矢量微分算子
矢量微分算子（vector differential operator），\(\nabla\),英文名称为“del”或者“nabla”，其在三维空间中的定义为：
\[
\nabla = \frac{\partial}{\partial x} \mathbf{i} + \frac{\partial}{\partial y} \mathbf{j} + \frac{\partial}{\partial z} \mathbf{k}
\]
梯度(\(grad\))
将上述算符应用于标量场，则得到的结果是标量场的梯度（\(grad\)）：
\[
grad f(x,y,z) = \nabla f(x,y,z) = \frac{\partial f}{\partial x} \mathbf{i} + \frac{\partial f}{\partial y} \mathbf{j} + \frac{\partial f}{\partial z} \mathbf{k}
\]
散度(\(div\))
该算符与向量场\(\mathbf{F}(x,y,z)\)的标量积为该向量场的散度：
\[
div \mathbf{F}(x,y,z) = \nabla \cdot \mathbf{F} = \frac{\partial F_1}{\partial x} \mathbf{i} + \frac{\partial F_2}{\partial y} \mathbf{j} + \frac{\partial F_3}{\partial z} \mathbf{k}
\]
旋度(\(curl\))
该算符与矢量场\(\mathbf{F}(x,y,z)\)的矢量积称为该矢量场的旋度：
\[
\begin{align*}
curl \mathbf{F}(x,y,z) &= \nabla \times \mathbf{F} \\
&= (\frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z})\mathbf{i} - (\frac{\partial F_3}{\partial x} - \frac{\partial F_1}{\partial z}) \mathbf{j} + (\frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y})\mathbf{k} \\
&=
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
F_1 & F_2 &F_3
\end{vmatrix}
\end{align*}
\]
