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% Chapter 2

\chapter{定理盒子}

\begin{outline}
\index{zhaiyao@摘要盒子}
  \item 这是一个摘要盒子。
  \item 它的标题是可选参数,默认标题是摘要。
\end{outline}

\section{定理}
\index{d@定理}
\begin{theorem}[(高斯公式\footnote{选自高等数学公式。})]
\label{theorem:gauss formula}
\index{d@定理!g@高斯公式}
设空间闭区域$\Omega$是由分片光滑的闭曲面$\Sigma$围成,若函数$P(x,y,z)$,$Q(x,y,z)$,$R(x,y,z)$在$\Omega$上具有一节连续偏导数,则有
\begin{align}
  \iiint_\Omega\left(\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial x}\right)\mathrm{d}V & =\oiint_\Sigma P\mathrm{d}y\mathrm{d}z + Q\mathrm{d}z\mathrm{d}x + R\mathrm{d}x\mathrm{d}y
  \\
  & =\oiint_\Sigma \left(P\cos\alpha + Q\cos\beta + R\cos\gamma\right)\mathrm{d}S
\end{align}
这里$\Sigma$是整个边界曲面$\Omega$的外侧,$\cos\alpha,\cos\beta,\cos\gamma$是$\Sigma$在点$(x,y,z)$处的法向量的方向余弦。引用这个公式如\cref{theorem:gauss formula}。
\end{theorem}

\begin{definition}[(Stokes formula)]
\index{d@定理!s@Stokes formula}
Let $\Gamma$ be a piecewise smooth directed closed curve, $\Sigma$ is a piecewise smooth directed surface bounded by $\Gamma$, the side of $\Gamma$ and the positive direction of $\Sigma$ According to the right-hand rule, if the function $P(x,y,z)$, $Q(x,y,z)$, $R(x,y,z)$ has a first-order continuous deviation on the curve $\Sigma$ Derivative, then
\begin{multline}
  \iint_\Sigma\left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}\right)\mathrm{d}y\mathrm{d}z + \left(\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}\right)\mathrm{d}z\mathrm{d}x
  \\
  + \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\mathrm{d}x\mathrm{d}y = \oint_\Gamma P\mathrm{d}x + Q\mathrm{d}y + R\mathrm{d}z
\end{multline}
The Stokes formula is an extension of the basic calculus formula in the case of surface integration. It is also an extension of the Green formula. This formula gives the second type of surface integral on the surface block and the second type of curve on the boundary curve  Link between points.
\end{definition}

\clearpage
\begin{lemma}
定理环境的边框和习题标题盒子可使用 \lstinline{thmbox = false} 关闭。
\end{lemma}

\begin{corollary}
这是一个推论。
\end{corollary}

\begin{proposition}
这是一个性质。
\end{proposition}

\begin{example}
这是一个例。
\end{example}

\begin{remark}
这是一个注。
\end{remark}

\begin{proof}
这是一个证明,末尾自动添加证明结束符。
\end{proof}

\begin{mybox}*[My title][MintCream]
\index{z@自定义盒子}
  \zhlipsum*[3][name = aspirin]
  \tcblower
  \zhlipsum*[8][name = aspirin]
\end{mybox}

\zhlipsum*[3][name = aspirin]
\begin{exercise}[LightYellow][1.][习题]
\index{x@习题环境}
  \item 设$w = f(x + y + z,xyz)$,$f$具有二阶连续偏导数,求$\dfrac{{\partial w}}{{\partial x}}$和$\dfrac{{{\partial ^2}w}}{{\partial x\partial z}}$。
  \item 已知$y = y(x)$在任意点$x$处的增量$\Delta y = \dfrac{y\Delta x}{1 + x^2} + \alpha$,其中$\alpha$是$\Delta x$的高阶无穷小($\Delta x\to 0$时),$y(0) = \pi$,则$y(1) = \uline{\mbox{\hspace{2em}}}$。
  \item 设函数$f(x)$在$( - \infty, + \infty)$上有定义,则下述命题中正确的是 \mbox{(\hspace{1.5em})}
    \begin{tasks}
      \task 若$f(x)$在$( - \infty, + \infty)$上可导且单调增加,则对一切$x\in ( - \infty, + \infty)$,都有$f'(x) > 0$。
      \task 若$f(x)$在点$x_0$处取得极值,则$f'(x_0) = 0$。
      \task 若$f''(x_0) = 0$,则$(x_0,f(x_0))$是曲线$y = f(x)$的拐点坐标。
      \task 若$f'(x_0) = 0$, $f''(x_0) = 0$,$f'''(x_0)\ne 0$,则$x_0$一定不是$f(x)$的极值点。
    \end{tasks}
\end{exercise}