# Notes On Relation Between Latex And Latex

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documentclass[11pt]{article} usepackage{graphicx} usepackage{a4wide} ewtheorem{theorem}{Theorem}[section] ewtheorem{corollary}[theorem]{Corollary} ewtheorem{lemma}[theorem]{Lemma} ewtheorem{proposition}[theorem]{Proposition} ewtheorem{definition}[theorem]{Definition} ewtheorem{remark}[theorem]{Remark} ewtheorem{assumption}[theorem]{Assumption} ewtheorem{conjecture}[theorem]{Conjecture} ewtheorem{example}{underline{Example}} setlength{parindent}{0mm} %============================================= % % Comments in latex are marked with a % and do not print % %============================================= % % To include figures save then either as eps or as jpg or pdf. % %=============================================…show more content…
vspace{0.5cm} The flowmap denoted \$phi_{t,t_{0}}in Diff(R^{N})\$ where \$Diff(R^{N})\$ denotes the group of diffeomorphisms of \$R^{N}\$ is defined as a map that takes the initial solution, that is the solution at \$t_{0}\$ to the solution at any time \$t\$, this can be expressed mathematically by egin{center} \$phi_{t,t_{0}}: Y_{t_{0}}longmapsto Y_{t}\$ end{center} That is to say, given any initial data \$Y_{0}in R^{N}\$, the solution \$ Y_{t}\$ at any later time can be easily specified. This can be done by applying the action of the flowmap to the initial data \$Y_{0}\$ in order to \$Y_{t}=phi_{t,t_{0}} circ Y_{t_{0}}\$. Consider a function \$fin Diff(R^{N})\$, by using the chain rule, we obtain \$ frac{d}{dt}(f(Y))=V(Y).partial_{Y} f(Y)\$ This means that action of the vector fields on \$Diff(R^{N})\$ is as first order partial differential operators since \$V(Y).partial_{Y} f(Y)\$ is considered as first order partial differential operators. vspace{0.5cm} The evolution of \$fcircphi_{t}\$ is given by egin{center} \$ frac{d}{dt}(fcircphi_{t})=Vcirc fcircphi_{t}\$ end{center} The above equation is an autonomous linear functional differential equation for \$fcircphi_{t}\$. Such equation has a solution \$fcircphi_{t}=exp(tV)circ fcircphi_{t}\$ as \$phi_{0}=id\$ Setting \$f=id\$ provides the representation of the flowmap as follows egin{center} \$phi_{t}=exp(tV)\$ end{center} Hence, in this considered case, the flowmap is the exponential of the vector field. By compositing the above equation