##### 向量算子 - 向量算子 - **向量算子**是[[向量场]]中使用的微分算子, 包括[[梯度]], [[散度]], [[旋度]], [[拉普拉斯算子]], 在不同[[坐标系]]有不同形式 - 梯度作用于标量场, 产生向量场 - $\displaystyle \text{grad}\ f = \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)$ - 散度作用于向量场, 产生标量场 - $\displaystyle \text{div}\ \mathbf{F} = \nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}$ - 旋度作用于向量场, 产生向量场 - $\displaystyle \text{curl}\ \mathbf{F} = \nabla \times \mathbf{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}, \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}, \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right)$ - 拉普拉斯算子可作用于标量场或向量场 - $\displaystyle \Delta f = \nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}$ - $\Delta \mathbf{F} = (\Delta F_x, \Delta F_y, \Delta F_z)$