From b60b6e47f7033781c4df907a6f11e1370f261df1 Mon Sep 17 00:00:00 2001 From: Marcello Seri Date: Thu, 19 Jan 2023 10:48:01 +0100 Subject: [PATCH] Fix typos, improve wording Signed-off-by: Marcello Seri --- 7-integration.tex | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/7-integration.tex b/7-integration.tex index 76ac397..81f6466 100644 --- a/7-integration.tex +++ b/7-integration.tex @@ -155,7 +155,7 @@ \section{Orientation on manifolds} where $\alpha_{lk}\in C^{\infty}(U_k\cap U_l)$, $a_{lk} > 0$ and there is no implied sum\footnote{All indices are low indices.}. Since the covering $\{U_i\}$ is locally finite\footnote{See Theorem~\ref{thm:partitionof1}.}, a point $p\in M$ belongs only to a finite number of open sets, let's call them $U_{i_0}, U_{i_1}, \ldots, U_{i_N}$. That is, \begin{equation} - \omega(p) = \sum_{k=1}^N \omega_{i_k}(p) = \left( 1 + \sum_{k = 1}^N a_{i_k i_0} \right) \omega_{i_0}(p) \neq 0 + \omega(p) = \sum_{k=0}^N \omega_{i_k}(p) = \left( 1 + \sum_{k = 1}^N a_{i_k i_0} \right) \omega_{i_0}(p) \neq 0 \end{equation} since $\omega_{i_0}(p) \neq 0$ and $a_{i_k i_0} > 0$. That is, for all $p\in M$ we have that $\omega(p) \neq 0$. @@ -261,7 +261,7 @@ \section{Orientation on manifolds} \varphi_{U}:\pi^{-1}(U)\to U\times \R^{n}, \end{equation} with $\R^n$ equipped with its standard orientation, is fiberwise orientation-preserving. - \marginnote[-2em]{Otherwise said, we can cover the manifold by (continuous) local frames whose local trivializations are orientation preserving.} + \marginnote[-2em]{Otherwise said, we can cover the manifold by (continuous) local frames such that the local trivializations are orientation preserving.} With this definition, the orientability of $M$ coincides with the orientability of the bundle $TM\to M$. \end{remark} @@ -361,7 +361,7 @@ \section{Integrals on manifolds} \begin{definition}\label{def:intnform:chart} Let $M$ be a smooth $n$-manifold and $(U,\varphi)$ be a chart from an oriented atlas of $M$ with coordinates $(x^i)$. - If $\omega\in\Omega^n(M)$ be a $n$-form, $n > 0$, with compact support in $U$, we define the integral of $\omega$ as\sidenote[][-1em]{Recall that for a diffeomorphism $\phi$, $\phi_* = (\phi^{-1})^*$.} + If $\omega\in\Omega^n(M)$ is a $n$-form, $n > 0$, with compact support in $U$, we define the integral of $\omega$ as\sidenote[][-1em]{Recall that for a diffeomorphism $\phi$, $\phi_* = (\phi^{-1})^*$.} \begin{equation} \int_M \omega = \int_U \omega := \int_{\varphi(U)} \varphi_*\omega := \int_{\R^n} \omega(x) d x^1\cdots dx^n, \end{equation} @@ -768,10 +768,10 @@ \section{Stokes' Theorem} In fact this example is a particular case of the following corollary of Stokes' theorem. \begin{corollary} - Suppose $M$ is a smooth $m$-manifold with or without boundary,$N\subseteq M$ is an oriented compact smooth $n$-submanifold (without boundary) and $\omega$ is a closed $n$-form on $M$. + Suppose $M$ is a smooth $m$-manifold with or without boundary, $N\subseteq M$ is an oriented compact smooth $n$-submanifold and $\omega$ is a closed $n$-form on $M$. If $\int_N\omega \neq 0$ then the following are true: \begin{enumerate} - \item $\omega$ is not exact on $M$; + \item if $\partial N = \emptyset$, $\omega$ is not exact on $M$; \item $N$ is not the boundary of an oriented compact smooth submanifold with boundary in $M$. \end{enumerate} \end{corollary}