From d50ada037466d7156da6339362fe8a9db30ffc25 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?V=C3=ADctor?= Date: Wed, 8 Nov 2023 17:25:39 +0100 Subject: [PATCH] reoplaced ODE with space --- .../Differential_equations.tex | 110 +++++++++--------- .../Differential_geometry.tex | 4 +- .../Dynamical_systems/Dynamical_systems.tex | 6 +- .../Harmonic_analysis/Harmonic_analysis.tex | 4 +- .../Numerical_calculus/Numerical_calculus.tex | 2 +- .../Partial_differential_equations.tex | 4 +- .../Stochastic_processes.tex | 2 +- .../Montecarlo_methods/Montecarlo_methods.tex | 2 +- .../Classical_mechanics.tex | 10 +- 9 files changed, 72 insertions(+), 72 deletions(-) diff --git a/Mathematics/3rd/Differential_equations/Differential_equations.tex b/Mathematics/3rd/Differential_equations/Differential_equations.tex index d5e44bb..86046e5 100644 --- a/Mathematics/3rd/Differential_equations/Differential_equations.tex +++ b/Mathematics/3rd/Differential_equations/Differential_equations.tex @@ -33,21 +33,21 @@ where $\vf{g}:\Omega\subseteq\RR\times\RR^{m\cdot n}\rightarrow\RR^m$\footnote{Sometimes we will write $\vf{x}^{(n)}=\vf{g}\left(t,\vf{x},\vf{x}',\ldots,\vf{x}^{(n-1)}\right)$ instead of $\vf{x}^{(n)}(t)=\vf{g}\left(t,\vf{x}(t),\vf{x}'(t),\ldots,\vf{x}^{(n-1)}(t)\right)$ in order to simplify the notation.}. \end{definition} \begin{definition} - Consider the followingODE of $m$ unknowns and of order $n$: + Consider the following ODE of $m$ unknowns and of order $n$: \begin{equation}\label{DE:ode1} \vf{x}^{(n)}(t)=\vf{f}\left(t,\vf{x}(t),\vf{x}'(t),\ldots,\vf{x}^{(n-1)}(t)\right) \end{equation} - We say that $\vf{\varphi}:I\subseteq\RR\rightarrow\RR^m$ is a \emph{solution of theODE} if: + We say that $\vf{\varphi}:I\subseteq\RR\rightarrow\RR^m$ is a \emph{solution of the ODE} if: \begin{itemize} \item $\vf{\varphi}$ is $n$ times differentiable on $I$. \item $\displaystyle\left\{\left(t,\vf{\varphi}(t),\vf{\varphi}'(t),\ldots,\vf{\varphi}^{(n-1)}(t)\right):t\in I\right\}\subseteq\domain \vf{f}$ \item For all $t\in I$ we have: $$\vf{\varphi}^{(n)}(t)=\vf{f}\left(t,\vf{\varphi}(t),\vf{\varphi}'(t),\ldots,\vf{\varphi}^{(n-1)}(t)\right)$$ \end{itemize} - The set of all solutions of anODE is called \emph{general solution of theODE}. + The set of all solutions of an ODE is called \emph{general solution of the ODE}. \end{definition} \begin{proposition} - Consider anODE of $m$ unknowns and order $n$ of the form of \mcref{DE:ode1}. Then, we can transform thisODE to anODE of $m\cdot n$ unknowns and order 1 in the following way\footnote{Therefore, we will mainly study theODEs of order 1.}. Define $\vf{y}_i=\vf{x}^{(i-1)}$ for $i=1,\ldots,n$. Therefore, the functions $\vf{y}_i$ must satisfy: + Consider an ODE of $m$ unknowns and order $n$ of the form of \mcref{DE:ode1}. Then, we can transform this ODE to an ODE of $m\cdot n$ unknowns and order 1 in the following way\footnote{Therefore, we will mainly study the ODEs of order 1.}. Define $\vf{y}_i=\vf{x}^{(i-1)}$ for $i=1,\ldots,n$. Therefore, the functions $\vf{y}_i$ must satisfy: \begin{equation*} \left\{ \begin{aligned} @@ -62,25 +62,25 @@ This is called a \emph{system of ordinary differential equations} (of order 1) or a \emph{differential system}. \end{proposition} \begin{definition} - We say that anODE is \emph{autonomous} if it doesn't depend on the independent variable, that is, if it is of the form: $$\vf{x}'=\vf{f}(\vf{x})$$ Otherwise, we say that anODE is \emph{non-autonomous}. + We say that an ODE is \emph{autonomous} if it doesn't depend on the independent variable, that is, if it is of the form: $$\vf{x}'=\vf{f}(\vf{x})$$ Otherwise, we say that an ODE is \emph{non-autonomous}. \end{definition} \begin{definition} - We say that anODE of order $n$ is \emph{linear} if it is of the form: + We say that an ODE of order $n$ is \emph{linear} if it is of the form: \begin{equation}\label{DE:linear0} a_0(t)\vf{x}+a_1(t)\vf{x}'+\cdots+a_n(t)\vf{x}^{(n)}=\vf{b}(t) \end{equation} - where $a_i\in\mathcal{C}(I,\RR)$ for $i=0,\ldots,n$ and $\vf{b}\in\mathcal{C}(I,\RR^m)$ are arbitrary functions which do not need to be linear. We say that the linearODE of \mcref{DE:linear0} is \emph{homogeneous} if $\vf{b}(t)=\vf{0}$ $\forall t\in I$. We say that linearODE of \mcref{DE:linear0} is of \emph{constant coefficients} if $a_i(t):=a_{i0}\in\RR$ $\forall t\in I$ and $\forall i=0,\ldots,n$. + where $a_i\in\mathcal{C}(I,\RR)$ for $i=0,\ldots,n$ and $\vf{b}\in\mathcal{C}(I,\RR^m)$ are arbitrary functions which do not need to be linear. We say that the linear ODE of \mcref{DE:linear0} is \emph{homogeneous} if $\vf{b}(t)=\vf{0}$ $\forall t\in I$. We say that linear ODE of \mcref{DE:linear0} is of \emph{constant coefficients} if $a_i(t):=a_{i0}\in\RR$ $\forall t\in I$ and $\forall i=0,\ldots,n$. \end{definition} \begin{definition}[Initial value problem] - Let $U\subseteq\RR\times\RR^n$ be an open set and $\vf{f}:U\rightarrow\RR^n$ be a function. Given $(t_0,\vf{x}_0)\in U$, the \emph{initial value problem} (\emph{ivp}) (or \emph{Cauchy problem}) consists in finding a solution of theODE $$\vf{x}'=\vf{f}(t,\vf{x})$$ with \emph{initial conditions} $\vf{x}(t_0)=\vf{x}_0$. + Let $U\subseteq\RR\times\RR^n$ be an open set and $\vf{f}:U\rightarrow\RR^n$ be a function. Given $(t_0,\vf{x}_0)\in U$, the \emph{initial value problem} (\emph{ivp}) (or \emph{Cauchy problem}) consists in finding a solution of the ODE $$\vf{x}'=\vf{f}(t,\vf{x})$$ with \emph{initial conditions} $\vf{x}(t_0)=\vf{x}_0$. \end{definition} - \subsubsection{Methods for solvingODEs} + \subsubsection{Methods for solving ODEs} \begin{proposition}[Separation of variables] - Let $f:(a,b)\rightarrow\RR$, $g:(c,d)\rightarrow\RR$ be continuous functions such that $f(x)\ne 0$ $\forall x\in (a,b)$. Consider theODE $x'=f(x)g(t)$. To find the solution of thisODE, proceed as follows: - $$x'=f(x)g(t)\iff \int\frac{\dd{x}}{f(x)}=C+\int g(t)\dd{t}$$ where the constant $C$ is determined with the initial conditions of theODE. + Let $f:(a,b)\rightarrow\RR$, $g:(c,d)\rightarrow\RR$ be continuous functions such that $f(x)\ne 0$ $\forall x\in (a,b)$. Consider the ODE $x'=f(x)g(t)$. To find the solution of this ODE, proceed as follows: + $$x'=f(x)g(t)\iff \int\frac{\dd{x}}{f(x)}=C+\int g(t)\dd{t}$$ where the constant $C$ is determined with the initial conditions of the ODE. \end{proposition} \begin{proposition}[Variation of constants] - Let $I\subset \RR$ be an interval, $a,b\in\mathcal{C}(I,\RR)$. Consider theODE $x'=a(t)x+b(t)$. To find the solution of thisODE, proceed as follows: + Let $I\subset \RR$ be an interval, $a,b\in\mathcal{C}(I,\RR)$. Consider the ODE $x'=a(t)x+b(t)$. To find the solution of this ODE, proceed as follows: \begin{enumerate} \item Find the solution of the associated homogeneous system with the separation of variables method. Let's say that is $\varphi(t)c$, where $c\in\RR$. \item Try to find a general solution of the form $\varphi(t)c(t)$: @@ -91,12 +91,12 @@ \end{enumerate} \end{proposition} \begin{proposition}[Characteristic equation] - Consider the followingODE of order $n$ of constant coefficients: + Consider the following ODE of order $n$ of constant coefficients: \begin{equation}\label{DE:char} x^{(n)} + a_{n-1}x^{(n-1)} + \cdots + a_1 x' + a_0 x = 0 \end{equation} We define the \emph{characteristic equation} of that system as the equation: $$p(r):=r^n + a_{n-1}r^{n-1} + \cdots + a_1 r + a_0 = 0$$ - In order to find the solution of thisODE, we need to find the solutions to $p(r)=0$. So suppose $p$ has $s$ distinct real roots and $2(m-s)$ distinct complex roots. + In order to find the solution of this ODE, we need to find the solutions to $p(r)=0$. So suppose $p$ has $s$ distinct real roots and $2(m-s)$ distinct complex roots. $$\lambda_1,\ldots,\lambda_s,\lambda_{s+1},\overline{\lambda_{s+1}},\ldots,\lambda_{m},\overline{\lambda_m}$$ Here, $\lambda_i\in\RR$ $\forall i=1,\ldots,s$ and $\lambda_{i}=\alpha_i+\ii\beta_i\in\CC$ $\forall i=s+1,\ldots,m$. Assume, each of these roots have multiplicity $k_i\in\NN$. Then, the general solution to \mcref{DE:char} is: \begin{multline*} @@ -118,7 +118,7 @@ Then, the characteristic equation is precisely the characteristic polynomial of $\vf{A}$. \end{proposition} \begin{corollary} - Consider the followingODE of order $n$ of constant coefficients: + Consider the following ODE of order $n$ of constant coefficients: \begin{equation}\label{DE:char2} x'' + px' + q = 0 \end{equation} @@ -130,8 +130,8 @@ \item If $p^2-4q<0$, then $\lambda_1=\alpha+\ii\beta\in\CC$ and the solution is: $$\varphi(t)=\exp{\alpha t}\left[c_1\cos(\beta t)+c_2\sin(\beta t)\right]$$ \end{itemize} \end{corollary} - \begin{proposition}[Reducible linearODE of second order] - Let $I\subset \RR$ be an interval, $a,b,c,d\in\mathcal{C}(I,\RR)$. Consider the system ofODEs: + \begin{proposition}[Reducible linear ODE of second order] + Let $I\subset \RR$ be an interval, $a,b,c,d\in\mathcal{C}(I,\RR)$. Consider the system of ODEs: \begin{equation}\label{DE:ode-complex} \left\{ \begin{aligned} @@ -140,27 +140,27 @@ \end{aligned} \right. \end{equation} - In order to find the solution of thisODE, consider the change of variable $z=x+\ii y$. Then, \mcref{DE:ode-complex} becomes: - $$z'=[a(t)+\ii b(t)]z+c(t)+\ii d(t)$$ which is a linearODE of order 1 and can be easily solved. + In order to find the solution of this ODE, consider the change of variable $z=x+\ii y$. Then, \mcref{DE:ode-complex} becomes: + $$z'=[a(t)+\ii b(t)]z+c(t)+\ii d(t)$$ which is a linear ODE of order 1 and can be easily solved. \end{proposition} \begin{proposition}[Bernoulli differential equation] Let $p,q\in\mathcal{C}((a,b),\RR)$ and $\alpha\in\RR$. Consider the \emph{Bernoulli differential equation}: \begin{equation}\label{DE:bernoulli} x'+p(t)x=q(t)x^\alpha \end{equation} - If $\alpha=0,1$ theODE is linear. So suppose $\alpha\ne 0,1$. In order to solve it, consider the change of variable $y=x^{1-\alpha}$. Then, \mcref{DE:bernoulli} becomes: - $$y'+(1-\alpha)p(t)y=(1-\alpha)q(t)$$ which is a linearODE of order 1 and can be easily solved. + If $\alpha=0,1$ the ODE is linear. So suppose $\alpha\ne 0,1$. In order to solve it, consider the change of variable $y=x^{1-\alpha}$. Then, \mcref{DE:bernoulli} becomes: + $$y'+(1-\alpha)p(t)y=(1-\alpha)q(t)$$ which is a linear ODE of order 1 and can be easily solved. \end{proposition} \begin{proposition}[Riccati differential equation] Let $q_0,q_1,q_2\in\mathcal{C}((a,b),\RR)$. Consider the \emph{Riccati differential equation}: \begin{equation}\label{DE:riccati} x'=q_0(t)+q_1(t)x+q_2(t)x^2 \end{equation} - Suppose we have found a particular solution $x_1(t)$ of theODE of \mcref{DE:riccati}. In order to find the general solution, consider the change of variable $x=x_1(t)+\frac{1}{y}$. Then, \mcref{DE:riccati} becomes: - $$y'+[q_1(t)+2q_2(t)x_1(t)]y=-q_2(t)$$ which is a linearODE of order 1 and can be easily solved. + Suppose we have found a particular solution $x_1(t)$ of the ODE of \mcref{DE:riccati}. In order to find the general solution, consider the change of variable $x=x_1(t)+\frac{1}{y}$. Then, \mcref{DE:riccati} becomes: + $$y'+[q_1(t)+2q_2(t)x_1(t)]y=-q_2(t)$$ which is a linear ODE of order 1 and can be easily solved. \end{proposition} \begin{proposition}[Integrating factor] - Consider theODE: $$p(t,x)+q(t,x)x'=0\iff p(t,x)\dd{t}+q(t,x)\dd{x}=0$$ where $p,q\in\mathcal{C}^1(U,\RR)$ and $U\subseteq\RR^2$ is an open set. + Consider the ODE: $$p(t,x)+q(t,x)x'=0\iff p(t,x)\dd{t}+q(t,x)\dd{x}=0$$ where $p,q\in\mathcal{C}^1(U,\RR)$ and $U\subseteq\RR^2$ is an open set. An \emph{integrating factor} $\mu(t,x)\in\mathcal{C}^1(U)$, $\mu(t,x)\ne 0$, is a function so that $$\mu(t,x)p(t,x)\dd{t}+\mu(t,x)q(t,x)\dd{x}$$ is an exact differential ($\dd{\Phi(t,x)}$) of a function $\Phi(t,x)$, that is: \begin{gather} \label{DE:ifactor1}\frac{\partial\Phi}{\partial t}(t,x)=\mu(t,x)p(t,x)\\ @@ -363,7 +363,7 @@ We say that linear equation of \mcref{DE:linear} is \emph{homogeneous} if $\vf{b}(t)=\vf{0}$ $\forall t\in I$. We say that linear equation of \mcref{DE:linear} is of \emph{constant coefficients} if $\vf{A}(t)=\vf{A}$ $\forall t\in I$, where $\vf{A}\in\mathcal{M}_n(\RR)$. \end{definition} \begin{definition} - Let $I\subseteq\RR$ be an interval, $t_0\in I$, $\vf{x}_0\in\RR^n$ and consider theODE of \mcref{DE:linear}. We define the \emph{flow of the linearODE} as the function: + Let $I\subseteq\RR$ be an interval, $t_0\in I$, $\vf{x}_0\in\RR^n$ and consider the ODE of \mcref{DE:linear}. We define the \emph{flow of the linear ODE} as the function: $$ \function{\vf{\phi}}{I\times I\times \RR^n}{\RR^n}{(t,t_0,\vf{x}_0)}{\vf{\varphi}_{(t_0,\vf{x}_0)}(t)} $$ @@ -383,7 +383,7 @@ \end{proposition} \subsubsection{Homogeneous systems} \begin{theorem} - Let $I\subseteq\RR$ be an interval and $\vf{A}\in\mathcal{C}(I,\mathcal{L}(\RR^n))$. We define $\mathcal{A}_n$ as the set of all solutions of the linearODE: + Let $I\subseteq\RR$ be an interval and $\vf{A}\in\mathcal{C}(I,\mathcal{L}(\RR^n))$. We define $\mathcal{A}_n$ as the set of all solutions of the linear ODE: \begin{equation}\label{DE:homo} \vf{x}'=\vf{A}(t)\vf{x} \end{equation} Then, $\mathcal{A}_n$ is a vector space of dimension $n$ and for each $t_0\in I$, the function @@ -415,37 +415,37 @@ \end{enumerate} \end{corollary} \begin{definition} - Let $I\subseteq \RR$ be an interval, $\vf{A}\in\mathcal{C}(I,\mathcal{L}(\RR^n))$ and $\vf{M}(t)=(m_{ij}(t))\in\mathcal{M}_n(\RR)$. We say that $\vf{M}(t)$ is a \emph{matrix solution} of theODE of \mcref{DE:homo} if $\vf{\varphi}_j={(m_{1j}(t),\ldots,m_{nj}(t))}^\mathrm{T}\in\mathcal{A}_n$ for $j=1,\ldots,n$. We say that $\vf{M}(t)$ is a \emph{fundamental matrix solution} of theODE of \mcref{DE:homo} if $\vf{M}(t)$ is a matrix solution and $\vf{\varphi}_1,\ldots,\vf{\varphi}_n$ are linearly independent. + Let $I\subseteq \RR$ be an interval, $\vf{A}\in\mathcal{C}(I,\mathcal{L}(\RR^n))$ and $\vf{M}(t)=(m_{ij}(t))\in\mathcal{M}_n(\RR)$. We say that $\vf{M}(t)$ is a \emph{matrix solution} of the ODE of \mcref{DE:homo} if $\vf{\varphi}_j={(m_{1j}(t),\ldots,m_{nj}(t))}^\mathrm{T}\in\mathcal{A}_n$ for $j=1,\ldots,n$. We say that $\vf{M}(t)$ is a \emph{fundamental matrix solution} of the ODE of \mcref{DE:homo} if $\vf{M}(t)$ is a matrix solution and $\vf{\varphi}_1,\ldots,\vf{\varphi}_n$ are linearly independent. \end{definition} \begin{proposition} Let $I\subseteq \RR$ be an interval, $\vf{A}\in\mathcal{C}(I,\mathcal{L}(\RR^n))$ and $\vf{M}(t)\in\mathcal{M}_n(\RR)$. Then: \begin{enumerate} - \item $\vf{M}(t)$ is a matrix solution of theODE of \mcref{DE:homo} $\iff\vf{M}'(t)=\vf{A}(t)\vf{M}(t)$\footnote{By definition, if $\vf{M}(t)=(m_{ij}(t))$, then $\vf{M}'(t):=({m_{ij}}'(t))$.}. - \item $\vf{M}(t)$ is a matrix solution of theODE of \mcref{DE:homo} $\iff\forall \vf{c}\in\RR^n$, $\vf{M}(t)\vf{c}\in\mathcal{A}_n$. - \item If $\vf{M}(t)$ is a matrix solution of theODE of \mcref{DE:homo}, then $\forall \vf{C}\in\mathcal{M}_n(\RR)$, $\vf{M}(t)\vf{C}$ is a matrix solution of theODE of \mcref{DE:homo}. - \item If $\vf{M}(t)$ is a fundamental matrix solution of theODE of \mcref{DE:homo}, then $\det\vf{M}(t)\ne 0$ $\forall t\in I$. - \item $\vf{M}(t)$ is a fundamental matrix solution of theODE of \mcref{DE:homo} $\iff\vf{M}(t)$ is a matrix solution of theODE of \mcref{DE:homo} and $\exists t_0\in I$ such that $\det\vf{M}(t_0)\ne 0$. + \item $\vf{M}(t)$ is a matrix solution of the ODE of \mcref{DE:homo} $\iff\vf{M}'(t)=\vf{A}(t)\vf{M}(t)$\footnote{By definition, if $\vf{M}(t)=(m_{ij}(t))$, then $\vf{M}'(t):=({m_{ij}}'(t))$.}. + \item $\vf{M}(t)$ is a matrix solution of the ODE of \mcref{DE:homo} $\iff\forall \vf{c}\in\RR^n$, $\vf{M}(t)\vf{c}\in\mathcal{A}_n$. + \item If $\vf{M}(t)$ is a matrix solution of the ODE of \mcref{DE:homo}, then $\forall \vf{C}\in\mathcal{M}_n(\RR)$, $\vf{M}(t)\vf{C}$ is a matrix solution of the ODE of \mcref{DE:homo}. + \item If $\vf{M}(t)$ is a fundamental matrix solution of the ODE of \mcref{DE:homo}, then $\det\vf{M}(t)\ne 0$ $\forall t\in I$. + \item $\vf{M}(t)$ is a fundamental matrix solution of the ODE of \mcref{DE:homo} $\iff\vf{M}(t)$ is a matrix solution of the ODE of \mcref{DE:homo} and $\exists t_0\in I$ such that $\det\vf{M}(t_0)\ne 0$. \end{enumerate} \end{proposition} \begin{proposition} - Let $I\subseteq \RR$ be an interval, $\vf{A}\in\mathcal{C}(I,\mathcal{L}(\RR^n))$ and $\vf{\Phi}(t),\vf{\psi}(t)\in\mathcal{M}_n(\RR)$ be matrix solutions of theODE of \mcref{DE:homo} such that $\vf{\Phi}(t)$ is fundamental. Then, $\exists! \vf{C}\in\mathcal{M}_n(\RR)$ such that $\vf{\psi}(t)=\vf{\Phi}(t)\vf{C}$. Moreover, $\vf{\psi}(t)$ is fundamental if and only if $\det \vf{C}\ne 0$. + Let $I\subseteq \RR$ be an interval, $\vf{A}\in\mathcal{C}(I,\mathcal{L}(\RR^n))$ and $\vf{\Phi}(t),\vf{\psi}(t)\in\mathcal{M}_n(\RR)$ be matrix solutions of the ODE of \mcref{DE:homo} such that $\vf{\Phi}(t)$ is fundamental. Then, $\exists! \vf{C}\in\mathcal{M}_n(\RR)$ such that $\vf{\psi}(t)=\vf{\Phi}(t)\vf{C}$. Moreover, $\vf{\psi}(t)$ is fundamental if and only if $\det \vf{C}\ne 0$. \end{proposition} \subsubsection{Non-homogeneous linear systems} \begin{proposition} - Let $I\subseteq \RR$ be an interval, $\vf{A}\in\mathcal{C}(I,\mathcal{L}(\RR^n))$ and $\vf{b}\in\mathcal{C}(I,\RR^n)$. Suppose $\vf{\phi}(t,t_0,\vf{x}_0)$ is the flow of theODE of \mcref{DE:linear}. Then, $$\vf{\phi}(t,t_0,\vf{x}_0)=\Phi(t)\left[{\Phi(t_0)}^{-1}\vf{x}_0+\int_{t_0}^t{\Phi(s)}^{-1}\vf{b}(s)\dd{s}\right]$$ where $\Phi(t)$ is a fundamental matrix of the associated homogeneous system. + Let $I\subseteq \RR$ be an interval, $\vf{A}\in\mathcal{C}(I,\mathcal{L}(\RR^n))$ and $\vf{b}\in\mathcal{C}(I,\RR^n)$. Suppose $\vf{\phi}(t,t_0,\vf{x}_0)$ is the flow of the ODE of \mcref{DE:linear}. Then, $$\vf{\phi}(t,t_0,\vf{x}_0)=\Phi(t)\left[{\Phi(t_0)}^{-1}\vf{x}_0+\int_{t_0}^t{\Phi(s)}^{-1}\vf{b}(s)\dd{s}\right]$$ where $\Phi(t)$ is a fundamental matrix of the associated homogeneous system. \end{proposition} \begin{corollary} - Let $I\subseteq \RR$ be an interval, $\vf{A}\in\mathcal{C}(I,\mathcal{L}(\RR^n))$ and $\vf{b}\in\mathcal{C}(I,\RR^n)$. Then, the general solution $\vf{\varphi}(t)$ of theODE of \mcref{DE:homo} can be written as: $$\vf{\varphi}(t)=\vf{\varphi}_\mathrm{h}(t)+\vf{\varphi}_\mathrm{p}(t)$$ where $\vf{\varphi}_\mathrm{h}(t)$ is the general solution to the associated homogeneous system and $\vf{\varphi}_\mathrm{p}(t)$ is a particular solution of \mcref{DE:homo}. + Let $I\subseteq \RR$ be an interval, $\vf{A}\in\mathcal{C}(I,\mathcal{L}(\RR^n))$ and $\vf{b}\in\mathcal{C}(I,\RR^n)$. Then, the general solution $\vf{\varphi}(t)$ of the ODE of \mcref{DE:homo} can be written as: $$\vf{\varphi}(t)=\vf{\varphi}_\mathrm{h}(t)+\vf{\varphi}_\mathrm{p}(t)$$ where $\vf{\varphi}_\mathrm{h}(t)$ is the general solution to the associated homogeneous system and $\vf{\varphi}_\mathrm{p}(t)$ is a particular solution of \mcref{DE:homo}. \end{corollary} \begin{proposition}[Liouville's formula] - Let $I\subseteq \RR$ be an interval, $\vf{A}\in\mathcal{C}(I,\mathcal{L}(\RR^n))$, $\vf{\Phi}(t)\in\mathcal{M}_n(\RR)$ be a matrix solution of theODE of \mcref{DE:homo} and $t_0\in I$. Then, for all $t\in I$ we have: $$\det(\Phi(t))=\det (\Phi(t_0))\exp{\int_{t_0}^t\trace(\vf{A}(s))\dd{s}}$$ + Let $I\subseteq \RR$ be an interval, $\vf{A}\in\mathcal{C}(I,\mathcal{L}(\RR^n))$, $\vf{\Phi}(t)\in\mathcal{M}_n(\RR)$ be a matrix solution of the ODE of \mcref{DE:homo} and $t_0\in I$. Then, for all $t\in I$ we have: $$\det(\Phi(t))=\det (\Phi(t_0))\exp{\int_{t_0}^t\trace(\vf{A}(s))\dd{s}}$$ \end{proposition} \subsubsection{Constant coefficients linear systems} \begin{lemma} Let $I\subseteq\RR$ be a compact interval and $\vf{f}:I\times\RR^n\rightarrow\RR^n$ be a continuous function and Lipschitz continuous with respect to the second variable. Let $\vf{\varphi}:I\rightarrow\RR^n$ be the solution of the ivp of \mcref{DE:ivp}. Then, $\forall\vf{\psi}\in\mathcal{C}(I,\RR^n)$ the sequence $(\vf{T}^m\vf{\psi})$ converges uniformly to $\vf{\varphi}$ on $I$. \end{lemma} \begin{theorem} - Let $\vf{A}\in\mathcal{M}_n(\RR)$ and $\vf{\Phi}(t)\in\mathcal{M}_n(\RR)$ be a matrix solution of theODE + Let $\vf{A}\in\mathcal{M}_n(\RR)$ and $\vf{\Phi}(t)\in\mathcal{M}_n(\RR)$ be a matrix solution of the ODE \begin{equation}\label{DE:coef-constants} \vf{x}'=\vf{A}\vf{x} \end{equation} @@ -463,13 +463,13 @@ \end{equation} \end{definition} \begin{proposition} - Let $\vf{A}\in\mathcal{M}_n(\RR)$ and $t,s\in \RR$. Then, the matrix exponential $\exp{\vf{A}t}$ is a fundamental matrix of theODE of \mcref{DE:coef-constants} and has the following properties: + Let $\vf{A}\in\mathcal{M}_n(\RR)$ and $t,s\in \RR$. Then, the matrix exponential $\exp{\vf{A}t}$ is a fundamental matrix of the ODE of \mcref{DE:coef-constants} and has the following properties: \begin{enumerate} \item $\exp{\vf{A}\cdot 0}=\vf{I}_n$ \item $\exp{\vf{A}(t+s)}=\exp{\vf{A}t}\exp{\vf{A}s}$ \item ${\left(\exp{\vf{A}t}\right)}^{-1}=\exp{-\vf{A}t}$ \item $\left(\exp{\vf{A}t}\right)'=\vf{A}\exp{\vf{A}t}=\exp{\vf{A}t}\vf{A}$ - \item If $\vf{\Phi}(t)$ is an arbitrary fundamental matrix of theODE of \mcref{DE:coef-constants}, then: $$\exp{\vf{A}t}=\vf{\Phi}(t){\vf{\Phi}(0)}^{-1}$$ + \item If $\vf{\Phi}(t)$ is an arbitrary fundamental matrix of the ODE of \mcref{DE:coef-constants}, then: $$\exp{\vf{A}t}=\vf{\Phi}(t){\vf{\Phi}(0)}^{-1}$$ \end{enumerate} \end{proposition} \begin{lemma} @@ -492,7 +492,7 @@ $$ \end{proposition} \begin{corollary} - Let $\vf{A}\in\mathcal{M}_n(\RR)$ and $t\in\RR$ and consider the linearODE of \mcref{DE:coef-constants}. If $(\vf{v}_1,\ldots,\vf{v}_n)$ is a basis of eigenvectors with associated eigenvalues $\lambda_1,\ldots,\lambda_n$, respectively, then $(\vf{\varphi}_1,\ldots,\vf{\varphi}_n)$, where $\vf{\varphi}_i=\exp{\lambda_it}\vf{v}_i$ for $i=1,\ldots,n$, is a basis of $\mathcal{A}_n$. + Let $\vf{A}\in\mathcal{M}_n(\RR)$ and $t\in\RR$ and consider the linear ODE of \mcref{DE:coef-constants}. If $(\vf{v}_1,\ldots,\vf{v}_n)$ is a basis of eigenvectors with associated eigenvalues $\lambda_1,\ldots,\lambda_n$, respectively, then $(\vf{\varphi}_1,\ldots,\vf{\varphi}_n)$, where $\vf{\varphi}_i=\exp{\lambda_it}\vf{v}_i$ for $i=1,\ldots,n$, is a basis of $\mathcal{A}_n$. \end{corollary} \begin{lemma} Let $\vf{A}=\diag(\lambda_1,\ldots,\lambda_n)\in\mathcal{M}_n(\RR)$ and $t\in\RR$. Then: @@ -503,7 +503,7 @@ \begin{multline*} \exp{\vf{A}t}\vf{v}=\exp{\vf{A}t}\vf{u}+\ii\exp{\vf{A}t}\vf{w}=\exp{\alpha t}\left[\cos(\beta t)\vf{u}-\sin(\beta t)\vf{w}\right]+\\+\ii\exp{\alpha t}\left[\sin(\beta t)\vf{u}+\cos(\beta t)\vf{w}\right] \end{multline*} - and $\exp{\vf{A}t}\vf{u}$, $\exp{\vf{A}t}\vf{w}$ are linearly independent solutions of theODE of \mcref{DE:coef-constants} with initial conditions $\vf{x}(0)=\vf{u}$ and $\vf{x}(0)=\vf{w}$, respectively. + and $\exp{\vf{A}t}\vf{u}$, $\exp{\vf{A}t}\vf{w}$ are linearly independent solutions of the ODE of \mcref{DE:coef-constants} with initial conditions $\vf{x}(0)=\vf{u}$ and $\vf{x}(0)=\vf{w}$, respectively. \end{proposition} \begin{definition} Let $\vf{A}\in\mathcal{M}_n(\RR)$. A vector $\vf{w}\in\RR^n$ is a \emph{generalized eigenvector} of rank $m$ of $\vf{A}$ corresponding to the eigenvalue $\lambda\in\RR$ if: $${(\vf{A}-\lambda\vf{I}_n)}^m\vf{w}=0\quad\text{but}\quad{(\vf{A}-\lambda\vf{I}_n)}^{m-1}\vf{w}\ne 0$$ @@ -526,7 +526,7 @@ \end{aligned} \right. $$ - are solutions of theODE of \mcref{DE:coef-constants}. Furthermore, if $\vf{v}_1,\ldots,\vf{v}_k$ are linearly independent, then so are $\vf{\varphi}_1,\ldots,\vf{\varphi}_k$. + are solutions of the ODE of \mcref{DE:coef-constants}. Furthermore, if $\vf{v}_1,\ldots,\vf{v}_k$ are linearly independent, then so are $\vf{\varphi}_1,\ldots,\vf{\varphi}_k$. \end{lemma} \begin{corollary} Let $\vf{A}\in\mathcal{M}_n(\RR)$ and $\sigma(\vf{A})=\{\lambda_1,\ldots,\lambda_n\}$ be the spectrum of $\vf{A}$ such that: @@ -534,7 +534,7 @@ \item $\lambda_1,\ldots,\lambda_{2k}\in\CC\setminus\RR$, $\lambda_{k+i}=\overline{\lambda_i}$ and $\lambda_i=\alpha_i+\ii\beta_i$, $\alpha_i,\beta_i\in\RR$ for $i=1,\ldots,k$. \item $\lambda_{2k+1},\ldots,\lambda_n\in\RR$ \end{itemize} - Then, the general solution of theODE of \mcref{DE:coef-constants} is of the form: + Then, the general solution of the ODE of \mcref{DE:coef-constants} is of the form: \begin{multline*} \vf{\varphi}(t)=\sum_{i=1}^k\exp{\alpha_i t}\left(\vf{P}_i(t)\cos(\beta_i t)+\vf{Q}_i(t)\sin(\beta_i t)\right)+\\+\sum_{i=2k+1}^n\exp{\lambda_i t}\vf{R}_i(t) \end{multline*} @@ -549,7 +549,7 @@ \vf{x}(t_0) =\vf{x}_0 \end{cases} \end{equation} - has a unique maximal solution $\vf{\varphi}_{(t_0,\vf{x}_0,\vf{\lambda})}(t)$ defined on an interval $I_{(t_0,\vf{x}_0,\vf{\lambda})}$. We define the \emph{flow} of theODE $\vf{x}'=\vf{f}(t,\vf{x},\vf{\lambda})$ as: $$\function{\vf{\phi}}{I_{(t_0,\vf{x}_0,\vf{\lambda})}\times\RR\times\RR^n\times\RR^p}{\RR^n}{(t,t_0,\vf{x}_0,\vf{\lambda})}{\vf{\varphi}_{(t_0,\vf{x}_0,\vf{\lambda})}(t)}$$ + has a unique maximal solution $\vf{\varphi}_{(t_0,\vf{x}_0,\vf{\lambda})}(t)$ defined on an interval $I_{(t_0,\vf{x}_0,\vf{\lambda})}$. We define the \emph{flow} of the ODE $\vf{x}'=\vf{f}(t,\vf{x},\vf{\lambda})$ as: $$\function{\vf{\phi}}{I_{(t_0,\vf{x}_0,\vf{\lambda})}\times\RR\times\RR^n\times\RR^p}{\RR^n}{(t,t_0,\vf{x}_0,\vf{\lambda})}{\vf{\varphi}_{(t_0,\vf{x}_0,\vf{\lambda})}(t)}$$ \end{definition} \subsubsection{Continuous and Lipschitz continuous dependence} \begin{lemma} @@ -577,7 +577,7 @@ If, moreover, $w\in\mathcal{C}^1((a,b))$, then: $$u(t)\leq w(a)\exp{\int_a^tv(r)\dd{r}}+\int_a^tw'(s)\exp{\int_s^tv(r)\dd{r}}\dd{s}\quad\forall t\in[a,b)$$ \end{lemma} \begin{proposition} - Let $U\subseteq\RR\times\RR^n$ be an open set and $\vf{f}:U\rightarrow\RR^n$ be a continuous function and Lipschitz continuous with respect to the second variable with Lipschitz constant $L$. Let $\vf{\phi}$ be the flow of theODE $\vf{x}'=\vf{f}(t,\vf{x})$. Then, $\forall (t_0,\vf{x}_1),(t_0,\vf{x}_2)\in U$ and $\forall t\in I_{(t_0,\vf{x}_1)}\cap I_{(t_0,\vf{x}_2)}$, we have: + Let $U\subseteq\RR\times\RR^n$ be an open set and $\vf{f}:U\rightarrow\RR^n$ be a continuous function and Lipschitz continuous with respect to the second variable with Lipschitz constant $L$. Let $\vf{\phi}$ be the flow of the ODE $\vf{x}'=\vf{f}(t,\vf{x})$. Then, $\forall (t_0,\vf{x}_1),(t_0,\vf{x}_2)\in U$ and $\forall t\in I_{(t_0,\vf{x}_1)}\cap I_{(t_0,\vf{x}_2)}$, we have: $$\|\vf{\phi}(t,t_0,\vf{x}_2)-\vf{\phi}(t,t_0,\vf{x}_1)\|\leq\exp{L|t-t_0|}\|\vf{x}_2-\vf{x}_1\|$$ Thus, $\vf{\phi}$ is locally Lipschitz continuous with respect to the third variable. \end{proposition} @@ -651,7 +651,7 @@ which creates a partition of $X$, called \emph{phase portrait}. \end{definition} \begin{definition} - The \emph{phase space} of anODE or system ofODEs is the space in which all possible states of a system are represented with each possible state corresponding to one unique point in the phase space. + The \emph{phase space} of an ODE or system of ODEs is the space in which all possible states of a system are represented with each possible state corresponding to one unique point in the phase space. \end{definition} \begin{center} \begin{minipage}[b]{0.475\linewidth} @@ -676,16 +676,16 @@ Let $(G,X,\Psi)$ be a dynamical system and $x\in X$. We define the following function: $$\function{\Psi_x}{G}{\gamma(x)}{t}{\Psi(t,x)}$$ \end{definition} \begin{lemma} - Let $\vf{f}:\RR^n\rightarrow\RR^n$ be a continuous function such that the flow $\vf{\phi}(t,t_0,\vf{x}_0)$ of theODE $\vf{x}' =\vf{f}(\vf{x})$ is defined for all $t\in\RR$. Then, $(\RR,\RR^n,\vf{\Psi})$ is a dynamical system, where $\vf{\Psi}(t,\vf{x})=\vf{\phi}(t,0,\vf{x})$. Furthermore, note that $\vf{\gamma}(\vf{x})=\im(\vf{\phi}(\cdot,0,\vf{x}))$. + Let $\vf{f}:\RR^n\rightarrow\RR^n$ be a continuous function such that the flow $\vf{\phi}(t,t_0,\vf{x}_0)$ of the ODE $\vf{x}' =\vf{f}(\vf{x})$ is defined for all $t\in\RR$. Then, $(\RR,\RR^n,\vf{\Psi})$ is a dynamical system, where $\vf{\Psi}(t,\vf{x})=\vf{\phi}(t,0,\vf{x})$. Furthermore, note that $\vf{\gamma}(\vf{x})=\im(\vf{\phi}(\cdot,0,\vf{x}))$. \end{lemma} \begin{lemma} - Let $\vf{f}:\RR^n\rightarrow\RR^n$ be a continuous function such that $\exists M,N\in\RR_{\geq 0}$ with $\|\vf{f}(\vf{x})\|\leq M\|\vf{x}\|+N$. Then, the solutions of theODE $\vf{x}' =\vf{f}(\vf{x})$ are defined for all $t\in\RR$. + Let $\vf{f}:\RR^n\rightarrow\RR^n$ be a continuous function such that $\exists M,N\in\RR_{\geq 0}$ with $\|\vf{f}(\vf{x})\|\leq M\|\vf{x}\|+N$. Then, the solutions of the ODE $\vf{x}' =\vf{f}(\vf{x})$ are defined for all $t\in\RR$. \end{lemma} \begin{definition} - Let $\vf{f},\vf{g}:\RR^n\rightarrow\RR^n$ be continuous functions and $\vf{x}' =\vf{f}(\vf{x})$, $\vf{x}' =\vf{g}(\vf{x})$ be twoODEs for which we have existence and uniqueness of solutions. We say that these twoODEs are \emph{equivalent} if there exists $\vf{h}:\RR^n\rightarrow\RR^n$ such that $\vf{h}(\vf{x})\geq 0$ and $\vf{f}(\vf{x})=\vf{h}(\vf{x})\vf{g}(\vf{x})$ $\forall \vf{x}\in\RR^n$. Therefore, $\vf{f}$ and $\vf{g}$ have the same orbits oriented in the same way. + Let $\vf{f},\vf{g}:\RR^n\rightarrow\RR^n$ be continuous functions and $\vf{x}' =\vf{f}(\vf{x})$, $\vf{x}' =\vf{g}(\vf{x})$ be two ODEs for which we have existence and uniqueness of solutions. We say that these two ODEs are \emph{equivalent} if there exists $\vf{h}:\RR^n\rightarrow\RR^n$ such that $\vf{h}(\vf{x})\geq 0$ and $\vf{f}(\vf{x})=\vf{h}(\vf{x})\vf{g}(\vf{x})$ $\forall \vf{x}\in\RR^n$. Therefore, $\vf{f}$ and $\vf{g}$ have the same orbits oriented in the same way. \end{definition} \begin{corollary} - Let $\vf{f}:\RR^n\rightarrow\RR^n$ be a continuous function such that theODE $\vf{x}' =\vf{f}(\vf{x})$ has existence and uniqueness of solutions for all initial conditions. Then, there exists a continuous function $\vf{g}:\RR^n\rightarrow\RR^n$ such that the autonomousODEs induced by $\vf{f}$ and $\vf{g}$ are equivalent and the flow of theODE $\vf{x}' =\vf{g}(\vf{x})$ is defined $\forall t\in\RR$. + Let $\vf{f}:\RR^n\rightarrow\RR^n$ be a continuous function such that the ODE $\vf{x}' =\vf{f}(\vf{x})$ has existence and uniqueness of solutions for all initial conditions. Then, there exists a continuous function $\vf{g}:\RR^n\rightarrow\RR^n$ such that the autonomous ODEs induced by $\vf{f}$ and $\vf{g}$ are equivalent and the flow of the ODE $\vf{x}' =\vf{g}(\vf{x})$ is defined $\forall t\in\RR$. \end{corollary} \begin{lemma} Let $H$ be a proper subgroup of $\RR$ which is closed. Then, $\exists T\in\RR_{\geq 0}$ such that $H=T\ZZ$. @@ -705,7 +705,7 @@ Let $(\RR,\RR^n,\vf{\Psi})$ be a dynamical system and $\vf{\gamma}(\vf{x})$ be an orbit of $(\RR,\RR^n,\vf{\Psi})$. We say that $\vf{\gamma}(\vf{x})$ is \emph{periodic} of period $T>0$ if $\vf{\gamma}(\vf{x})\cong \S^1$ and $\ker\vf{\Psi_{\vf{x}}}=T\ZZ$. \end{definition} \begin{proposition} - Let $(\RR,\RR^n,\vf{\Psi})$ be a dynamical system such that $\vf{\Psi}(t,\vf{x})=\vf{\phi}(t,0,\vf{x})$, where $\vf{\phi}(t,t_0,\vf{x}_0)$ is the flow of theODE $\vf{x}' =\vf{f}(\vf{x})$. Let $\vf{p}\in\RR^n$. Then, the following statements are equivalent: + Let $(\RR,\RR^n,\vf{\Psi})$ be a dynamical system such that $\vf{\Psi}(t,\vf{x})=\vf{\phi}(t,0,\vf{x})$, where $\vf{\phi}(t,t_0,\vf{x}_0)$ is the flow of the ODE $\vf{x}' =\vf{f}(\vf{x})$. Let $\vf{p}\in\RR^n$. Then, the following statements are equivalent: \begin{enumerate} \item $\{\vf{p}\}$ is a critical point. \item $\vf{\phi}(t,0,\vf{p})=\vf{p}$. @@ -776,7 +776,7 @@ Let $(G,X,\Psi_1)$ and $(G,X,\Psi_2)$ be dynamical systems and $h$ be a conjugacy of class $\mathcal{C}^r$ between them. Then, $h$ is an equivalence of class $\mathcal{C}^r$ between $(G,X,\Psi_1)$ and $(G,X,\Psi_2)$. \end{proposition} \begin{proposition} - Two dynamical systems induced by two equivalentODEs are equivalent (as a dynamical systems). + Two dynamical systems induced by two equivalent ODEs are equivalent (as a dynamical systems). \end{proposition} \begin{proposition} Let $(G,X,\Psi_1)$ and $(G,X,\Psi_2)$ be dynamical systems and $h:X\rightarrow Y$ be an equivalence of class $\mathcal{C}^r$ between them. Then: @@ -805,10 +805,10 @@ x^{\beta/\alpha} & \text{if }x\geq 0 \\ -{|x|}^{\beta/\alpha} & \text{if }x< 0 \end{cases}$$ - Then, $h$ is a topological conjugation between the systems induced by theODEs $x'=\alpha x$ and $y'=\beta y$. + Then, $h$ is a topological conjugation between the systems induced by the ODEs $x'=\alpha x$ and $y'=\beta y$. \end{proposition} \begin{proposition} - Let $\vf{A},\vf{B}\in\mathcal{M}_n(\RR)$ be similar matrices, that is, $\exists\vf{P}\in\mathcal{M}_n(\RR)$ such that $\vf{B}=\vf{P}\vf{A}\vf{P}^{-1}$. Then, the function $$\function{\vf{h}}{\RR^n}{\RR^n}{\vf{x}}{\vf{Px}}$$ is a conjugation between the systems induced by theODEs $\vf{x}'=\vf{A}\vf{x}$ and $\vf{y}'=\vf{B}\vf{y}$. + Let $\vf{A},\vf{B}\in\mathcal{M}_n(\RR)$ be similar matrices, that is, $\exists\vf{P}\in\mathcal{M}_n(\RR)$ such that $\vf{B}=\vf{P}\vf{A}\vf{P}^{-1}$. Then, the function $$\function{\vf{h}}{\RR^n}{\RR^n}{\vf{x}}{\vf{Px}}$$ is a conjugation between the systems induced by the ODEs $\vf{x}'=\vf{A}\vf{x}$ and $\vf{y}'=\vf{B}\vf{y}$. \end{proposition} \subsubsection{Local equivalence and conjugacy of dynamical systems} \begin{definition} @@ -1106,7 +1106,7 @@ \subsection{Qualitative theory of planar differential systems} \subsubsection{Polynomial vectors fields} \begin{definition} - Let $p,q\in\RR[x,y]$. The system ofODEs + Let $p,q\in\RR[x,y]$. The system of ODEs \begin{equation}\label{DE:poly} \left\{ \begin{aligned} diff --git a/Mathematics/3rd/Differential_geometry/Differential_geometry.tex b/Mathematics/3rd/Differential_geometry/Differential_geometry.tex index 30a0be6..dd377be 100644 --- a/Mathematics/3rd/Differential_geometry/Differential_geometry.tex +++ b/Mathematics/3rd/Differential_geometry/Differential_geometry.tex @@ -885,7 +885,7 @@ Let $S\subseteq\RR^3$ be a surface and $\vf{X}$, $\vf{Y}$ be vector fields tangent to $S$ along a curve $\vf\alpha:I\rightarrow S$ of class $\mathcal{C}^\infty$ such that they are parallel. Then, $t\mapsto\langle \vf{X}(t),\vf{Y}(t)\rangle$ is constant. In particular, the norms $\|\vf{X}(t)\|$, $\|\vf{Y}(t)\|$ as well as the angle between $\vf{X}(t)$ and $\vf{Y}(t)$ are constant. \end{proposition} \begin{proposition} - Let $S\subseteq\RR^3$ be a surface, $(V,\vf\varphi(u,v))$ is a parametrization of $S$ and $\vf\alpha: I\rightarrow S$ be a parametrized curve of class $\mathcal{C}^\infty$ such that $\vf\alpha=\vf\varphi(u(t),v(t))$. Then, given $t_0\in I$ and $\vf{w}\in T_{\vf\alpha(t_0)}S$ there exists a unique parallel vector field $\vf{X}=a\vf\varphi_u+b\vf\varphi_v$ along $\vf\alpha$ such that $\vf{X}(t_0)=\vf{w}$. This vector field is called \emph{parallel transport} of the vector $\vf{w}$ along $\vf{\alpha}$, and it is defined on the entire interval $I$. It can be found by solving this system ofODEs: + Let $S\subseteq\RR^3$ be a surface, $(V,\vf\varphi(u,v))$ is a parametrization of $S$ and $\vf\alpha: I\rightarrow S$ be a parametrized curve of class $\mathcal{C}^\infty$ such that $\vf\alpha=\vf\varphi(u(t),v(t))$. Then, given $t_0\in I$ and $\vf{w}\in T_{\vf\alpha(t_0)}S$ there exists a unique parallel vector field $\vf{X}=a\vf\varphi_u+b\vf\varphi_v$ along $\vf\alpha$ such that $\vf{X}(t_0)=\vf{w}$. This vector field is called \emph{parallel transport} of the vector $\vf{w}$ along $\vf{\alpha}$, and it is defined on the entire interval $I$. It can be found by solving this system of ODEs: $$\left\{ \begin{aligned} a'+\Gamma_{11}^1au'+\Gamma_{12}^1av'+\Gamma_{21}^1bu'+\Gamma_{22}^1bv' & =0 \\ @@ -991,7 +991,7 @@ \end{lemma} \begin{definition} Let $U\subseteq\RR^n$ be an open set and $\vf{X}=\sum X^i\pdv{}{x^i}\in\mathcal{X}(U)$. We say that a parametrized curve $\vf{\gamma}:I\rightarrow\RR^n$ is an \emph{integral curve} of $\vf{X}$ if: $$\vf\gamma'(t)=\vf{X}(\vf\gamma(t))\qquad \forall t\in I$$ - That is, the integral curve $\vf\gamma(t)=(x^1(t),\ldots,x^n(t))$ of $\vf{X}$ satisfies the following system ofODEs: + That is, the integral curve $\vf\gamma(t)=(x^1(t),\ldots,x^n(t))$ of $\vf{X}$ satisfies the following system of ODEs: $$ \left\{ \begin{aligned} diff --git a/Mathematics/4th/Dynamical_systems/Dynamical_systems.tex b/Mathematics/4th/Dynamical_systems/Dynamical_systems.tex index e32113c..353ab54 100644 --- a/Mathematics/4th/Dynamical_systems/Dynamical_systems.tex +++ b/Mathematics/4th/Dynamical_systems/Dynamical_systems.tex @@ -36,7 +36,7 @@ \end{enumerate} \end{theorem} \begin{definition} - Let $f,g:\RR^2\rightarrow\RR$ be two functions and consider the system ofODEs: + Let $f,g:\RR^2\rightarrow\RR$ be two functions and consider the system of ODEs: \begin{equation}\label{DS:plane} \left\{ \begin{aligned} @@ -645,14 +645,14 @@ A semiestable limit cycle $\Gamma_\mu$ of a family of rotated vector fields splits into two simple limit cycles, one stable and one unstable, as the parameter $\mu$ is varied in one sense and it disappears as $\mu$ is varied in the opposite sense. \end{theorem} \begin{theorem}[Melnikov's method] - Let $\vf{f}\in\mathcal{C}^1(\RR^2)$, $\vf{g}\in\mathcal{C}^1(\RR^2\times\RR^m)$ and $\varepsilon\simeq 0$. Consider the followingODE: + Let $\vf{f}\in\mathcal{C}^1(\RR^2)$, $\vf{g}\in\mathcal{C}^1(\RR^2\times\RR^m)$ and $\varepsilon\simeq 0$. Consider the following ODE: \begin{equation}\label{DS:melnikov} \vf{x}'=\vf{f}(\vf{x})+\varepsilon\vf{g}(\vf{x},\vf{\mu}) \end{equation} Suppose that for $\varepsilon =0$ the system has a one-parameter family of periodic orbits $\vf\gamma_h(t)$ of period $T_h$. Then for any simple zero $(\vf\mu_0,h_0)$ of the function $$M(\vf\mu, h)=\int_{0}^{T_h}\vf{f}(\vf\gamma_h(t))\times \vf{g}(\vf\gamma_h(t))\dd{t}$$ there exists a unique limit cycle $\vf\Gamma_\varepsilon$ for $\varepsilon\simeq 0$ such that $\displaystyle\lim_{\varepsilon\to 0}\vf\Gamma_\varepsilon=\vf\gamma_{h_0}$. On the other hand, if $M(\vf\mu_0,h_0)\ne 0$, for sufficiently small $\varepsilon$, the system of \mcref{DS:melnikov} with $\vf\mu=\vf\mu_0$ has no limit cycle in any sufficiently small neighborhood of $\vf\gamma_{h_0}$. \end{theorem} \begin{corollary}[Melnikov's method] - Let $H\in\mathcal{C}^2(\RR^2)$, $P,Q\in\mathcal{C}^1(\RR^2\times\RR^m)$ and $\varepsilon\simeq 0$. Consider the following system ofODEs: + Let $H\in\mathcal{C}^2(\RR^2)$, $P,Q\in\mathcal{C}^1(\RR^2\times\RR^m)$ and $\varepsilon\simeq 0$. Consider the following system of ODEs: \begin{equation*} \left\{ \begin{aligned} diff --git a/Mathematics/4th/Harmonic_analysis/Harmonic_analysis.tex b/Mathematics/4th/Harmonic_analysis/Harmonic_analysis.tex index 834c688..f4b48d1 100644 --- a/Mathematics/4th/Harmonic_analysis/Harmonic_analysis.tex +++ b/Mathematics/4th/Harmonic_analysis/Harmonic_analysis.tex @@ -166,7 +166,7 @@ Let $f(x)=\exp{-a x^2}$. Then, $\F f(\xi)=\sqrt{\frac{\pi}{a}}\exp{-\frac{{(\pi \xi)}^2}{a}}$ and moreover $\F^2f=f$. In particular if $a=\pi$, then $\F f=f$. \end{lemma} \begin{sproof} - $f$ satisfies theODE $y'=-2a x y$. Taking $\ \widehat{}\ $ on this expression and using \mcref{HA:diffFourierXf,HA:diffFourierTransf} we obtain that $\widehat{f}$ must satisfy the followingODE: + $f$ satisfies the ODE $y'=-2a x y$. Taking $\ \widehat{}\ $ on this expression and using \mcref{HA:diffFourierXf,HA:diffFourierTransf} we obtain that $\widehat{f}$ must satisfy the following ODE: $$y'=-\frac{2\pi^2\xi}{a} y$$ with initial condition $y(0)=\int_{-\infty}^{+\infty}\exp{-a x^2}\dd{x}=\sqrt{\frac{\pi}{a}}$. \end{sproof} @@ -1266,7 +1266,7 @@ $$ \partial_t \widehat{E}+4\pi^2a^2\norm{\vf\xi}^2\widehat{E}=\delta_t $$ - because $\delta=\delta_{\vf{x}}\delta_t$. It can be seen that a solution of thisODE is: + because $\delta=\delta_{\vf{x}}\delta_t$. It can be seen that a solution of this ODE is: $$ \widehat{E}(t,\xi)=\indi{[0,\infty)}(t)\exp{-4\pi^2a^2\norm{\vf\xi}^2t} $$ diff --git a/Mathematics/4th/Numerical_calculus/Numerical_calculus.tex b/Mathematics/4th/Numerical_calculus/Numerical_calculus.tex index 1724aca..d90f392 100644 --- a/Mathematics/4th/Numerical_calculus/Numerical_calculus.tex +++ b/Mathematics/4th/Numerical_calculus/Numerical_calculus.tex @@ -84,7 +84,7 @@ Moreover, we say that the algorithm has \emph{order of convergence} $p$ if $\norm{\vf{e}_n}=\O{h^p}$. \end{definition} \begin{remark} - Note that in a consistent method the difference equation for the method approaches theODE as the step size goes to zero, whereas in a convergent method is the solution to the difference equation that approaches the solution to theODE as the step size goes to zero. + Note that in a consistent method the difference equation for the method approaches the ODE as the step size goes to zero, whereas in a convergent method is the solution to the difference equation that approaches the solution to the ODE as the step size goes to zero. \end{remark} \begin{theorem}\label{NC:errorLipschitz} Consider a consistent one-step explicit method such that its incremental function $\vf\phi$ is Lipschitz continuous (with constant $L$) with respect to $\vf{x}$. Then: diff --git a/Mathematics/4th/Partial_differential_equations/Partial_differential_equations.tex b/Mathematics/4th/Partial_differential_equations/Partial_differential_equations.tex index db14b0b..6fc16d0 100644 --- a/Mathematics/4th/Partial_differential_equations/Partial_differential_equations.tex +++ b/Mathematics/4th/Partial_differential_equations/Partial_differential_equations.tex @@ -135,7 +135,7 @@ \end{definition} \begin{proposition}[Fermat's principle] \emph{Fermat's principle} states that the path taken by a ray between two given points $a$ and $b$ is the path that can be traveled in the least time. Mathematically, we want to minimize the functional: $$\mathcal{T}(\vf{x})=\int_a^b\frac{\abs{\dd{\vf{x}}}}{v(\vf{x})}$$ - So we shall solve the equation $\delta \mathcal{T}=0$, which is equivalent to solve: $$\delta\int_a^bn(\vf{x})\dd{s}=0$$ where $s$ is the arc-length parameter. From the Euler-Lagrange equations, we get the followingODE: $$\dv{}{s}\left(n\dv{\vf{x}}{s}\right)=\grad{n}$$ + So we shall solve the equation $\delta \mathcal{T}=0$, which is equivalent to solve: $$\delta\int_a^bn(\vf{x})\dd{s}=0$$ where $s$ is the arc-length parameter. From the Euler-Lagrange equations, we get the following ODE: $$\dv{}{s}\left(n\dv{\vf{x}}{s}\right)=\grad{n}$$ \end{proposition} \begin{proposition}[Eikonal equation] The time $T(x)$ taken by the light to travel from a fixed point $x_0$ to $x$ in a medium of refractive index $n$ is given by: $${\norm{\grad{T}}}^2=n^2$$ @@ -635,7 +635,7 @@ for certain constants $C_1, C_2\in\RR$. \end{proposition} \begin{sproof} - Observe that $u(x,t) = f(\frac{x}{\sqrt{t}})=:f(s)$ and the heat equation is transformed into $-f's=2\alpha f''$. The solution of thisODE is straightforward. + Observe that $u(x,t) = f(\frac{x}{\sqrt{t}})=:f(s)$ and the heat equation is transformed into $-f's=2\alpha f''$. The solution of this ODE is straightforward. \end{sproof} \subsubsection{Distributions} \begin{definition} diff --git a/Mathematics/4th/Stochastic_processes/Stochastic_processes.tex b/Mathematics/4th/Stochastic_processes/Stochastic_processes.tex index 35871f9..48bd584 100644 --- a/Mathematics/4th/Stochastic_processes/Stochastic_processes.tex +++ b/Mathematics/4th/Stochastic_processes/Stochastic_processes.tex @@ -1165,7 +1165,7 @@ Let ${(X_t)}_{t\geq 0}$ be a CTHMC. Then, ${(X_t)}_{t\geq 0}$ is said to be \emph{stable} if $\forall i\in I$, $q_i<\infty$, and is said to be \emph{conservative} if $\forall i\in I$, $q_i=\sum_{\substack{k\in I\\k\ne i}}q_{ik}$. \end{definition} \begin{theorem} - Let ${(X_t)}_{t\geq 0}$ be a CTHMC and a regular jump process. Then, the two KolmogorovODEs are satisfied. + Let ${(X_t)}_{t\geq 0}$ be a CTHMC and a regular jump process. Then, the two Kolmogorov ODEs are satisfied. \end{theorem} \subsubsection{Limit and stationary distributions} \begin{definition} diff --git a/Mathematics/5th/Montecarlo_methods/Montecarlo_methods.tex b/Mathematics/5th/Montecarlo_methods/Montecarlo_methods.tex index f051b5a..5eb4210 100644 --- a/Mathematics/5th/Montecarlo_methods/Montecarlo_methods.tex +++ b/Mathematics/5th/Montecarlo_methods/Montecarlo_methods.tex @@ -320,7 +320,7 @@ Note that Euler scheme reduces to generating independent increments $\vf{B}_{t_{i+1}}-\vf{B}_{t_i}\sim \sqrt{t_{i+1}-t_i}N_d(0,\vf{I}_d)$. \end{remark} \begin{remark} - Trying to build an implicit Euler scheme for SDEs is much more complicated than for ODEs, as we need to ensure that the process is still adapted. + Trying to build an implicit Euler scheme for SDEs is much more complicated than for ODEs, as we need to ensure that the process is still adapted. \end{remark} \begin{definition} Let $\vf{X}$ be the solution to \mcref{MM:SDE}. We define the \emph{continuous Euler scheme} as: diff --git a/Physics/Basic/Classical_mechanics/Classical_mechanics.tex b/Physics/Basic/Classical_mechanics/Classical_mechanics.tex index 1c0b3d9..1e9267e 100644 --- a/Physics/Basic/Classical_mechanics/Classical_mechanics.tex +++ b/Physics/Basic/Classical_mechanics/Classical_mechanics.tex @@ -300,7 +300,7 @@ \label{CM_RLC-genS} \end{minipage} \end{center} - TheODE of the system for the charge $q(t)$ is: $$\ddot{q}+\frac{R}{L}\dot{q}+\frac{q}{LC}=\frac{V_\text{in}}{L}$$ Thus, in steady-state part we have: $$q(t)=\frac{\epsilon_0/\omega}{\sqrt{{\left(L\omega-\frac{1}{C\omega}\right)}^2+R^2}}\cos(\omega t-\delta)$$ where $\delta=-\arctan\left(\frac{R}{L\omega-\frac{1}{C\omega}}\right)$. + The ODE of the system for the charge $q(t)$ is: $$\ddot{q}+\frac{R}{L}\dot{q}+\frac{q}{LC}=\frac{V_\text{in}}{L}$$ Thus, in steady-state part we have: $$q(t)=\frac{\epsilon_0/\omega}{\sqrt{{\left(L\omega-\frac{1}{C\omega}\right)}^2+R^2}}\cos(\omega t-\delta)$$ where $\delta=-\arctan\left(\frac{R}{L\omega-\frac{1}{C\omega}}\right)$. And finally: \begin{equation}\label{CM_Vout} V_\text{out}=RI=-\frac{R}{\sqrt{{\left(L\omega-\frac{1}{C\omega}\right)}^2+R^2}}\epsilon_0\sin(\omega t-\delta) @@ -321,7 +321,7 @@ % \label{CM_RLC-genP} % \end{minipage} % \end{center} - % TheODE of the system for the charge $q(t)$ is: $$\ddot{q}+\frac{R}{L}\dot{q}+\frac{q}{LC}=\frac{V_\text{in}}{L}$$ Thus, in steady-state part we have: $$q(t)=\frac{\epsilon_0/\omega}{\sqrt{{\left(L\omega-\frac{1}{C\omega}\right)}^2+\frac{1}{R^2}}}\cos(\omega t-\delta)$$ where $\delta=-\arctan\left(\frac{1/R}{L\omega-\frac{1}{C\omega}}\right)$. + % The ODE of the system for the charge $q(t)$ is: $$\ddot{q}+\frac{R}{L}\dot{q}+\frac{q}{LC}=\frac{V_\text{in}}{L}$$ Thus, in steady-state part we have: $$q(t)=\frac{\epsilon_0/\omega}{\sqrt{{\left(L\omega-\frac{1}{C\omega}\right)}^2+\frac{1}{R^2}}}\cos(\omega t-\delta)$$ where $\delta=-\arctan\left(\frac{1/R}{L\omega-\frac{1}{C\omega}}\right)$. % And finally: % \begin{equation}\label{CM_Vout} % V_\text{out}=RI=-\frac{R}{\sqrt{{\left(L\omega-\frac{1}{C\omega}\right)}^2+R^2}}\epsilon_0\sin(\omega t-\delta) @@ -341,11 +341,11 @@ f_0 & \text{if }0\leq t\leq \Delta t \\ 0 & \text{if }t>\Delta t \end{cases}$$ - That is, $f$ is a piecewise function\footnote{We shall suppose that the system was at equilibrium for $t<0$.}. Moreover, assuming that $x(0)=\dot{x}(0)=0$, the general solution to thisODE when $0\Delta t$, then the general solution to theODE is: + If $t>\Delta t$, then the general solution to the ODE is: $$x(t)=\exp{-\beta t}\left(k_1\cos(\tilde{\omega}t)+k_2\sin(\tilde{\omega}t)\right)$$ where $k_1=\text{O}({\Delta t}^2)$, $k_2=\frac{f_0\Delta t}{\tilde{\omega}}+\text{O}({\Delta t}^2)$ Therefore, $\forall t>\Delta t$: $$x(t)=f_0\Delta t\frac{\exp{-\beta t}}{\tilde{\omega}}\sin(\tilde{\omega}t)+\text{O}({\Delta t}^2)$$ \end{proposition} @@ -391,7 +391,7 @@ \end{theorem} \subsubsection{Non linear oscillations} \begin{definition} - Consider a pendulum whose rod (of length $L$) is in a non-small angle $\theta_0$ at initial time. TheODE that satisfies $\theta(t)$ is: + Consider a pendulum whose rod (of length $L$) is in a non-small angle $\theta_0$ at initial time. The ODE that satisfies $\theta(t)$ is: $$\ddot{\theta}+\frac{g}{L}\sin\theta=0$$ Then, the period of the pendulum does depend on $\theta_0$. Indeed: \begin{multline*}