最后更新: 2025-04-05 11:06 查看: 34 次反馈刷题

三种算子的性质与迭代

## 散度 梯度与涡度
散度，梯度，涡度都是綫性算子，也就是把它們运用在一綫性組合上，仍然得出相仿的綫性組合，師

$$
\begin{aligned}
\operatorname{div}(\lambda a +\mu b ) & =\lambda \operatorname{div} a +\mu \operatorname{div} b \\
\operatorname{grad}(\lambda \varphi+\mu \psi) & =\lambda \operatorname{grad} \varphi+\mu \operatorname{grad} \psi \\
\operatorname{rot}(\lambda a +\mu b ) & =\lambda \operatorname{rot} a +\mu \operatorname{rot} b
\end{aligned}
$$

这儿 $\lambda, \mu$ 是任澺的常数。
又关于各种乘积有以下的公式：
首先两函数之积的梯度

$$
\operatorname{grad}(\varphi \psi)=\left(\frac{\partial(\varphi \psi)}{\partial x}, \frac{\partial(\varphi \psi)}{\partial y}, \frac{\partial(\varphi \psi)}{\partial z}\right)=\varphi \operatorname{grad} \psi+\psi \operatorname{grad} \varphi .
$$

函数 $\varphi$ 的函数 $F$ 的梯度等于

$$
\operatorname{grad} F(\varphi)=\left(\frac{\partial F(\varphi)}{\partial x}, \frac{\partial F(\varphi)}{\partial y} \cdot \frac{\partial F(\varphi)}{\partial z}\right)=F^{\prime}(\varphi) \operatorname{grad} \varphi
$$

二矢量 $a =\left(a_x, a_y, a_z\right), b =\left(b_x, b_y, b_z\right)$ 的內积的梯度等于

$$
\begin{aligned}
& \operatorname{grad} a \cdot b =\left(\frac{\partial}{\partial x}( a \cdot b ), \frac{\partial}{\partial y}( a \cdot b ), \frac{\partial}{\partial z}( a \cdot b )\right) \\
&=\left(\left(\frac{\partial}{\partial x} a \right) \cdot b + a \cdot\left(\frac{\partial}{\partial x} b \right),\left(\frac{\partial}{\partial y} a \right) \cdot b + a \cdot\left(\frac{\partial}{\partial y} b \right)\right. \\
&\left.\left(\frac{\partial}{\partial z} a \right) \cdot b + a \cdot\left(\frac{\partial}{\partial z} b \right)\right)=\left(a_x \frac{\partial}{\partial x}+a_y \frac{\partial}{\partial y}+a_z \frac{\partial}{\partial z}\right) b + \\
&+\left(b_x \frac{\partial}{\partial x}+b_y \frac{\partial}{\partial y}+b_z \frac{\partial}{\partial z}\right) a + a \times \operatorname{rot} b + b \times \operatorname{rot} a
\end{aligned}
$$

図数 $\varphi$ 乘矢量 $a$ 的散度是

$$
\operatorname{div} \varphi a =\frac{\partial\left(\varphi a_x\right)}{\partial x}+\frac{\partial\left(\varphi a_y\right)}{\partial y}+\frac{\partial\left(\varphi a_z\right)}{\partial z}=\varphi \operatorname{div} a +\operatorname{grad} \varphi \cdot a
$$

两矢量 $a , b$ 的外积的散度为
$\operatorname{div} a \times b =\frac{\partial}{\partial x}\left(a_y b_z-a_z b_y\right)+\frac{\partial}{\partial y}\left(a_z b_x-a_x b_z\right)+\frac{\partial}{\partial z}\left(a_x b_y-a_v b_x\right)$

$$
\begin{aligned}
= & -a_x\left(\frac{\partial b_z}{\partial y}-\frac{\partial b_v}{\partial z}\right)-a_y\left(\frac{\partial b_x}{\partial z}-\frac{\partial b_x}{\partial x}\right)-a_x\left(\frac{\partial b_y}{\partial x}-\frac{\partial b_x}{\partial y}\right) \\
& +b_x\left(\frac{\partial a_z}{\partial y}-\frac{\partial a_y}{\partial z}\right)+b_y\left(\frac{\partial a_x}{\partial z}-\frac{\partial a_z}{\partial x}\right)+b_x\left(\frac{\partial a _y}{\partial x}-\frac{\partial a_x}{\partial y}\right) \\
= & b \cdot \operatorname{rot} a - a \cdot \operatorname{rot} b .
\end{aligned}
$$

函数乘矢量的涡度是

$$
\operatorname{rot} \varphi a =\varphi \operatorname{rot} a +\operatorname{grad} \varphi \times a
$$

二矢量之外积的涡度是

$$
\begin{aligned}
\operatorname{rot} a \times b = & \left(b_x \frac{\partial}{\partial x}+b_y \frac{\partial}{\partial y}+b_z \frac{\partial}{\partial z}\right) a \\
& -\left(a_x \frac{\partial}{\partial x}+a_y \frac{\partial}{\partial y}+a_z \frac{\partial}{\partial z}\right) b \\
& +(\operatorname{div} b ) a -(\operatorname{div} a ) b .
\end{aligned}
$$

## 三种算子的迭代

运算 $\operatorname{grad} \varphi$ 把一純量場变为矢量場，运算 $\operatorname{div} R$ 把一矢量場变为純量場，而运算 $\operatorname{rot} R$ 复把一矢量場变为矢量場。这三种运算的迭用可得以下的一些公式。

首先易証
$\operatorname{div} \operatorname{rot} R = 0$
及
$\operatorname{rot} \operatorname{grad} \varphi= 0$.

其次，

$$
\begin{aligned}
\operatorname{div} \operatorname{grad} \varphi & =\frac{\partial^2 \varphi}{\partial x^2}+\frac{\partial^2 \varphi}{\partial y^2}+\frac{\partial^2 \varphi}{\partial z^2} \\
& =\Delta \varphi(\text { Laplace 算子 })
\end{aligned}
$$

最后由于

$$
\begin{gathered}
\operatorname{grad} \operatorname{div} R =\left(\frac{\partial}{\partial x}\left(\frac{\partial X}{\partial x}+\frac{\partial Y}{\partial y}+\frac{\partial Z}{\partial z}\right), \frac{\partial}{\partial y}\left(\frac{\partial X}{\partial x}+\frac{\partial Y}{\partial y}+\frac{\partial Z}{\partial z}\right)\right. \\
\left.\frac{\partial}{\partial z}\left(\frac{\partial X}{\partial x}+\frac{\partial Y}{\partial y}+\frac{\partial Z}{\partial z}\right)\right)
\end{gathered}
$$

及

$$
\begin{aligned}
\operatorname{rot} \operatorname{rot} R & =\operatorname{rot}\left(\frac{\partial Z}{\partial y}-\frac{\partial Y}{\partial z}, \frac{\partial X}{\partial z}-\frac{\partial Z}{\partial x}, \frac{\partial Y}{\partial x}-\frac{\partial X}{\partial y}\right) \\
& =\left(\frac{\partial^2 Y}{\partial x \partial y}-\frac{\partial^2 X}{\partial y^2}-\frac{\partial^2 X}{\partial z^2}+\frac{\partial^2 Z}{\partial x \partial z}, \frac{\partial^2 Z}{\partial y \partial z}-\frac{\partial^2 Y}{\partial z^2}-\frac{\partial^2 Y}{\partial x^2}+\frac{\partial^2 X}{\partial y \partial x}\right. \\
& \left.\frac{\partial^2 X}{\partial z \partial x}-\frac{\partial^2 Z}{\partial x^2}-\frac{\partial^2 Z}{\partial y^2}+\frac{\partial^2 Y}{\partial z \partial y}\right)
\end{aligned}
$$

相減可得

$$
\operatorname{grad} \operatorname{div} R -\operatorname{rot} \operatorname{rot} R =\Delta R
$$

其他版本
【数学分析】数量场和向量场

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