13.2.7 空间微分算子的回顾

13.2.7.1 空间微分算子的运算法则

$U, U_{1}, U_{2}$ 和 $F$ 是标量函数; $c$ 是常数; $\vec{V}, {\vec{V}}_{1}, {\vec{V}}_{2}$ 是向量函数:

\begin{matrix} (13.84) & grad (U_{1} + U_{2}) = grad U_{1} + grad U_{2} . \end{matrix}

\begin{matrix} (13.85) & grad (c U) = c grad U . \end{matrix}

\begin{matrix} (13.86) & grad (U_{1} U_{2}) = U_{1} grad U_{2} + U_{2} grad U_{1} . \end{matrix}

\begin{matrix} (13.87) & grad F (U) = F^{'} (U) grad U . \end{matrix}

\begin{matrix} (13.88) & div ({\vec{V}}_{1} + {\vec{V}}_{2}) = div {\vec{V}}_{1} + div {\vec{V}}_{2} . \end{matrix}

\begin{matrix} (13.89) & div (c \vec{V}) = c div \vec{V} \end{matrix}

\begin{matrix} (13.90) & div (U \vec{V}) = \vec{V} \cdot grad U + U div \vec{V} . \end{matrix}

\begin{matrix} (13.91) & rot ({\vec{V}}_{1} + {\vec{V}}_{2}) = rot {\vec{V}}_{1} + rot {\vec{V}}_{2} \end{matrix}

\begin{matrix} (13.92) & rot (c \vec{V}) = c rot \vec{V} \end{matrix}

\begin{matrix} (13.93) & rot (U \vec{V}) = U rot \vec{V} - \vec{V} \times grad U . \end{matrix}

\begin{matrix} (13.94) & div rot \vec{V} \equiv 0. \end{matrix}

\begin{matrix} (13.95) & rot grad U \equiv \vec{0} (零向量). \end{matrix}

\begin{matrix} (13.96) & div grad U = Δ U \end{matrix}

\begin{matrix} (13.97) & rot rot \vec{V} = grad div \vec{V} - Δ \vec{V} . \end{matrix}

\begin{matrix} (13.98) & div ({\vec{V}}_{1} \times {\vec{V}}_{2}) = {\vec{V}}_{2} \cdot rot {\vec{V}}_{1} - {\vec{V}}_{1} \cdot rot {\vec{V}}_{2} . \end{matrix}

表 13.2 笛卡儿、柱面和球面坐标系中向量分析的表达式

	笛卡儿坐标系	柱面坐标系	球面坐标系
$d \vec{s} = d \vec{r}$	${\vec{e}}_{x} d x + {\vec{e}}_{y} d y + {\vec{e}}_{z} d z$	${\vec{e}}_{ρ} d ρ + {\vec{e}}_{φ} ρ d φ + {\vec{e}}_{z} d z$	${\vec{e}}_{r} d r + {\vec{e}}_{ϑ} r d ϑ + {\vec{e}}_{φ} r \sin$ $ϑ d φ$
$grad U$	${\vec{e}}_{x} \frac{\partial U}{\partial x} + {\vec{e}}_{y} \frac{\partial U}{\partial y} + {\vec{e}}_{z} \frac{\partial U}{\partial z}$	${\vec{e}}_{ρ} \frac{\partial U}{\partial ρ} + {\vec{e}}_{φ} \frac{1}{ρ} \frac{\partial U}{\partial φ} + {\vec{e}}_{z} \frac{\partial U}{\partial z}$	${\vec{e}}_{r} \frac{\partial U}{\partial r} + {\vec{e}}_{ϑ} \frac{1}{r} \frac{\partial U}{\partial ϑ} + {\vec{e}}_{φ} \frac{1}{r \sin ϑ} \frac{\partial U}{\partial φ}$
$div \vec{V}$	$\frac{\partial V_{x}}{\partial x} + \frac{\partial V_{y}}{\partial y} + \frac{\partial V_{z}}{\partial z}$	$\frac{1}{ρ} \frac{\partial}{\partial ρ} (ρ V_{ρ}) + \frac{1}{ρ} \frac{\partial V_{φ}}{\partial φ} + \frac{\partial V_{z}}{\partial z}$	$\frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} V_{r}) + \frac{1}{r \sin ϑ} \frac{\partial}{\partial ϑ} (V_{ϑ} \sin ϑ)$ $+ \frac{1}{r \sin ϑ} \frac{\partial V_{φ}}{\partial φ}$
$rot \vec{V}$	${\vec{e}}_{x} (\frac{\partial V_{z}}{\partial y} - \frac{\partial V_{y}}{\partial z})$	${\vec{e}}_{ρ} (\frac{1}{ρ} \frac{\partial V_{z}}{\partial φ} - \frac{\partial V_{φ}}{\partial z})$	${\vec{e}}_{r} \frac{1}{r \sin ϑ} [\frac{\partial}{\partial ϑ} (V_{φ} \sin ϑ) - \frac{\partial V_{ϑ}}{\partial φ}]$
	$+ {\vec{e}}_{y} (\frac{\partial V_{x}}{\partial z} - \frac{\partial V_{z}}{\partial x})$	$+ {\vec{e}}_{φ} (\frac{\partial V_{ρ}}{\partial z} - \frac{\partial V_{z}}{\partial ρ})$	$+ {\vec{e}}_{ϑ} \frac{1}{r} [\frac{1}{\sin ϑ} \frac{\partial V_{r}}{\partial φ} - \frac{\partial}{\partial r} (r V_{φ})]$
	$+ {\vec{e}}_{z} (\frac{\partial V_{y}}{\partial x} - \frac{\partial V_{x}}{\partial y})$	$+ {\vec{e}}_{z} (\frac{1}{ρ} \frac{\partial}{\partial ρ} (ρ V_{φ}) - \frac{1}{ρ} \frac{\partial V_{ρ}}{\partial φ})$	$+ {\vec{e}}_{φ} \frac{1}{r} [\frac{\partial}{\partial r} (r V_{ϑ}) - \frac{\partial V_{r}}{\partial ϑ}]$
$Δ U$	$\frac{\partial^{2} U}{\partial x^{2}} + \frac{\partial^{2} U}{\partial y^{2}} + \frac{\partial^{2} U}{\partial z^{2}}$	$\frac{1}{ρ} \frac{\partial}{\partial ρ} (ρ \frac{\partial U}{\partial ρ}) + \frac{1}{ρ^{2}} \frac{\partial^{2} U}{\partial φ^{2}}$	$\frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} \frac{\partial U}{\partial r})$
$Δ U$		$+ \frac{\partial^{2} U}{\partial z^{2}}$	$+ \frac{1}{r^{2} \sin ϑ} \frac{\partial}{\partial ϑ} (\sin ϑ \frac{\partial U}{\partial ϑ})$ $+ \frac{1}{r^{2} \sin ϑ} \frac{\partial^{2} U}{\partial φ^{2}}$

13.2.7.3 基本关系式和结果 (表 13.3)

算子	符号	关系	变元	结果	意义
梯度	$grad U$	$\nabla U$	标量	向量	极大增加
向量梯度	$grad \vec{V}$	$\nabla \vec{V}$	向量	二阶张量
散度	$div \vec{V}$	$\nabla \cdot \vec{V}$	向量	标量	源, 汇
旋度	$rot \vec{V}$	$\nabla \times \vec{V}$	向量	向量	卷曲
拉普拉斯算子	$△ U$	$(\nabla \cdot \nabla) U$	标量	标量	位势场源
拉普拉斯算子	$△ \vec{V}$	$(\nabla \cdot \nabla) \vec{V}$	向量	向量

13.2.7 空间微分算子的回顾 ​

13.2.7.1 空间微分算子的运算法则 ​

13.2.7.3 基本关系式和结果 (表 13.3) ​

13.2.7 空间微分算子的回顾

13.2.7.1 空间微分算子的运算法则

13.2.7.3 基本关系式和结果 (表 13.3)