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LaTeX‘缺少$’找不到原因

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LaTeX‘缺少$’找不到原因

因此，我正在为考试写一些数学定义，出于某种原因，Texworks 说我的工作中“缺少$”，但我看不出有什么问题。

\subsection{Directional Derivative}
Let $\Omega \subset \mathbb{R}^n$ be open and let $f:\Omega \to \mathbb{R}$ be a continuosly differentiable scalar field on $\Omega$. Let $\vec{\hat{a}}$ be a unit vector in $\mathbb{R}^n$ Then:
$$D\hat{a}f(\vec{x_0})=\lim_{h \to 0}\frac{(f(\vec{x_0}+h\vec{\hat{a}}-f(\vec{x_0})}{h}$$
is the \bi{directional derivative} of fin the direction $\vec{\hat{a}}$

\subsection{ $\nabla$ Operator}
The \textit{\textbf{ $\nabla$ or del operator}} is written formally as
$$ \nabla = \frac{\partial}{\partial x_1}\vect{e_1} + \frac{\partial}{\partial x_2}\vect{e_2}+ \dots \frac{\partial}{\partial x_n}\vect{e_n}$$

\subsection{Gradient}
Let $f:\Omega \to \mathbb{R}$ be a continuously differentiable scalar field on an open set $\Omega \subset \mathbb{R}^n$ Then:
$$grad \, f\equiv \nabla f:=\frac{\partial f}{\partial x_1}\vect{e}_1+\dots+\frac{\partial f}{\partial x_n}\vec{e}_n$$ is the \bi{gradient} on $\Omega$ which is itself a vector field. And is obatained by applying the $\nabla$ Operator to the function.

\subsection{Divergence of F}

如果我忽略错误并强制它完成，我会得到一个奇怪的重复，e_1 越来越小，然后在最后得到一个偏导数。

而事实上它应该是偏导数的总和。

有人知道我做错什么了吗？

答案1

您的打字不一致。

  1. \vec{x}_0在或之间进行选择\vec{x_0}并遵循惯例；在示例中使用了第一种形式，这似乎更常见。

  2. 一旦定义了\bi“粗体斜体”，就一直使用它。

  3. 请小心使用参数中的空格。

  4. 始终使用\[...\]和不是 $$...$$（看为什么 \[ ... \] 比 $$ ... $$ 更可取？）。

  5. \colon优于:函数符号。

  6. 标点符号应该一致；“Then”后面的冒号似乎没有必要，而最后的句号是是否也将它们添加到段落或句子结尾处显示的方程式中则取决于个人喜好。

  7. 不要使用诸如这样的“手工制作的”操作符名称grad\,，而应使用更高级的\DeclareMathOperator方式。

\documentclass{article}
\usepackage{amsmath}
\usepackage{amssymb}

\DeclareMathOperator{\grad}{grad}
\newcommand{\bi}[1]{\textbf{\textit{#1}}} % guess

\begin{document}

\subsection{Directional Derivative}

Let $\Omega \subset \mathbb{R}^n$ be open and let $f\colon\Omega \to \mathbb{R}$ be
a continuosly differentiable scalar field on $\Omega$. Let $\vec{\hat{a}}$ be a unit
vector in $\mathbb{R}^n$. Then
\[
D\hat{a}f(\vec{x}_0)=\lim_{h \to 0}\frac{(f(\vec{x}_0+h\vec{\hat{a}}-f(\vec{x}_0)}{h}
\]
is the \bi{directional derivative} of fin the direction $\vec{\hat{a}}$.

\subsection{$\nabla$ Operator}
The \bi{$\nabla$ or del operator} is written formally as
\[
\nabla = \frac{\partial}{\partial x_1}\vec{e}_1 + 
  \frac{\partial}{\partial x_2}\vec{e}_2+ \dots \frac{\partial}{\partial x_n}\vec{e}_n.
\]

\subsection{Gradient}

Let $f\colon\Omega \to \mathbb{R}$ be a continuously differentiable scalar field
on an open set $\Omega \subset \mathbb{R}^n$. Then
\[
\grad f\equiv \nabla f:=
  \frac{\partial f}{\partial x_1}\vec{e}_1+\dots+\frac{\partial f}{\partial x_n}\vec{e}_n
\]
is the \bi{gradient} on $\Omega$ which is itself a vector field. And is obtained
by applying the $\nabla$ Operator to the function.

\subsection{Divergence of $F$}
\end{document}

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