最后更新: 2025-04-05 11:06 查看: 76 次反馈同步训练

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}

其他版本
【数学分析】数量场和向量场

免费注册看余下 50%

非VIP会员每天15篇文章，开通VIP 无限制查看

上一篇：等值线

下一篇：多元函数的极值

本文对您是否有用？有用 (0) 无用 (0)