\[ \vec{v} \\ \vec{AB} \\ \bar{v} \\ \bar{p_0} \\ \overline{p_0} \\ \overline{v} \\ \overline{AB} \\ \overrightarrow{AB} \\ \]
        \[
            \vec{v}             \\ 
            \vec{AB}            \\ 
            \bar{v}             \\ 
            \bar{p_0}           \\ 
            \overline{p_0}      \\ 
            \overline{v}        \\ 
            \overline{AB}       \\ 
            \overrightarrow{AB} \\ 
        \]
                    
\[ \cos{\alpha} = \frac{\vec{BA} \vec{BC}}{|\vec{BA}| | \vec{BC} |} \\ \cos{\alpha} = \frac{\vec{BA} \, \vec{BC}}{|\vec{BA}| | \vec{BC} |} \\ \cos{\alpha} = \frac{\vec{BA} \quad \vec{BC}}{|\vec{BA}| | \vec{BC} |} \\ \cos{\alpha} = \frac{\vec{BA} \qquad \vec{BC}}{|\vec{BA}| | \vec{BC} |} \\ \]

        \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{equation} \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} \end{equation}
        \begin{equation}
        \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}
        \end{equation} 
                    
\[ I_3 = \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix} \]
     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)
                
\[ \begin{equation} \begin{aligned} M_{z}(\beta) & =\begin{bmatrix} \cos\beta & -\sin\beta & 0\\ \sin\beta & \cos\beta & 0\\ 0 & 0 & 1 \end{bmatrix} \\ M_{y}(\beta) & =\begin{bmatrix} \cos \beta & \sin\beta & 0\\ 0 & 1 & 0 \\ -sin\beta & \cos\beta & 0 \end{bmatrix} \\ M_{x}(\beta) & =\begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\beta & -\sin\beta\\ 0 & \sin\beta& \cos\beta \end{bmatrix} \end{aligned} \end{equation} \]
\begin{equation}
\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}
\end{equation}
                
\[ \text{Equation} \] \[ \begin{equation} \begin{aligned} x & = y + 1 \\ y & = z + 1 \\ z & = x + 1 \end{aligned} \end{equation} \]
                    \begin{equation}
                    \begin{aligned}
                    x & = y + 1 \\
                    y & = z + 1 \\
                    z & = x + 1 
                    \end{aligned}
                    \end{equation}
                
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
\[ \begin{equation} \begin{aligned} \sin x & = \sum_{k=0}^{\infty} (-1)^k \frac{x^{2k+1}}{(2k+1)!} \\ \sin x & = \frac{x^1}{1!} - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \\ \cos x & = \sum_{k=0}^{\infty} (-1)^k \frac{x^{2k}}{2k!} \\ \cos x & = \frac{x^0}{0!} - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \end{aligned} \end{equation} \]
\begin{equation}
\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}
\end{equation}
                
Derivative evaluated at
\begin{equation} \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} \end{equation}
\begin{equation}
\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}
\end{equation}
                
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
                
\[ \mid \\ \nmid \\ \vert \\ \lvert \\ \left\langle \right\rangle \\ \left< \right> \\ \left( \right) \\ \]
                \mid \\
                \nmid \\
                \vert \\
                \lvert \\
                \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{equation} \begin{aligned} F ={} & \{F_{x} \in F_{c} : (|S| > |C|) \\ & \cap (\mathrm{minPixels} < |S| < \mathrm{maxPixels}) \\ & \cap (|S_{\mathrm{conected}}|> |S| - \epsilon)\} \end{aligned} \end{equation}
                \begin{equation}
                \begin{aligned}
                F ={} & \{F_{x} \in  F_{c} : (|S| > |C|) \\
                      & \cap (\mathrm{minPixels}  < |S| < \mathrm{maxPixels}) \\
                      & \cap (|S_{\mathrm{conected}}| > |S| - \epsilon)\}
                \end{aligned}
                \end{equation}
                
% 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{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\}
\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*}

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