导数
导数是一元函数的概念.
函数\(y=f(x)\)在点\(x_0\)的某个邻域内有定义,自变量\(x\)在\(x_0\)处每取得\(\Delta x\)增量,因变量\(y\)取得\(\Delta y=f(x_0+\Delta x)-f(x_0)\)增量.
如果\(\Delta x\to 0\)时,极限\(\lim\limits_{\Delta x\to 0}\frac{\Delta y}{\Delta x}\)存在,则称\(y=f(x)\)在\(x_0\)处可导,该极限称为\(f(x)\)在\(x_0\)处导数,记为\(f'(x_0),y'\mid_{x=x_0}\)或\(\frac{df(x)}{dx}\mid_{x=x_0}\).
\[f'(x_0)=\lim\limits_{\Delta x\to 0}\frac{\Delta y}{\Delta x}=\lim\limits_{\Delta x\to 0}\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}
\]
另外2种等价写法:
\[\begin{aligned}
f'(x_0)&=\lim\limits_{h\to 0}\frac{f(x_0+h)-f(x_0)}{h}\\
f'(x_0)&=\lim\limits_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}
\end{aligned}
\]
如果\(y=f(x)\)在定义域内都可导,则可记作
\[f'(x),y', \space or \space \frac{df(x)}{dx}
\]
导数几何意义是函数的变化率、切线斜率.
偏导数
偏导数是多元函数的概念. 定义偏导数时,只有1个自变量变化,固定其他自变量,进而研究因变量的变化.
以二元函数\(z=f(x,y)\)为例:
设函数\(z=f(x,y)\)在点\((x_0,y_0)\)的某一邻域内有定义,\(y\)固定在\(y_0\)、\(x\)在\(x_0\)处增量\(\Delta x\),那么函数增量:
\[\Delta z = f(x_0+\Delta x, y_0)-f(x_0, y_0)
\]
如果极限
\[\lim\limits_{\Delta x\to 0}\frac{\Delta z}{\Delta x}=\lim\limits_{\Delta x\to 0}\frac{f(x_0+\Delta x, y_0)-f(x_0, y_0)}{\Delta x}
\]
存在,则称该极限为\(z=f(x,y)\)在点\((x_0,y_0)\)处对\(x\)的偏导数.
该偏导数3种记法:
\[\frac{\partial z}{\partial x}\mid_{\begin{aligned}x=x_0\\y=y_0\end{aligned}},
\frac{\partial f}{\partial x}\mid_{\begin{aligned}x=x_0\\y=y_0\end{aligned}}\space or \space f_x(x_0,y_0)
\]
类似,\(z=f(x,y)\)在点\((x_0,y_0)\)处对y的偏导数:
\[f_y(x_0,y_0)=\lim\limits_{\Delta y\to 0}\frac{\Delta z}{\Delta y}=\lim\limits_{\Delta y\to 0}\frac{f(x_0, y_0+\Delta y)-f(x_0, y_0)}{\Delta y}
\]
如果\(z=f(x,y)\)在定义域内对\(x\)的偏导数都存在,则可写作:
\[\frac{\partial z}{\partial x},\frac{\partial f}{\partial x}\space or \space f_x(x,y)
\]
通常简写:\(f_x\)
如果\(z=f(x,y)\)在定义域内对\(y\)的偏导数都存在,则可写作:
\[\frac{\partial z}{\partial y},\frac{\partial f}{\partial y}\space or \space f_y(x,y)
\]
通常简写:\(f_y\)
偏导数的几何意义是函数沿着坐标轴方向的变化率.
全微分
偏导数是只让一个自变量变化,固定其他自变量,从而得到变化率;全微分是让所有自变量同时变化,从而得到因变量增量.
以二元函数\(z=f(x,y)\)为例,
\[\begin{aligned}
f(x+\Delta x,y)-f(x,y)≈f_x(x,y)\Delta x\\
f(x,y+\Delta y)-f(x,y)≈f_y(x,y)\Delta x\\
\end{aligned}
\]
左边叫二元函数对x、y的偏增量,右边叫对x、y的偏微分.
对应自变量x、y变化,可得到全增量:
\[\Delta z = f(x+\Delta x, y+\Delta y)-f(x,y)
\]
因为直接计算全增量\(\Delta z\)较为复杂,所以用\(\Delta x, \Delta y\)的线性函数近似替代. 如果\(\Delta z\)能表示成:
\[\Delta z = A\Delta x + B\Delta y + o(ρ)
\]
其中,A、B不依赖\(\Delta x, \Delta y\)且仅与x、y有关,\(ρ=\sqrt{(\Delta x)^2+(\Delta y)^2}\),则称\(z=f(x,y)\)在点\((x,y)\)处可微分,线性函数\(A\Delta x+B\Delta y\)为z在\((x,y)\)处的全微分,记作\(dz\).
\[dz=A\Delta x+B\Delta y
\]
如果z在区域D内所有点处都可微分,则称z在D内可微分.
\(z=f(x,y)\)在\((x,y)\)处可微分,则在\((x,y)\)处一定连续.
\[\lim\limits_{ρ\to 0}\Delta z = \lim\limits_{ρ\to 0}[A\Delta x + B\Delta y + o(ρ)]= \lim\limits_{(\Delta x,\Delta y)\to 0} [A\Delta x+B\Delta y]+\lim\limits_{ρ\to 0}o(ρ)=0
\]
∴
\[\lim\limits_{ρ\to 0}[f(x+\Delta x, y+\Delta y)]=\lim\limits_{ρ\to 0}[z+f(x,y)]=f(x,y)
\]
∴\(z=f(x,y)\)在\((x,y)\)处连续
定理(必要) 如果\(z=f(x,y)\)在点\((x,y)\)可微分,则\(z=f(x,y)\)在该点处偏导数\(\frac{\partial z}{\partial x}、\frac{\partial z}{\partial y}\)存在,且全微分为:
\[dz=\frac{\partial z}{\partial x}\Delta x + \frac{\partial z}{\partial y}\Delta y
\]
证:
\(P(x,y)\)处可微分 => P的某个邻域内任一点\(P'(x+\Delta x,y+\Delta y)\),都有\(\Delta z = A\Delta x + B\Delta y + o(ρ)\)成立
令\(\Delta y=0\),即\(P'(x+\Delta x,y)\)也成立
∴
\[\begin{aligned}
\Delta z &= A\Delta x + o(|\Delta x|)\\
f(x+\Delta x, y)-f(x,y)&=A\Delta x+o(|\Delta x|)\\
\lim\limits_{\Delta x\to 0}\frac{f(x+\Delta x, y)-f(x,y)}{\Delta x}&=A=\frac{\partial z}{\partial x}
\end{aligned}
\]
同理,
\[B=\frac{\partial z}{\partial y}
\]
故
\[dz=A\Delta x + B\Delta y=\frac{\partial z}{\partial x}\Delta x+\frac{\partial z}{\partial y}\Delta y
\]
定理(充分) 如果函数\(z=f(x,y)\)的偏导数\(\frac{\partial z}{\partial x}、\frac{\partial z}{\partial y}\)在点\((x,y)\)处连续,则函数在该点可微分.
证:
\[\begin{aligned}
\Delta z &= f(x+\Delta x, y+\Delta y)-f(x,y)\\
&=[f(x+\Delta x,y+\Delta y)-f(x,y+\Delta y)]+[f(x,y+\Delta y)-f(x,y)]
\end{aligned}
\]
由拉格朗日中值定理:如果\(f(x)\)在\([a,b]\)上连续,在\((a,b)\)上可导,那么在\((a,b)\)内至少存在一点\(ξ\),使得\(f'(ξ)=\frac{f(b)-f(a)}{b-a}\)成立.
而\(z=f(x,y)\)在\((x,y)\)处偏导数存在且连续
∴存在\(θ_1,θ_2\in(0,1)\)使得下式成立:
\[\begin{aligned}
f(x+\Delta x,y+\Delta y)-f(x,y+\Delta y)&=f_x(x+θ_1\Delta x,y+\Delta y)\Delta x\\
f(x,y+\Delta y)-f(x,y)&=f_y(x,y+θ_2\Delta y)\Delta y
\end{aligned}
\]
又
\[\begin{aligned}
\lim\limits_{\Delta x\to 0,\Delta y\to 0}\frac{f(x+\Delta x,y+\Delta y)-f(x,y+\Delta y)}{\Delta x}
&=f_x(x,y)\\
\implies f(x+\Delta x,y+\Delta y)-f(x,y+\Delta y) &=f_x(x,y)\Delta x+ε_1\Delta x\\
\lim\limits_{\Delta y\to 0}\frac{f(x,y+\Delta y)-f(x,y)}{\Delta y}&=f_y(x,y)\\
\implies f(x,y+\Delta y)-f(x,y)&=f_y(x,y)\Delta y+ε_2\Delta y
\end{aligned}
\]
其中,\(ε_1\)为\(\Delta x,\Delta y\)的函数,当\(\Delta x\to 0,\Delta y\to 0\)时,\(ε_1\to 0\);\(ε_2\)为\(\Delta y\)的函数,当\(\Delta y\to 0\)时,\(ε_2\to 0\).
∴
\[\Delta z = f_x(x,y)\Delta x+f_y(x,y)\Delta y+ε_1\Delta x+ε_2\Delta y
\]
而
\[|\frac{ε_1\Delta x+ε_2\Delta y}{ρ}|=|\frac{ε_1\Delta x+ε_2\Delta y}{\sqrt{x^2+y^2}}|\le |\frac{ε_1\Delta x}{\sqrt{x^2+y^2}}|+|\frac{ε_2\Delta y}{\sqrt{x^2+y^2}}|\le |ε_1|+|ε_2|
\]
也就是说,\((\Delta x,\Delta y)\to (0,0)\)时,\(ρ\to 0\),\(ε_1\Delta x+ε_2\Delta y\to 0\)
综上,\(z=f(x,y)\)在\((x,y)\)处可微分.
方向导数
偏导数是函数沿着坐标轴方向的变化率,但有时,需要沿着任意方向的变化率,这就要用到方向导数.
![img](https://img2023.cnblogs.com/blog/741401/202405/741401-20240510212834346-1271270517.png)
设\(l\)是xOy平面上的一条射线,起点\(P_0(x_0,y_0)\),同向单位向量\(\bm{e_l}=(\cos α,\cos β)\). 则\(l\)参数方程:
\[\begin{aligned}
x&=x_0+t\cos α,\\
y&=y_0+t\cos β,t\ge 0
\end{aligned}
\]
设函数\(z=f(x,y)\)在点\(P_0\)的某个邻域\(U(P_0)\)内有定义,该邻域内点\(P(x_0+t\cos α,y_0+t\cos β)\)是\(l\)上一点,那么\(|PP_0|=t\),函数增量:
\(\Delta z =f(x,y)-f(x_0,y_0)=f(x_0+t\cos α,y_0+t\cos β)-f(x_0,y_0)\)
如果\(P\to P_0(t\to 0^+)\)时,极限:
\[\lim\limits_{t\to 0^+}\frac{\Delta z}{t}
\]
存在,则该极限称为函数\(f(x,y)\)在点\(P_0\)沿方向\(l\)的方向导数,即
\[\frac{\partial f}{\partial l}\mid_{(x_0,y_0)}=\lim\limits_{t\to 0^+}\frac{\Delta z}{t}
\]
方向导数几何意义:函数在\(P_0\)点处沿着\(l\)方向的变化率.
如果函数在\(P_0\)处偏导数存在,那么函数在\(P_0\)沿着\(\bm{e_l}=(1,0)=(\cos α,\cos β)\)方向的方向导数:
\[\frac{\partial f}{\partial l}\mid_{(x_0,y_0)}=\lim\limits_{t\to 0^+}\frac{\Delta z}{t}=\lim\limits_{t\to 0^+}\frac{f(x_0+t,y_0)-f(x_0,y_0)}{t}=f_x(x_0,y_0)
\]
同理,沿着\(\bm{e_l}=(0,1)=(\cos α,\cos β)\)方向的方向导数:
\[\frac{\partial f}{\partial l}\mid_{(x_0,y_0)}=f_y(x_0,y_0)
\]
也就是说,偏导数存在时,沿着对应坐标轴方向的方向导数一定存在;反过来,不一定成立.
定理 如果函数\(f(x,y)\)在点\(P_0(x_0,y_0)\)可微分,那么函数在该点的方向导数存在,记作\(\frac{\partial f}{\partial l}\mid_{(x_0,y_0)}\),且有
\[\frac{\partial f}{\partial l}\mid_{(x_0,y_0)}=f_x(x_0,y_0)\cos \alpha + f_y(x_0,y_0)\cos \beta
\]
其中,\(\cos \alpha, \cos \beta\)是\(l\)的方向余弦.
方向余弦:向量\(\bm{e_l}=(\cos \alpha, \cos \beta)\)是\(l\)与同方向的单位向量.
证:\(z=f(x,y)\)在\(P_0(x_0,y_0)\)可微分
∴偏导数\(f_x(x_0,y_0),f_y(x_0,y_0)\)存在
∴有
\[\begin{aligned}
\Delta z &= A\Delta x+B\Delta y+o(ρ)\\
&=f_x(x_0,y_0)\Delta x+f_y(x_0,y_0)\Delta y+o(\sqrt{(\Delta x)^2+(\Delta y)^2})
\end{aligned}
\]
沿着\(l\)方向,
\[\Delta x = t\cos α,\Delta y = t\cos β
\]
∴\(ρ=t\)
∴\(\Delta z = f_xt\cos α+f_yt\cos β\)
\[\begin{aligned}
\frac{\partial f}{\partial l}\mid_{(x_0,y_0)}&=\lim\limits_{t\to 0^+}\frac{\Delta z}{t}\\
&=\lim\limits_{t\to 0^+}\frac{f_xt\cos α+f_yt\cos β+o(t)}{t}\\
&=f_x(x_0,y_0)\cos α+f_y(x_0,y_0)\cos β
\end{aligned}
\]
梯度
方向导数是沿着某个方向的变化率,梯度是方向导数中变化率最大的那个方向向量.
设\(f(x,y)\)在平面区域D内具有一阶连续偏导数,对于任一点\(P_0(x_0,y_0)\in D\),向量
\[f_x(x_0,y_0)\bm{i}+f_y(x_0,y_0)\bm{j}
\]
称为\(f(x,y)\)在点\(P_0\)的梯度. 写作:
\[\bm{grad} \space f(x_0,y_0)=▽f(x_0,y_0)=f_x(x_0,y_0)\bm{i}+f_y(x_0,y_0)\bm{j}
\]
其中,\(▽=\frac{\partial}{\partial x}\bm{i}+\frac{\partial}{\partial y}\bm{j}\)称为(二维)向量微分算子或Nabla算子,\(▽f=\frac{\partial f}{\partial x}\bm{i}+\frac{\partial f}{\partial y}\bm{j}\). \(\bm{i,j}\)是直角坐标系\(x,y\)轴单位向量
如果\(f(x,y)\)在\(P_0(x_0,y_0)\)处可微分,射线\(l\)的方向向量\(\bm{e_l}=(\cos α,\cos β)\),根据方向导数与全微分关系,有
\[\begin{aligned}
\frac{\partial f}{\partial l}\mid_{(x_0,y_0)}&=f_x(x_0,y_0)\cos α+f_y(x_0,y_0)\cos β\\
&=(f_x,f_y)(\cos α,\cos β)\\
&=(f_x,f_y)\cdot \bm{e_l}\\
&=\bm{grad}\space f(x_0,y_0)\cdot \bm{e_l}\\
&=|\bm{grad}\space f(x_0,y_0)|\cos θ
\end{aligned}
\]
其中,\(θ\)是梯度\(\bm{grad}\space f(x_0,y_0)\)与\(\bm{e_l}\)的夹角.
∵\(f_x(x_0,y_0),f_y(x_0,y_0)\)是确定的
∴\(|\bm{grad}\space f(x_0,y_0)|\)大小确定
∴方向导数\(\frac{\partial f}{\partial l}\mid_{(x_0,y_0)}\)大小取决于\(θ\)大小
-
\(θ=0\),即\(\bm{e_l}\)与\(\bm{grad}\space f(x_0,y_0)\)同向,\(f(x,y)\)增加最快,函数在该方向的方向导数达到最大值\(|\bm{grad}\space f(x_0,y_0)|\).
-
\(θ=π\),反向,\(f(x,y)\)减少最快,方向导数达到最小值\(-|\bm{grad}\space f(x_0,y_0)|\).
-
\(θ=\frac{π}{2}\),垂直,变化率=0,方向导数为0
参考
[1] 同济大学数学系.高等数学(第六版 下册)[M].高等教育出版社,2007.
[2] 简谈:导数、偏导数、方向导数、梯度向量