\documentclass[letterpaper, twoside, 12pt]{book}
\usepackage{amsmath,amsfonts,amsthm}
\usepackage{notes}
\usepackage{hyperref}
\title{MATH 3142 Notes | Spring 2016}
\date{Updated: \today}
\author{Your Name Here\\ UNC Charlotte}
\begin{document}
\maketitle
\setcounter{chapter}{9}
\chapter{The Euclidean Space \texorpdfstring{\(\mathbb R^n\)}{Rn}}
\section{The Linear Structure of \texorpdfstring{\(\mathbb R^n\)}{Rn}
and the Scalar Product}
\begin{proposition}[10.2]
Let \(\mathbf u,\mathbf v,\mathbf w\in\mathbb R^n\)
and \(\alpha,\beta\in\mathbb R\). Then both of the following hold:
\[
\langle \mathbf u,\mathbf v\rangle=\langle \mathbf v,\mathbf u\rangle
\]
\[
\langle \alpha\mathbf u+\beta\mathbf w,v\rangle
=
\alpha\langle \mathbf u,\mathbf v\rangle+\beta\langle \mathbf w,\mathbf v\rangle
\]
\end{proposition}
\begin{proof}
\end{proof}
\begin{lemma}[10.4]
For \(\mathbf u,\mathbf v\in\mathbb R^n\), \(\mathbf u,\mathbf v\) are
orthogonal if and only if
\(\|\mathbf u+\mathbf v\|=\|\mathbf u\|^2+\|\mathbf v\|^2\).
\end{lemma}
\begin{proof}
\end{proof}
\begin{lemma}[10.5]
For \(\mathbf u,\mathbf v\in\mathbb R^n\) where \(\mathbf v\not=\mathbf 0\),
define \(\lambda=\frac{\langle \mathbf u,\mathbf v\rangle}{\langle \mathbf v,\mathbf v\rangle}\)
and \(\mathbf w=\mathbf u-\lambda\mathbf v\). Then \(\mathbf v,\mathbf w\)
are orthogonal and \(\mathbf u=\mathbf w+\lambda\mathbf v\).
\end{lemma}
\begin{proof}
\end{proof}
\begin{theorem}[10.6, The Cauchy-Schwarz Inequality]
For any two vectors \(\mathbf u,\mathbf v\in\mathbb R^n\),
\[
|\langle \mathbf u,\mathbf v\rangle|
\leq
\|\mathbf u\|\|\mathbf v\|
.\]
\end{theorem}
\begin{proof}
\end{proof}
\begin{theorem}[10.7, The Triangle Inequality]
For any two vectors \(\mathbf u,\mathbf v\in\mathbb R^n\),
\[
\|\mathbf u+\mathbf v\|
\leq
\|\mathbf u\|+\|\mathbf v\|
.\]
\end{theorem}
\begin{exercise}[3]
Show that for \(\mathbf u\in\mathbb R^n\), \(\alpha\in\mathbb R\):
\begin{enumerate}[(a)]
\item \(\|\mathbf u\|=0\) if and only if \(\mathbf u=\mathbf 0\).
\item \(\|\alpha\mathbf u\|=|\alpha|\|\mathbf u\|\).
\end{enumerate}
\end{exercise}
\begin{proof}
\end{proof}
\begin{exercise}[4]
For \(\mathbf u,\mathbf v\in\mathbb R^n\) verify the identity
\[
\|\mathbf u-\mathbf v\|^2
=
\|\mathbf u\|^2+\|\mathbf v\|^2-2\langle \mathbf u,\mathbf v\rangle
.\]
\end{exercise}
\begin{proof}
\end{proof}
\begin{exercise}[9]
Let \(\mathbf u\in\mathbb R^n\) and suppose \(\|\mathbf u\|<1\).
Show that for \(\mathbf v\in\mathbb R^n\),
\(\|\mathbf v-\mathbf u\|<1-\|\mathbf u\|\) implies
\(\|\mathbf v\|<1\).
\end{exercise}
\begin{proof}
\end{proof}
\begin{exercise}[10]
Let \(\mathbf u\in\mathbb R^n\) and \(r>0\). Suppose \(\mathbf v,\mathbf w\in\mathbb R^n\)
are at a distance less than \(r\) from \(\mathbf u\). Prove that if
\(0\leq t\leq 1\), then the point \(t\mathbf v+(1-t)]\mathbf w\) is also
at a distance less than \(r\) from \(\mathbf u\).
\end{exercise}
\begin{proof}
\end{proof}
\end{document}
上述标记导致了标题中奇怪的错误：LATEX ERROR: BAD MATH ENVIRONMENT DELIMITER.
Errors:
./foo.tex:121: LATEX ERROR: BAD MATH ENVIRONMENT DELIMITER. [\end{document}]
./foo.tex:121: Extra }, or forgotten $. [\end{document}]
./foo.tex:121: Extra }, or forgotten $. [\end{document}]
./foo.tex:121: Missing $ inserted. [\end{document}]
./foo.tex:121: Missing } inserted. [\end{document}]
./foo.tex:121: Missing } inserted. [\end{document}]
请注意，它是完全大写的，不像我在文档中添加额外内容时遇到的错误$。实际上，添加任何类型的环境似乎都会导致错误。更奇怪的是，在 Overleaf 等平台上编译时似乎不会发生此错误。
由于它是系统特定的，我不确定能否找到解决方案。不管怎么说，我使用的是 Ubuntu 14.04 LTS。至少，有人知道为什么我得到的错误都是大写的吗？
答案1
尽管\(和是强大的命令，但在移动章节标题等参数时\)使用它们更为安全。这似乎是这些命令不够强大的一个案例。$
答案2
由于这里没有提供示例，因此只能根据错误消息进行猜测
%\RequirePackage[2014/01/01]{latexrelease}
\documentclass{book}
\begin{document}
\tableofcontents
\chapter{a \(x\) b}
xxx
\pagebreak
xx
\section{x}
x
\end{document}
您的一个系统比 2015 年还老，因此\(很脆弱，所以必须像标题一样使用\protect\(。另一个系统是 2015 年或 2016 年的，\(非常强大。
您可以通过取消注释 requirepackage 调用来模拟旧系统并查看错误。