \text{Maxwell’s equations} \end{equation*}

\left.\begin{aligned}
B’&=-\partial\times E,\\
E’&=\partial\times B - 4\pi j,
\end{aligned}
\right\}
\end{equation*}

\begin{equation*}
\left\{\begin{aligned}
B’&=-\partial\times E,\\
E’&=\partial\times B - 4\pi j,
\end{aligned}
\right\}

$f(x)= \begin{cases} \frac{x^2-x}{x},& \text{if } x\geq 1\\ 0, & \text{otherwise} \end{cases}$
f(x)=

$\int_{0}^{2\pi} \sin{\theta} d \theta \bigg|_{\theta = 0}^{\theta = 2\pi}$
$\int_{0}^{2\pi} \sin{\theta} d \theta \bigg|_{\theta = 0}^{\theta = 2\pi}$