矢量基本操作和运算法则

北京理工大学 | 李明健

$\varphi,\psi$ $\boldsymbol F,\boldsymbol G$ 等加粗表示矢量。

$\boldsymbol F= F_1 \boldsymbol x+ F_2\boldsymbol y+ F_3 \boldsymbol z$ ，则有：

矢量的模：

|\boldsymbol F|=\sqrt{F_1^2+F_2^2+F_3^2}

点积（标量积）：

\boldsymbol F \cdot \boldsymbol G= |\boldsymbol F||\boldsymbol G|\cos \theta = F_1 G_1+F_2 G_2+F_3 G_3

叉积（矢量积）：

\boldsymbol F \times \boldsymbol G= (F_2 G_3-F_3 G_2)\boldsymbol x +(F_3 G_1-F_1 G_3)\boldsymbol y+ (F_1 G_2-F_2 G_1)\boldsymbol z

标量三重积：

[\boldsymbol F,\boldsymbol G,\boldsymbol H]=\boldsymbol F\cdot \boldsymbol G\times \boldsymbol H = \boldsymbol F \times \boldsymbol G \cdot \boldsymbol H =\boldsymbol G\cdot \boldsymbol H\times \boldsymbol F

矢量三重积：

\boldsymbol F\times ( \boldsymbol G\times \boldsymbol H)=(\boldsymbol F\cdot \boldsymbol H) \boldsymbol G - (\boldsymbol F\cdot \boldsymbol G) \boldsymbol H

$\nabla$ ，表示以下操作：

标量的梯度：

\nabla \varphi = \frac{\partial \varphi}{\partial x} \boldsymbol x + \frac{\partial \varphi}{\partial y} \boldsymbol y+ \frac{\partial \varphi}{\partial z} \boldsymbol z

矢量的散度：

\nabla \cdot \boldsymbol F =\frac{\partial F_1}{\partial x}+\frac{\partial F_2}{\partial y}+\frac{\partial F_3}{\partial z}

$\nabla \cdot \boldsymbol F =0$ $\nabla \times \boldsymbol A = \boldsymbol F$ .

矢量的旋度：

\nabla \times \boldsymbol F =\left( \frac{\partial F_3}{\partial y}-\frac{\partial F_2}{\partial z} \right) \boldsymbol x + \left( \frac{\partial F_1}{\partial z}-\frac{\partial F_3}{\partial x} \right) \boldsymbol y + \left( \frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y} \right) \boldsymbol z

$\nabla \times \boldsymbol F =0$ 的是无旋（irrotational）有源场。

$(\boldsymbol{F} \cdot \boldsymbol{\nabla})$ 是一个矢量微分算符，如：

\begin{array}{l} (\boldsymbol{F} \cdot \boldsymbol{\nabla}) \varphi=F_{1} \dfrac{\partial \varphi}{\partial x}+F_{2} \dfrac{\partial \varphi}{\partial y}+F_{3} \dfrac{\partial \varphi}{\partial z} \\ (\boldsymbol{F} \cdot \boldsymbol{\nabla}) \boldsymbol{G}= (\boldsymbol{F} \cdot \boldsymbol{\nabla}) G_{1}\boldsymbol x + (\boldsymbol{F} \cdot \nabla) G_{2}\boldsymbol y + (\boldsymbol{F} \cdot \nabla) G_{3} \boldsymbol z \end{array}

拉普拉斯算符：

\nabla^2 = \nabla \cdot \nabla = \Delta= \frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial ^2}{\partial z^2}

下面是矢量常用操作：

\begin{array}{l} \nabla \cdot(\nabla \varphi)=\nabla^{2} \varphi \\ \nabla \cdot(\nabla \times \boldsymbol F)=0 \\ \nabla \times(\nabla \varphi)=0 \\ \nabla(\varphi \psi)=\varphi \nabla \psi+\psi \nabla \varphi \\ \nabla \cdot(\varphi \boldsymbol F)=\varphi \nabla \cdot \boldsymbol F+\boldsymbol F \cdot \nabla \varphi \\ \nabla \times(\varphi \boldsymbol F)=\varphi \nabla \times \boldsymbol F+\nabla \varphi \times \boldsymbol F \\ \nabla \times(\nabla \times \boldsymbol F)=\nabla(\nabla \cdot \boldsymbol F)-\nabla^{2} \boldsymbol F \\ \nabla(\boldsymbol F \cdot \boldsymbol G)=\boldsymbol F \times(\nabla \times \boldsymbol G)+\boldsymbol G \times(\nabla \times \boldsymbol F)+(\boldsymbol F \cdot \nabla) \boldsymbol G+(\boldsymbol G \cdot \nabla) \boldsymbol F \\ \nabla \cdot(\boldsymbol F \times \boldsymbol G)=\boldsymbol G \cdot(\nabla \times \boldsymbol F)-\boldsymbol F \cdot(\nabla \times \boldsymbol G) \\ \nabla \times(\boldsymbol F \times \boldsymbol G)=\boldsymbol F(\nabla \cdot \boldsymbol G)-\boldsymbol G(\nabla \cdot \boldsymbol F)+(\boldsymbol G \cdot \nabla) \boldsymbol F-(\boldsymbol F \cdot \nabla) \boldsymbol G \end{array}

以上。