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)

——————————————————–

\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

——————————————————–

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'

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