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### 课程: 多变量微积分>单元 2

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# 参数曲面的偏导数, 第二部分

## 我们要做什么

• 在设置中，我们有一些带有二维输入和三维输出的矢量值函数:
$\stackrel{\to }{\mathbf{\text{v}}}\left(s,t\right)=\left[\begin{array}{c}x\left(s,t\right)\\ y\left(s,t\right)\\ z\left(s,t\right)\end{array}\right]$

$\begin{array}{rl}\phantom{\rule{1em}{0ex}}\frac{\partial \stackrel{\to }{\mathbf{\text{v}}}}{\partial t}\left(s,t\right)& =\left[\begin{array}{c}\frac{\partial x}{\partial t}\left(s,t\right)\\ \frac{\partial y}{\partial t}\left(s,t\right)\\ \frac{\partial z}{\partial t}\left(s,t\right)\end{array}\right]\end{array}$
$\begin{array}{rl}\phantom{\rule{1em}{0ex}}\frac{\partial \stackrel{\to }{\mathbf{\text{v}}}}{\partial s}\left(s,t\right)& =\left[\begin{array}{c}\frac{\partial x}{\partial s}\left(s,t\right)\\ \frac{\partial y}{\partial s}\left(s,t\right)\\ \frac{\partial z}{\partial s}\left(s,t\right)\end{array}\right]\end{array}$
• 您可以将这些偏导数解释为与定义向量$\stackrel{\to }{\mathbf{\text{v}}}$的参数曲面相切。

## 目标

$\stackrel{\to }{\mathbf{\text{v}}}\left(t,s\right)=\left[\begin{array}{c}3\mathrm{cos}\left(t\right)+\mathrm{cos}\left(t\right)\mathrm{cos}\left(s\right)\\ 3\mathrm{sin}\left(t\right)+\mathrm{sin}\left(t\right)\mathrm{cos}\left(s\right)\\ \mathrm{sin}\left(s\right)\end{array}\right]$

## 解释偏导数

#### 关于 $t$‍  的区别

$\begin{array}{rl}\phantom{\rule{1em}{0ex}}\frac{\partial \stackrel{\to }{\mathbf{\text{v}}}}{\partial t}\left(t,s\right)& =\frac{\partial }{\partial t}\left[\begin{array}{c}3\mathrm{cos}\left(t\right)+\mathrm{cos}\left(t\right)\mathrm{cos}\left(s\right)\\ \\ 3\mathrm{sin}\left(t\right)+\mathrm{sin}\left(t\right)\mathrm{cos}\left(s\right)\\ \\ \mathrm{sin}\left(s\right)\end{array}\right]\\ \\ & =\left[\begin{array}{c}\frac{\partial }{\partial t}\left(3\mathrm{cos}\left(t\right)+\mathrm{cos}\left(t\right)\mathrm{cos}\left(s\right)\right)\\ \\ \frac{\partial }{\partial t}\left(3\mathrm{sin}\left(t\right)+\mathrm{sin}\left(t\right)\mathrm{cos}\left(s\right)\right)\\ \\ \frac{\partial }{\partial t}\left(\mathrm{sin}\left(s\right)\right)\end{array}\right]\\ \\ & =\left[\begin{array}{c}-3\mathrm{sin}\left(t\right)-\mathrm{sin}\left(t\right)\mathrm{cos}\left(s\right)\\ \\ 3\mathrm{cos}\left(t\right)+\mathrm{cos}\left(t\right)\mathrm{cos}\left(s\right)\\ \\ 0\end{array}\right]\end{array}$

$ts$-平面上, $s$ 的常量值与水平线相对应。 $s=\pi /2$, 这条线在这以红色绘制:

$\begin{array}{rl}\phantom{\rule{1em}{0ex}}\stackrel{\to }{\mathbf{\text{v}}}\left(\frac{\pi }{4},\frac{\pi }{2}\right)& =\left[\begin{array}{c}3\mathrm{cos}\left(\pi /4\right)+\mathrm{cos}\left(\pi /4\right)\mathrm{cos}\left(\pi /2\right)\\ 3\mathrm{sin}\left(\pi /4\right)+\mathrm{sin}\left(\pi /4\right)\mathrm{cos}\left(\pi /2\right)\\ \mathrm{sin}\left(\pi /2\right)\end{array}\right]\\ \\ & =\left[\begin{array}{c}3\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}\left(0\right)\\ 3\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}\left(0\right)\\ 1\end{array}\right]\\ \\ & =\left[\begin{array}{c}\frac{3\sqrt{2}}{2}\\ \\ \frac{3\sqrt{2}}{2}\\ \\ 1\end{array}\right]\end{array}$

$\begin{array}{rl}\phantom{\rule{1em}{0ex}}\frac{\partial \stackrel{\to }{\mathbf{\text{v}}}}{\partial t}\left(\frac{\pi }{4},\frac{\pi }{2}\right)& =\left[\begin{array}{c}-3\mathrm{sin}\left(\pi /4\right)-\mathrm{sin}\left(\pi /4\right)\mathrm{cos}\left(\pi /2\right)\\ 3\mathrm{cos}\left(\pi /4\right)+\mathrm{cos}\left(\pi /4\right)\mathrm{cos}\left(\pi /2\right)\\ 0\end{array}\right]\\ \\ & =\left[\begin{array}{c}-3\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}\left(0\right)\\ 3\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}\left(0\right)\\ 0\end{array}\right]\\ \\ & =\left[\begin{array}{c}-\frac{3\sqrt{2}}{2}\\ \frac{3\sqrt{2}}{2}\\ 0\end{array}\right]\end{array}$

#### 关于 $s$‍  的区别

$s$相关的偏导数是相似的。 您可以通过计算$\stackrel{\to }{\mathbf{\text{v}}}$中的每个分量的偏导数来计算它：
$\begin{array}{rl}\phantom{\rule{1em}{0ex}}\frac{\partial \stackrel{\to }{\mathbf{\text{v}}}}{\partial s}\left(t,s\right)=\frac{\partial }{\partial s}& \left[\begin{array}{c}3\mathrm{cos}\left(t\right)+\mathrm{cos}\left(t\right)\mathrm{cos}\left(s\right)\\ 3\mathrm{sin}\left(t\right)+\mathrm{sin}\left(t\right)\mathrm{cos}\left(s\right)\\ \mathrm{sin}\left(s\right)\end{array}\right]\\ =& \left[\begin{array}{c}-\mathrm{cos}\left(t\right)\mathrm{sin}\left(s\right)\\ -\mathrm{sin}\left(t\right)\mathrm{sin}\left(s\right)\\ \mathrm{cos}\left(s\right)\end{array}\right]\end{array}$

## 总结

• 在设置中，我们有一些带有二维输入和三维输出的矢量值函数:
$\stackrel{\to }{\mathbf{\text{v}}}\left(s,t\right)=\left[\begin{array}{c}x\left(s,t\right)\\ y\left(s,t\right)\\ z\left(s,t\right)\end{array}\right]$

$\begin{array}{rl}\phantom{\rule{1em}{0ex}}\frac{\partial \stackrel{\to }{\mathbf{\text{v}}}}{\partial t}\left(s,t\right)& =\left[\begin{array}{c}\frac{\partial x}{\partial t}\left(s,t\right)\\ \frac{\partial y}{\partial t}\left(s,t\right)\\ \frac{\partial z}{\partial t}\left(s,t\right)\end{array}\right]\end{array}$
$\begin{array}{rl}\phantom{\rule{1em}{0ex}}\frac{\partial \stackrel{\to }{\mathbf{\text{v}}}}{\partial s}\left(s,t\right)& =\left[\begin{array}{c}\frac{\partial x}{\partial s}\left(s,t\right)\\ \frac{\partial y}{\partial s}\left(s,t\right)\\ \frac{\partial z}{\partial s}\left(s,t\right)\end{array}\right]\end{array}$
• 您可以将这些偏导数解释为与定义向量$\stackrel{\to }{\mathbf{\text{v}}}$的参数曲面相切。
• 比如，想象在输入空间中沿着$t$方向推一个点，比如从坐标$\left(s,t\right)$推向坐标$\left(s,t+h\right)$的某个小$h$这导致输出沿表面有一个小的推力，这个推理由向量$h\frac{\partial \stackrel{\to }{\mathbf{\text{v}}}}{\partial t}\left(s,t\right)$表示。