# 向量算子

${\displaystyle \operatorname {grad} \equiv \nabla }$
${\displaystyle \operatorname {div} \ \equiv \nabla \cdot }$
${\displaystyle \operatorname {curl} \equiv \nabla \times }$

${\displaystyle \nabla ^{2}\equiv \operatorname {div} \ \operatorname {grad} \equiv \nabla \cdot \nabla }$

${\displaystyle \nabla f}$

${\displaystyle f\nabla }$

## 三維空間中的純量函數與向量函數

### 純量函數

${\displaystyle U}$  為空間位置 ${\displaystyle (x,y,z)}$  ，例如：

${\displaystyle U(x,y,z)=x^{2}+y^{2}+z^{2}=r^{2}}$

### 向量函數

${\displaystyle V}$  為空間位置 ${\displaystyle (x,y,z)}$  ，它可以被拆成三個分量，寫成以下的向量形式：

${\displaystyle V(x,y,z)=V_{x}(x,y,z){\hat {i}}+V_{y}(x,y,z){\hat {j}}+V_{z}(x,y,z){\hat {k}}}$

## 梯度與Nabla算子的定義

${\displaystyle {\frac {\partial U}{\partial x}},{\frac {\partial U}{\partial y}},{\frac {\partial U}{\partial z}}}$

${\displaystyle {\frac {\partial U}{\partial x}}{\hat {i}},{\frac {\partial U}{\partial y}}{\hat {j}},{\frac {\partial U}{\partial z}}{\hat {k}}}$

${\displaystyle {\frac {\partial U}{\partial x}}{\hat {i}}+{\frac {\partial U}{\partial y}}{\hat {j}}+{\frac {\partial U}{\partial z}}{\hat {k}}}$

${\displaystyle \nabla U={\frac {\partial U}{\partial x}}{\hat {i}}+{\frac {\partial U}{\partial y}}{\hat {j}}+{\frac {\partial U}{\partial z}}{\hat {k}}=V(x,y,z)}$

${\displaystyle \nabla U}$  本身是一個向量函數。在幾何與物理上，它指向變化速率最大的那個方向，在這個意義上，它被稱為 ${\displaystyle U}$  的梯度、或斜率。

### Nabla算子的單獨使用

${\displaystyle \nabla ={\frac {\partial }{\partial x}}{\hat {i}}+{\frac {\partial }{\partial y}}{\hat {j}}+{\frac {\partial }{\partial z}}{\hat {k}}}$ ，因此：
${\displaystyle \nabla U=\left({\frac {\partial }{\partial x}}{\hat {i}}+{\frac {\partial }{\partial y}}{\hat {j}}+{\frac {\partial }{\partial z}}{\hat {k}}\right)(U)={\frac {\partial U}{\partial x}}{\hat {i}}+{\frac {\partial U}{\partial y}}{\hat {j}}+{\frac {\partial U}{\partial z}}{\hat {k}}=\operatorname {grad} U}$

${\displaystyle \nabla }$  當作一個形式上的向量，則可以用向量內積叉積導出散度旋度

## 散度：Nabla算子與向量函數的內積

${\displaystyle \nabla }$  當作一個形式向量，與向量函數 ${\displaystyle V}$  做內積：

${\displaystyle \nabla \cdot V=\left({\frac {\partial }{\partial x}}{\hat {i}}+{\frac {\partial }{\partial y}}{\hat {j}}+{\frac {\partial }{\partial z}}{\hat {k}}\right)\cdot \left(V_{x}{\hat {i}}+V_{y}{\hat {j}}+V_{z}{\hat {k}}\right)={\frac {\partial V_{x}}{\partial x}}+{\frac {\partial V_{y}}{\partial y}}+{\frac {\partial V_{z}}{\partial z}}=U(x,y,z)}$

${\displaystyle \nabla \cdot V=\operatorname {div} V={\hat {i}}\cdot {\frac {\partial V}{\partial x}}+{\hat {j}}\cdot {\frac {\partial V}{\partial y}}+{\hat {k}}\cdot {\frac {\partial V}{\partial z}}}$

## 旋度：Nabla算子與向量函數的叉積

${\displaystyle \nabla }$  當作一個形式向量，與向量函數 ${\displaystyle V}$  做叉積：

{\displaystyle {\begin{aligned}\nabla \times V&=\left({\frac {\partial }{\partial x}}{\hat {i}}+{\frac {\partial }{\partial y}}{\hat {j}}+{\frac {\partial }{\partial z}}{\hat {k}}\right)\times \left(V_{x}{\hat {i}}+V_{y}{\hat {j}}+V_{z}{\hat {k}}\right)\\&={\hat {i}}\left({\frac {\partial V_{z}}{\partial y}}-{\frac {\partial V_{y}}{\partial z}}\right)+{\hat {j}}\left({\frac {\partial V_{x}}{\partial z}}-{\frac {\partial V_{z}}{\partial x}}\right)+{\hat {k}}\left({\frac {\partial V_{y}}{\partial x}}-{\frac {\partial V_{x}}{\partial y}}\right)={\begin{vmatrix}{\hat {i}}&{\hat {j}}&{\hat {k}}\\{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\\V_{x}&V_{y}&V_{z}\end{vmatrix}}\end{aligned}}}

${\displaystyle \nabla \times V=\operatorname {curl} V={\hat {i}}\times {\frac {\partial V}{\partial x}}+{\hat {j}}\times {\frac {\partial V}{\partial y}}+{\hat {k}}\times {\frac {\partial V}{\partial z}}}$

## 拉普拉斯算子

${\displaystyle \nabla \cdot \nabla =\left({\frac {\partial }{\partial x}}{\hat {i}}+{\frac {\partial }{\partial y}}{\hat {j}}+{\frac {\partial }{\partial z}}{\hat {k}}\right)\cdot \left({\frac {\partial }{\partial x}}{\hat {i}}+{\frac {\partial }{\partial y}}{\hat {j}}+{\frac {\partial }{\partial z}}{\hat {k}}\right)={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}}$

## 延伸阅读

• H. M. Schey (1996) Div, Grad, Curl, and All That: An Informal Text on Vector Calculus, ISBN 0-393-96997-5.