# LaTeX q

$N_2={M\ddot{o}}_1\cup _{\partial}{M\ddot{o}}_2$

$S={M\ddot{o}}_1\cap {M\ddot{o}}_2\cong S^1$

${\cal{N}}(S)\cong S\times I\quad {\rm{rel}}\quad N_2$

but if $D=$ corë of $M\ddot{o}$ the ${\cal{N}}(D)\cong D\tilde{\times} I\cong M\ddot{o}\quad {\rm{rel}}\quad N_2$

but if $C\ddot{o}=$ corë of $M\ddot{o}$ the ${\cal{N}}(C\ddot{o})\cong C\ddot{o}\tilde{\times} I\cong M\ddot{o}\quad {\rm{rel}}\quad N_2$

$\partial N(C)=C\times\{0,1\}\cong S^1\sqcup S^1$

$\partial M\ddot{o}\cong S^1$

MATH-LATEX

$\left(\begin{array}{c}x\\y\end{array}\right)\stackrel{T}\mapsto\left(\begin{array}{cc}1 & 2\\3 & 4\\-6 & 5 \end{array}\right)\left(\begin{array}{c}x\\y\end{array}\right)=\left(\begin{array}{c}x+2y\\3x+4y\\-6x+5y\end{array}\right)$

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$\Box$

$\Xi$

$\sqsubset\Box\overbrace{\underbrace{\Omega\qquad\Lambda\Phi\Box\Box\Box\Phi\Lambda\qquad\Omega}_{\sqcap}}^{\sqcup} \Box\sqsupset$

$\cdot$

$\cdot$

$\cdot$

$\cdot$

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$D_TT=\left(\begin{array}{ccc}\frac{\partial}{\partial x}(\frac{dx}{dt})&\frac{\partial}{\partial y}(\frac{dx}{dt})&\frac{\partial}{\partial z}(\frac{dx}{dt})\\\frac{\partial}{\partial x}(\frac{dy}{dt})&\frac{\partial}{\partial y}(\frac{dy}{dt})&\frac{\partial}{\partial z}(\frac{dy}{dt})\\\frac{\partial}{\partial x}(\frac{dz}{dt})&\frac{\partial}{\partial y}(\frac{dz}{dt})&\frac{\partial}{\partial z}(\frac{dz}{dt})\end{array}\right)\left(\begin{array}{c}\frac{dx}{dt}\\\frac{dy}{dt}\\\frac{dz}{dt} \end{array}\right)$

$=\left(\begin{array}{c}\frac{\partial}{\partial x}(\frac{dx}{dt})\frac{dx}{dt}+\frac{\partial}{\partial y}(\frac{dx}{dt})\frac{dy}{dt}+\frac{\partial}{\partial z}(\frac{dx}{dt})\frac{dz}{dt}\\\frac{\partial}{\partial x}(\frac{dy}{dt})\frac{dx}{dt}+\frac{\partial}{\partial y}(\frac{dy}{dt})\frac{dy}{dt}+\frac{\partial}{\partial z}(\frac{dy}{dt})\frac{dz}{dt}\\\frac{\partial}{\partial x}(\frac{dz}{dt})\frac{dx}{dt}+\frac{\partial}{\partial y}(\frac{dz}{dt})\frac{dy}{dt}+\frac{\partial}{\partial z}(\frac{dz}{dt})\frac{dz}{dt}\end{array}\right)$

$=T'$