方向导数

方向导数

一阶方向导数

方向导数的定义

对函数\(f:v\to \mathbb{R}\)，记\(f_{\bm v}:t\to f(\bm x+t\bm v)\)，若\(f_{\bm v}\)对\(t\)的微分在\(t=0\)处存在，那么可定义\(f\)在\(\bm x\)处沿向量\(\bm v\)的方向导数为

\[\mathrm{D}_{\bm v}f(\bm x)=\dfrac{\partial f}{\partial \bm v}=\frac{\mathrm{d}f_{\bm v}}{\mathrm{d}t}\bigg|_{t=0}=\lim_{t\to 0}\dfrac{f(\bm x+t\bm v)-f(\bm x)}{t}\\ \]

方向导数的性质

• \(\mathrm{D}_{\bm v}(f+g)=\mathrm{D}_{\bm v}f+\mathrm{D}_{\bm v}g\)
• \(\mathrm{D}_{\bm v}(cf)=c\mathrm{D}_{\bm v}f\)
• \(\mathrm{D}_{\bm v}(fg)=g\mathrm{D}_{\bm v}f+f\mathrm{D}_{\bm v}g\)
• \(h:\mathbb{R}\to\mathbb{R},g:v\to \mathbb{R}\)，则\(\mathrm{D}_{\bm v}(h\circ g)(\bm x)=h'(g(\bm x))\mathrm{D}_{\bm v}g(\bm x)\)

如果 \(f\) 在 \(\bm x\) 处可微，则沿着任意非零向量 \(\bm v\) 的方向导数都存在，且

\[\mathrm{D}_{\bm v}f(\bm x)=\mathrm{d}f(\bm v)|_{\bm x}=\bm v\cdot\nabla f(\bm x)\\ \]

其中 \(\mathrm{d}f|_{\bm x}\) 表示 \(\bm x\) 处的全微分

方向导数与偏导数

对于 \(f:U\subset\mathbb{R^n}\to\mathbb{R}\)，由于 \(\dfrac{\partial f}{\partial \bm v}=\bm v\cdot \nabla f\) ，而

\[\nabla f=\begin{pmatrix}\dfrac{\partial f}{\partial x_1}&\dfrac{\partial f}{\partial x_2}&\cdots&\dfrac{\partial f}{\partial x_n}\end{pmatrix}^T \]

则

\[\dfrac{\partial f}{\partial \bm v}=v_1\dfrac{\partial f}{\partial x_1}+v_2\dfrac{\partial f}{\partial x_2}+\cdots+v_n\dfrac{\partial f}{\partial x_n}\\ \]

二阶方向导数

二阶方向导数的定义

记 \(g=\dfrac{\partial f}{\partial \bm v}\)，则 \(\dfrac{\partial g}{\partial \bm v}\)指 \(f\) 沿\(\bm v\)方向的二阶方向导数，记为 \(\mathrm{D}_{\bm{v}}^2 f(\bm{x})\) 或 \(\dfrac{\partial^2 f}{\partial \bm{v}^2}(\bm{x})\)。同时，记\(\dfrac{\partial g}{\partial \bm u}\)为 \(\dfrac{\partial^2 f}{\partial \bm v\partial \bm u}\)

二阶方向导数的性质

1. 如果 \(f\) 在 \(\bm{x}\) 处可二阶微分，那么 \(\mathrm{D}_{\bm{v}}^2 f(\bm{x})\) 存在

2. \(\mathrm{D}_{\bm{v}}^2 f(\bm{x})\) 是一个标量

3. 如果 \(f\) 在 \(\bm{x}\) 处的二阶偏导数连续，那么 \(\mathrm{D}_{\bm{v}}^2 f(\bm{x})\) 也连续

4. 如果 \(f\) 在 \(\bm{x}\) 处的二阶偏导数存在，那么 \(\mathrm{D}_{\bm{v}}^2 f(\bm{x}) = \bm{v}^T \bold{H}_f(\bm{x}) \bm{v}\)，其中 \(\bold{H}_f(\bm{x})\) 是 \(f\) 在 \(\bm{x}\) 处的 \(\text{Hessian}\) 矩阵

二阶方向导数与二阶偏导数的关系

\[\begin{aligned}g=\dfrac{\partial f}{\partial \bm v}&=(\nabla f)^T\bm v\\&=\begin{bmatrix}\dfrac{\partial }{\partial x_1}&\dfrac{\partial }{\partial x_2}&\cdots&\dfrac{\partial}{\partial x_n}\end{bmatrix}f\bm v\end{aligned} \]

\[\begin{aligned}\dfrac{\partial g}{\partial \bm v}&=\bm v\cdot\nabla g\\&=\bm v^T\begin{bmatrix}\dfrac{\partial}{\partial x_1}\\\dfrac{\partial }{\partial x_2}\\\vdots\\\dfrac{\partial }{\partial x_n}\end{bmatrix}g\\&=\bm v^T\begin{bmatrix}\dfrac{\partial}{\partial x_1}\\\dfrac{\partial }{\partial x_2}\\\vdots\\\dfrac{\partial }{\partial x_n}\end{bmatrix}\begin{bmatrix}\dfrac{\partial }{\partial x_1}&\dfrac{\partial }{\partial x_2}&\cdots&\dfrac{\partial}{\partial x_n}\end{bmatrix}f\bm v\\&=\bm v^T\bold H\bm v\end{aligned} \]