方向导数

Directional Derivative
偏导数反映的是函数 沿坐标轴方向 的变化率，但是适用范围不够
引入方向导数来讨论函数沿指定的任一方向变化率问题

\begin{aligned} {\frac{\partial f}{\partial l} |}_{(x_{0}, y_{0})} & = lim_{t \to 0^{+}} \frac{f (x_{0} + t \cos α, y_{0} + t \cos β) - f (x_{0}, y_{0})}{t} \end{aligned}

\begin{aligned} {\frac{\partial f}{\partial l} |}_{(x_{0}, y_{0})} & = f_{x} (x_{0}, y_{0}) \cos α + f_{y} (x_{0}, y_{0}) \cos β \end{aligned}

\begin{aligned} {\frac{\partial f}{\partial l} |}_{(x_{0}, y_{0}, z_{0})} & = lim_{t \to 0^{+}} \frac{f (x_{0} + t \cos α, y_{0} + t \cos β, z_{0} + t \cos γ) - f (x_{0}, y_{0}, z_{0})}{t} \end{aligned}

\begin{array}{r} {\frac{\partial f}{\partial l} |}_{(x_{0}, y_{0}, z_{0})} = f_{x} (x_{0}, y_{0}, z_{0}) \cos α + f_{y} (x_{0}, y_{0}, z_{0}) \cos β + f_{z} (x_{0}, y_{0}, z_{0}) \cos γ \end{array}