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Chapter 8 多元函数微分学¶
1. 多元函数的极限与连续性¶
1.3 二元函数的极限与连续¶
极限¶
设\(n\)元函数\(f(P)\)在\(U^0(P_0,\delta_0)\)内有定义,若存在一常数\(A,\forall \varepsilon>0,\exists\delta>0(\delta\le\delta_0)\),当\(0<\rho(P,P_0)<\delta\),都有\(|f(P)-A|<\varepsilon\)成立,则称\(A\)为\(n\)元函数\(f(P)\)当\(P\to P_0\)时的极限,记作 $$ \lim_{P\to P_0} f(x)=A 或 f(P)\to A(P\to P_0) $$ 注意:这个极限与点\(P(x,y)\)趋于点\(P_0(x_0,y_0)\)的方式无关,只要\(P\)距离\(P_0\)充分接近,点\(P\)趋于\(P_0\)点的方式可有无数种.
多元函数极限的归结原则¶
若\(f(P)\)在\(U^0(P_0,\delta_0)\)内有定义,则 $$ \lim_{P\to P_0}f(P)=A\iff 对于U^0(P_0,\delta_0)的任何子集E,\lim_{\stackrel{P\to P_0}{(P\in E)}}f(P)=A $$ 同样可用归结原则,若发现点\(P\)按两个特殊的路径趋于点\(P_0\)时,\(f(P)\)极限存在但不相等,则\(f(P)\)在该点\(P_0\)极限不存在. 这是判断多元函数极限不存在的重要方法之一.
题型:证明极限:用定义;求极限:夹逼定理
累次极限与二重极限¶
累次极限\(\lim_{x\to x_0}\lim_{y\to y_0}f(x,y),\lim_{y\to y_0}\lim_{x\to x_0}f(x,y)\)本质上是二次求一元函数的极限,累次极限存在性与二重极限存在性没有一定联系.
定理:若累次极限\(\lim_{x\to x_0}\lim_{y\to y_0}f(x,y),\lim_{y\to y_0}\lim_{x\to x_0}f(x,y)\)与二重极限都存在,则三者相等.
推论:若\(\lim_{x\to x_0}\lim_{y\to y_0}f(x,y),\lim_{y\to y_0}\lim_{x\to x_0}f(x,y)\)存在但不相等,则二重极限不存在.
连续¶
定义:设\(f(P)=f(x,y)\)在点\(P_0(x_0,y_0)\)的某邻域\(U(P_0)\)内有定义,且\(\pmb{\lim_{(x,y)\to(x_0,y_0)}f(x,y)=f(x_0,y_0)}\),则称函数\(f(P)=f(x,y)\)在点\(P_0(x_0,y_0)\)处连续,记 $$ \Delta z=f(x,y)-f(x_0,y_0)=f(x_0+\Delta x_,y_0+\Delta y)-f(x_0.y_0) $$ 称为函数(值)的全增量,则连读定义可写为\(\lim_{\Delta x\to0,\Delta y\to0}\Delta z=0\).
记\(\Delta_xz=f(x,y_0)-f(x_0,y_0)=f(x_0+\Delta x,y_0)-f(x_0,y_0)\)称为函数(值)对\(x\)的偏增量.
记\(\Delta_yz=f(x_0,y)-f(x_0,y_0)=f(x_0,y_0+\Delta y)-f(x_0,y_0)\)称为函数(值)对\(y\)的偏增量.
若\(f(P)\)在点\(P(x_0,y_0)\)不连续,称\(P_0\)是\(f(x,y)\)的间断点,若\(f(x,y)\)在某区域\(G\)上每一点都连续,则称\(f(x,y)\)在区域\(G\)上连续. 若\(f(x,y)\)在闭域\(G\)上每一内点都连续,并在\(G\)的边界点\(P_0(x_0,y_0)\)处成立\(\pmb{\lim_{P\to P_0,P\in G}f(P)=f(P_0)}\),则称\(f(P)=f(x,y)\)在闭域\(G\)上连续. 闭域上连续的二元函数的图形称为连续曲面.
注:初等多元函数在他们的定义域内都是连续的.
定理:设\(f(P)=f(x,y)\)在平面上一个有界闭区域\(G\)上连续,则
(1)\(f(P)\)必在\(G\)上取到最大值,最小值及其中间的一切值.
(2)\(f(P)\)在\(G\)上一致连续,即\(\pmb{\forall\varepsilon>0,\exists\delta>0\ \ (\delta\le\delta_0)}\),当\(\pmb{0<\rho(P_1,P_2)<\delta}\)时,都有\(\pmb{|f(P_1)-f(P_2)|<\varepsilon}\).
计算二元(或多元)函数的极限常用方法:
(1)利用不等式,使用两边夹逼定理.
(2)变量替换化为已知极限,或化为一元函数极限,利用极坐标.
(3)利用初等函数的连续性,利用极限的四则运算性质.
(4)利用初等变形,特别是指数形式常可先求其对数的极限.
(5)若事先能看出极限值,可用\(\varepsilon-\delta\)方法进行证明.
证明二元(或多元)函数极限不存在:
(1)证明径向路径的极限与辐角(或斜率)有关.
(2)证明某个特殊路径的极限不存在.
(3)证明两个特殊路径的极限存在但不相等.
(4)若二元函数在该点某空心邻域里连续,而两个累次极限存在但不相等,则该点极限不存在.
2 偏导数与全微分¶
2.1 偏导数¶
偏导数的定义¶
定义:设函数\(z=f(x,y)\)在点\(P_0(x_0,y_0)\)的某邻域内有定义,若极限 $$ \lim_{\Delta x\to0}\frac{\Delta_xz}{\Delta x}=\lim_{\Delta x\to0}\frac{f(x_0+\Delta x,y_0)-f(x_0,y_0)}{\Delta x}=\lim_{x\to x_0}\frac{f(x,y_0)-f(x_0,y_0)}{x-x_0} $$ 存在,则称该极限值为函数\(z=f(x,y)\)在点\(P_0(x_0,y_0)\)关于\(x\)的偏导数,记为 $$ f'x(x_0,y_0) 或 \frac{\partial z}{\partial x}\bigg| 或 z'}x=x_0\y=y_0\end{split}x| $$ 否则称}} 或 \lim_{\Delta y\to0}\frac{f(x,y+\Delta y)-f(x,y)}{\Delta y\(z=f(x,y)\)在点\(P_0(x_0,y_0)\)处对\(x\)的偏导数不存在.
若对于某一区域\(G\)上的每一点\((x,y)\),极限\(\lim_{\Delta x\to0}\frac{\Delta_xz}{\Delta x}=\lim_{\Delta x\to0}\frac{f(x+\Delta x,y)-f(x,y)}{\Delta x}\)都存在,它是\(x,y\)的函数,称为函数\(z=f(x,y)\)在\(G\)上关于\(x\)的偏导函数,简称偏导数,记作 $$ f'_x(x,y)=\frac{\partial }{\partial x}f(x,y)=z'_x=\frac{\partial z}{\partial x} $$ 同理可以定义函数对\(y\)的偏导数 $$ f'_y(x,y)=\frac{\partial }{\partial y}f(x,y)=z'_y=\frac{\partial z}{\partial y} $$
偏导数的计算¶
求\(f'_x(x_0,y_0)\)有三种方法:
(1)按定义.
(2)求导函数\(\frac{d}{dx}f(x,y_0)\),然后把\(x=x_0\)代入.
(3)求偏导函数\(f'_x(x,y)\),然后把\(x=x_0,y=y_0\)代入.
例1:\(f(x,y)=x+(y-1)\arcsin{\sqrt{\frac{x}{y}}}\),求\(f'_x(\frac{1}{2},1),f'_y(\frac{1}{2},1)\).
解:
(1) $$ \begin{align} f'x(\frac12,1)&=\lim=1\ f'}\frac{f(x,1)-f(\frac12,1)}{x-\frac12y(\frac12,1)&=\lim=\frac\pi4 \end{align} $$ (2) $$ \begin{align} f'}\frac{f(\frac12,y)-f(\frac12,1)}{y-1}=\lim_{y\to1}\frac{\frac12+(y-1)\arcsin{\sqrt\frac{1}{2y}}-\frac12}{y-1}=\lim_{y\to1}\arcsin{\sqrt\frac1{2y}x(\frac12,1)&=\frac{d}{dx}f(x,1)\bigg|=x'|{x=\frac12}=1 \f'_y(\frac12,1)&=\frac{d}{dy}f(\frac12,y)\bigg|=\frac\pi4 \end{align} $$ (3) $$ \begin{align} f'_x(x,y)&=1+\frac12(y-1)\frac{1}{\sqrt{1-\frac xy}}\cdot\frac{1}{\sqrt{\frac xy}}\cdot\frac{1}{y}\Longrightarrow f'_x(\frac12,1)=1 \f'_y(x,y)&=\arcsin{\sqrt{\frac xy}}+\frac12(y-1)\frac{1}{\sqrt{1-\frac xy}}\cdot\frac{1}{\sqrt{\frac xy}}\cdot\frac{-x}{y^2}\Longrightarrow f'_y(\frac12,1)=\arcsin{\frac{\sqrt2}{2}}=\frac\pi4 \end{align} $$}=\bigg[\frac12+(y-1)\arcsin{\sqrt{\frac{1}{2y}}}\bigg]'\Bigg|_{y=1
偏导数的几何意义¶
偏导数\(f'_x(x_0,y_0)\)的几何意义是表示\(\begin{cases}z=f(x,y)\\y=y_0\end{cases}\) 在\(M_0\)处的切线对\(Ox\)轴的斜率.
偏导数\(f'_y(x_0,y_0)\)的几何意义是表示\(\begin{cases}z=f(x,y)\\x=x_0\end{cases}\) 在\(M_0\)处的切线对\(Oy\)轴的斜率.
例1:讨论函数\(f(x,y)=\begin{cases}\frac{2xy}{x^2+y^2},&x^2+y^2\ne0\\0,&x^2+y^2=0\end{cases}\) 在\((0,0)\)点的偏导数是否存在,函数\(f\)是否连续?
解:
按定义求得 $$ f'x(0,0)=0,f'_y(0,0)=0\Longrightarrow函数f在点(0,0)处的偏导数都存在 $$ 但是 $$ \lim\Longrightarrow函数f在(0,0)处不连续 $$ }}f(x,y)=\lim_{\substack{x\to0\y=kx}}\frac{2k}{1+k^2注:多元函数中连续与可导无必然联系.
高阶偏导数¶
与一元函数的高阶导数一样,可以定义多元函数的高阶偏导数.
若函数\(z=f(x,y)\)的偏导数\(\frac{\partial z}{\partial x},\frac{\partial z}{\partial y}\)存在,称为一阶偏导数,它们仍是\(x,y\)的函数,如果它们对\(x,y\)的偏导仍存在,得到 $$ \frac{\partial }{\partial x}(\frac{\partial z}{\partial x}),\frac{\partial }{\partial y}(\frac{\partial z}{\partial x}),\frac{\partial }{\partial x}(\frac{\partial z}{\partial y}),\frac{\partial }{\partial y}(\frac{\partial z}{\partial y}) $$ 称为\(z=f(x,y)\)的二阶偏导数,记作 $$ \begin{align} z_{xx}''&=\frac{\partial ^2z}{\partial x^2}=\frac{\partial }{\partial x}(\frac{\partial z}{\partial x})=f''{xx}(x,y)\ z)=f''}''&=\frac{\partial ^2z}{\partial y^2}=\frac{\partial }{\partial y}(\frac{\partial z}{\partial y{yy}(x,y)\ z)=f''}''&=\frac{\partial ^2z}{\partial x\partial y}=\frac{\partial }{\partial y}(\frac{\partial z}{\partial x{xy}(x,y)\ z(x,y)\ \end{align} $$ 其中}''&=\frac{\partial ^2z}{\partial y\partial x}=\frac{\partial }{\partial x}(\frac{\partial z}{\partial y})=f''_{yx\(f_{xy}''(x,y)\)和\(f_{yx}''(x,y)\)称为二阶混合偏导数. 二阶及二阶以上的偏导数,统称为高阶偏导数.
定理:若函数\(z=f(x,y)\)的二阶偏导(函数)\(f_{xy}''(x,y),f_{yx}''(x,y)\)都在\((x_0,y_0)\)处连续,则\(f_{xy}''(x_0,y_0)=f_{yx}''(x_0,y_0)\).
2.2 全微分¶
全微分的概念¶
定义:若二元函数\(z=f(x,y)\)在点\((x,y)\)处的全增量\(\Delta z=f(x+\Delta x,y+\Delta y)-f(x,y)\)可以表示为 $$ \Delta z=A\Delta x+B\Delta y+o(\rho) (\rho=\sqrt{\Delta x^2+\Delta y^2}\to0) $$ 其中,\(A,B\)与变量的增量\(\Delta x,\Delta y\)无关,而仅与\(x,y\)有关,则称函数\(z=f(x,y)\)在点\((x,y)\)处可微. 其中\(\Delta z\)的线性主部\(A\Delta x+B\Delta y\)称为函数\(z=f(x,y)\)在点\((x,y)\)处的全微分,记作\(dz\),即 $$ dz=A\Delta x+B\Delta y $$ 函数\(z=f(x,y)\)在点\(P(x_0,y_0)\)处可微的充要条件 $$ \lim_{\substack{\Delta x\to0\\Delta y\to0}}\frac{\Delta z-\big[f'_x(x,y)\Delta x+f'_y(x,y)\Delta y\big]}{\sqrt{\Delta x^2+\Delta y^2}}=0 $$
可微的必要条件、充分条件¶
定理1:若\(z=f(x,y)\)在点\((x,y)\)处可微,则\(z=f(x,y)\)在点\((x,y)\)处连续,反之不成立.
定理2:若\(z=f(x,y)\)在点\((x,y)\)处可微,则\(z=f(x,y)\)在点\((x,y)\)的两个偏导数\(f'_x(x,y),f'_y(x,y)\)都存在,且\(A=f'_x(x,y),B=f'_y(x,y)\).
例1:研究\(f(x,y)=\begin{cases}\frac{2xy}{x^2+y^2},&x^2+y^2\ne0\\0,&x^2+y^2=0\end{cases}\) 在原点的可微性.
验证多元函数不可微有下述方法:
(1)若\(f(x,y)\)在点\((x,y)\)处不连续,则\(f\)在点\((x,y)\)处不可微.
(2)若\(f(x,y)\)在点\((x,y)\)处至少有一个偏导数不存在,则\(f\)在点\((x,y)\)处不可微.
(3)若\(f(x,y)\)在点\((x,y)\)处两个偏导数都存在,但\(\lim_{\substack{\Delta x\to0\\\Delta y\to0}}\frac{\Delta z-dz}{\rho}\)极限不存在或极限存在但不为零,则\(f\)在点\((x,y)\)处不可微.
定理3:若函数\(z=f(x,y)\)的偏导数\(f_x'(x,y),f_y'(x,y)\)在点\((x_0,y_0)\)处连续,则函数\(z=f(x,y)\)在点\((x_0,y_0)\)处可微.
全微分四则运算与计算¶
设\(u,v\)都是多元函数,且具有连续的偏导数,则
(1) \(d(u\pm v)=du\pm dv\).
(2) \(d(uv)=vdu+udv\);特别地\(d(cu)=cdu\)(\(c\)是常数).
(3) \(d(\frac{u}{v})=\frac{vdu-udv}{v^2}(v\ne0)\).
下面对(2)做出证明
设\(u=u(x,y),v=v(x,y)\),有条件知,\(uv\)可微,且 $$ \begin{align} d(uv)&=\frac{\partial }{\partial x}(uv)dx+\frac{\partial }{\partial y}(uv)dy=(\frac{\partial u}{\partial x}v+u\frac{\partial v}{\partial x})dx+(\frac{\partial u}{\partial y}v+u\frac{\partial v}{\partial y})dy\ &=(\frac{\partial u}{\partial x}dx+\frac{\partial u}{\partial y}dy)v+(\frac{\partial v}{\partial x}dx+\frac{\partial v}{\partial y}dy)u=vdu+udv \end{align} $$
全微分在近似计算和误差估计中的应用¶
由全微分的定义,当\(|\Delta x|,|\Delta y|\)很小时,有\(\Delta z\approx dz=f_x'(x,y)\Delta x+f_y'(x,y)\Delta y\). 得 $$ f(x+\Delta x,y+\Delta y)-f(x,y)\approx f_x'(x,y)\Delta x+f_y'(x,y)\Delta y $$ 设函数\(z=f(x,y)\),若测得\(x\)的近似值为\(x_0\),\(y\)的近似值为\(y_0\),用测量的近似值\(x_0,y_0\)分别代替\(x,y\)来计算函数值\(z\),就会引起绝对误差 $$ |\Delta z|=|f(x,y)-f(x_0,y_0)|\approx|dz|=|f_x'(x_0,y_0)\Delta x+f_y'(x_0,y_0)\Delta y|\ \le|f_x'(x_0,y_0)||\Delta x|+|f_y'(x_0,y_0)||\Delta y|\le|f_x'(x_0,y_0)|\delta_1 +|f_y'(x_0,y_0)|\delta_2 $$ 因此 $$ \bigg|\frac{\Delta z}{z_0}\bigg|\approx\bigg|\frac{dz}{z_0}\bigg|\le\Bigg|\frac{f_x'(x_0,y_0)}{f(x_0,y_0)}\Bigg|\delta_1+\Bigg|\frac{f_y'(x_0,y_0)}{f(x_0,y_0)}\Bigg|\delta_2 $$
例1:计算\(1.007^{2.98}\).
解:设\(f(x,y)=x^y\Longrightarrow f_x'(x,y)=yx^{y-1},f_y'(x,y)=x^y\ln{x}\),有 $$ \begin{align} 1.007^{2.98}&=f(1.007,2.98)=f(1+0.007,3-0.02)\ &\approx f(1,3)+f_x'(1,3)\times0.007+f_y'(1,3)\times(-0.02)\ &=1+3\times0.007+1^3\times\ln{1}\times(-0.02)=1.021 \end{align} $$
计算原函数¶
设二元函数\(u(x,y)\)存在二阶连续偏导,且 $$ du(x,y)=P(x,y)dx+Q(x,y)dy $$ 则\(P(x,y)=\frac{\partial u}{\partial x},Q(x,y)=\frac{\partial u}{\partial y}\),得 $$ \frac{\partial P}{\partial y}=\frac{\partial ^2u}{\partial x\partial y}=\frac{\partial ^2u}{\partial y\partial x}=\frac{\partial Q}{\partial x} $$ 反之,若\(P(x,y),Q(x,y)\)在区域D上存在一阶连续偏导,且\(\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}\),则存在二元函数\(u(x,y)\)满足 $$ du(x,y)=P(x,y)dx+Q(x,y)dy $$
例1:当\(y>0\)时,\(u(x,y)\)可微,且 $$ du(x,y)=(\frac{y}{x2+y2}+xy2)dx+(-\frac{x}{x2+y2}+x2y+2y)dy $$ 求\(u(x,y)\).
解:\(du(x,y)\stackrel{\triangle}{=}P(x,y)dx+Q(x,y)dy\),则 $$ \begin{align} &P(x,y)=\frac{\partial u}{\partial x}=\frac{y}{x2+y2}+xy^2\ &Q(x,y)=\frac{\partial u}{\partial y}=-\frac{x}{x2+y2}+x^2y+2y\ &\Longrightarrow u(x,y)=\int(\frac{y}{x2+y2}+xy2)dx=\arctan{\frac{x}{y}}+\frac{1}{2}x2y^2+\psi(y)\ &\Longrightarrow\frac{\partial u}{\partial y}=\frac1{1+(\frac xy)^2}\cdot(-\frac x{y2})+x2y+\psi'(y)=-\frac{x}{x2+y2}+x^2y+2y\ &\Longrightarrow\psi'(y)=2y\Longrightarrow\psi(y)=y^2+C\ &\Longrightarrow u(x,y)=\arctan{\frac xy}+\frac12x2y2+y^2+C \end{align} $$
3 复合函数微分法¶
3.1 复合函数的偏导公式¶
若\(z=f(u,v)\)定义在\(uv\)平面\(\Omega_{uv}\)上,函数\(u=\varphi(x,y),v=\psi(x,y)\)定义在\(xy\)平面\(\Omega_{xy}\)上 $$ \big{(u,v):u=\varphi(x,y),v=\psi(x,y),(x,y\in\Omega_{xy})\big}\subset\Omega_{uv} $$ 于是\(z\)是以\(f\)为外函数,\(\varphi,\psi\)为内函数的复合函数\(z=f[\varphi(x,y),\psi(x,y)]\),\(x,y\)为自变量,\(u,v\)为中间变量.
定理:若函数\(u=\varphi(x,y),v=\psi(x,y)\)在点\((x,y)\)处偏导数都存在,\(z=f(u,v)\)在点\((u,v)=(\varphi(x,y),\psi(x,y))\)处可微,则复合函数\(z=f[\varphi(x,y),\psi(x,y)]\)在点\((x,y)\)处的偏导数存在,且 $$ \frac{\partial z}{\partial x}=\frac{\partial z}{\partial u}\cdot\frac{\partial u}{\partial x}+\frac{\partial z}{\partial v}\cdot\frac{\partial v}{\partial x}\ \frac{\partial z}{\partial y}=\frac{\partial z}{\partial u}\cdot\frac{\partial u}{\partial y}+\frac{\partial z}{\partial v}\cdot\frac{\partial v}{\partial y} $$ 注:设\(z=f(x,u,v),u=u(x),v=v(x)\),这里\(z\)是通过三个中间变量的\(x\)的一元复合函数,则 $$ \frac{dz}{dx}=\frac{\partial f}{\partial x}+\frac{\partial f}{\partial u}\cdot\frac{du}{dx}+\frac{\partial f}{\partial v}\cdot\frac{dv}{dx} (全导数) $$
例1:设函数\(W=F(x,y,z),z=f(x,y),y=\varphi(x)\),其中\(F,f\)具有连续的偏导数,\(\varphi\)可导,求\(\large{\frac{dW}{dx}}\).
解: $$ \frac{dW}{dx}=F_x'+F_y'\varphi'+F_z'\big[f_x'+f_y'\cdot\varphi'\big] $$ 注:此处\(W\)是\(x\)的一元复合函数,因此\(\large{\frac{dW}{dx}=\frac{\partial W}{\partial x}}\).
例2:设\(z=f(x,y,u),u=u(x,y)\),请问:\(\Large{\frac{\partial z}{\partial x}=\frac{\large\partial f}{\partial x}}\)?
解: $$ \frac{\partial z}{\partial x}=\frac{\partial f}{\partial x}\cdot\frac{dx}{dx}+\frac{\partial f}{\partial y}\cdot\frac{\partial y}{\partial x}+\frac{\partial f}{\partial u}\cdot\frac{\partial u}{\partial x}=\frac{\partial f}{\partial x}+\frac{\partial f}{\partial u}\cdot\frac{\partial u}{\partial x} $$ 注:\(\large{\frac{\partial z}{\partial x}}\)是复合函数,\(z=f[x,y,u(x,y)]\)对自变量\(x\)的偏导,而\(\large{\frac{\partial f}{\partial x}}\)是外函数,\(f(x,y,u)\)对中间变量\(x\)的偏导,这时\(y,u\)都看成常数.
3.2 复合函数的全微分¶
一阶微分不变性¶
若\(z=z(x,y)\),\(x,y\)是自变量,且\(z=z(x,y)\)可微,则其全微分为 $$ dz=\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}dy $$ 设\(z=z(x,y),x=x(s,t),y=y(s,t)\)都具有连续的偏导数,则复合函数\(z=z(x(s,t),y(s,t))\)可微,且 $$ dz=\frac{\partial z}{\partial s}ds+\frac{\partial z}{\partial t}dt $$ 证明: $$ \begin{cases} \large\frac{\partial z}{\partial s}=\frac{\partial z}{\partial x}\cdot\frac{\partial x}{\partial s}+\frac{\partial z}{\partial y}\cdot\frac{\partial y}{\partial s}\ \large\frac{\partial z}{\partial t}=\frac{\partial z}{\partial x}\cdot\frac{\partial x}{\partial t}+\frac{\partial z}{\partial y}\cdot\frac{\partial y}{\partial t} \end{cases} $$
上述表明虽然\(x,y\)不是自变量,但全微分的形式与当\(x,y\)是自变量时是一样的,这就是全微分的一阶微分形式不变性.
换句话说,若\(z=z(u,v)\),且\(dz=\varphi(u,v)du+\psi(u,v)dv\),则 $$ \frac{\partial z}{\partial u}=\varphi(u,v),\frac{\partial z}{\partial v}=\psi(u,v) $$ 注:
(1)一元、多元函数都不具有高阶微分不变性.
(2)当\(x,y\)是自变量时,\(dx,dy\)各自取值,
当\(x,y\)是中间变量时,它们的取值由\(s,t,ds,dt\)确定.
4 隐函数的偏导数¶
4.1 隐函数的偏导数¶
隐函数存在定理¶
定理1:若函数\(F(x,y,z)\)满足下列条件
(1)函数\(F\)在以\(P_0(x_0,y_0,z_0)\)为内点的某一区域\(D\subset R^3\)上连续.
(2)\(F(x_0,y_0,z_0)=0\).(初始条件)
(3)在\(D\)内存在连续的偏导数\(F'_x(x,y,z),F_y'(x,y,z),F_z'(x,y,z)\).
(4)\(F_z'(x_0,y_0,z_0)\ne0\).
则在点\(P_0\)的某邻域\(U(P_0)\)内唯一确定一个具有连续导数的函数\(z=f(x,y)\),且满足\(z_0=f(x_0,y_0),F(x,y,f(x,y))\equiv0\),且有\(\large\frac{\partial z}{\partial x}=-\frac{F_x'(x,y,z)}{F_z'(x,y,z)},\frac{\partial z}{\partial y}=-\frac{F_y'(x,y,z)}{F_z'(x,y,z)}\).
说明:对方程\(F(x,y,z(x,y))\equiv0\)两边关于\(x,y\)求偏导得 $$ F_x'\cdot1+F_z'\cdot\frac{\partial z}{\partial x}=0\Longrightarrow\frac{\partial z}{\partial x}=-\frac{F_x'}{F_z'}\ F_y'\cdot1+F_z'\cdot\frac{\partial z}{\partial y}=0\Longrightarrow\frac{\partial z}{\partial y}=-\frac{F_y'}{F_z'} $$ 注:\(\large\frac{\partial z}{\partial x}=-\frac{F_x'}{F_z'},\frac{\partial z}{\partial y}=-\frac{F_y'}{F_z'}\)仅适用于\(F(x,y,z)=0\),对于一般形式不适用,一般形式只能用复合函数求偏导或一阶微分不变性来求.
4.2 隐函数组偏导数¶
隐函数组存在定理¶
定理:设函数\(F(x,y,u,v),G(x,y,u,v)\)在点\(P_0(x_0,y_0,u_0,v_0)\)的某邻域内满足
(1)\(F(x_0,y_0,u_0,v_0)=G(x_0,y_0,u_0,v_0)=0\).(初始条件)
(2)\(F,G\)存在一阶连续偏导数\(F_x',F_y',F_u',F_v',G_x',G_y',G_u',G_v'\).
(3)在点\(P_0\)处行列式\(J\stackrel{\small\triangle}{=}\frac{\partial (F,G)}{\partial (u,v)}=\left|\begin{array}{c} F_u' &F_v'\\G_u' &G_v'\end{array}\right|\ne0\).(Jacobi行列式)
则方程组\(\begin{cases}F(x,y,u,v)=0\\G(x,y,u,v)=0\end{cases}\) 在点\(P_0(x_0,y_0,u_0,v_0)\)的某邻域内唯一确定一组函数组\(\begin{cases}u=u(x,y)\\v=v(x,y)\end{cases}\)满足
(1)\(u_0=u(x_0,y_0),v_0=v(x_0,y_0)\).
(2)\(F(x,y,u(x,y),v(x,y))=0,G(x,y,u(x,y),v(x,y))=0\).
(3)\(u=u(x,y),v=v(x,y)\)具有一阶连续偏导数: $$ \frac{\partial u}{\partial x}=-\frac{1}{J}\cdot\frac{\partial (F,G)}{\partial (x,v)}=-\frac{\left|\begin{array}{c}F_x' &F_v'\G_x' &G_v'\end{array}\right|}{\left|\begin{array}{c} F_u' &F_v'\G_u' &G_v'\end{array}\right|}, \frac{\partial v}{\partial x}=-\frac{1}{J}\cdot\frac{\partial (F,G)}{\partial (u,x)}=-\frac{\left|\begin{array}{c}F_u' &F_x'\G_u' &G_x'\end{array}\right|}{\left|\begin{array}{c} F_u' &F_v'\G_u' &G_v'\end{array}\right|}\ \frac{\partial u}{\partial y}=-\frac{1}{J}\cdot\frac{\partial (F,G)}{\partial (y,v)}=-\frac{\left|\begin{array}{c}F_y' &F_v'\G_y' &G_v'\end{array}\right|}{\left|\begin{array}{c} F_u' &F_v'\G_u' &G_v'\end{array}\right|}, \frac{\partial v}{\partial y}=-\frac{1}{J}\cdot\frac{\partial (F,G)}{\partial (u,y)}=-\frac{\left|\begin{array}{c}F_u' &F_y'\G_u' &G_y'\end{array}\right|}{\left|\begin{array}{c} F_u' &F_v'\G_u' &G_v'\end{array}\right|} $$
以下仅有关\(\large\frac{\partial u}{\partial x},\frac{\partial u}{\partial y},\frac{\partial v}{\partial x},\frac{\partial v}{\partial y}\)计算公式推导: $$ \begin{cases} F(x,y,u,v)=0\ &\Longrightarrow\ G(x,y,u,v)=0 \end{cases} \begin{cases} \begin{align} F_x'+F_u'\cdot\frac{\partial u}{\partial x}+F_v'\cdot\frac{\partial v}{\partial x}&=0\ G_x'+G_u'\cdot\frac{\partial u}{\partial x}+G_v'\cdot\frac{\partial v}{\partial x}&=0 \end{align} \end{cases} $$
解得\(\large\frac{\partial u}{\partial x},\frac{\partial v}{\partial x}\),同理可得\(\large\frac{\partial u}{\partial y},\frac{\partial v}{\partial y}\).
注:在实际计算时,一般用公式直接求.
例1:设\(u=f(x,y,z),\psi(x^2,e^y,z)=0,y=\sin{x}\),其中\(f,\psi\)具有连续的偏导数且\(\psi_3'\ne0\),求\(\large\frac{du}{dx}\).
解:
法一
\[ \begin{align} u=f(x,y,z)&\Longrightarrow \frac{du}{dx}=f_x'+f_y'\frac{dy}{dx}+f_z'\frac{dz}{dx}&(1)\\ y=\sin x&\Longrightarrow \frac{dy}{dx}=\cos x &(2)\\ \psi(x^2,e^y,z)=0&\Longrightarrow\psi_1'\cdot2x+\psi_2'\cdot e^y\cdot\frac{dy}{dx}+\psi_3'\cdot\frac{dz}{dx}=0\\ &\Longrightarrow\frac{dz}{dx}=-\frac1{\psi_3'}(2x\psi_1'+e^y\cos x\psi_2')&(3) \end{align} \]把\((2),(3)\)代入\((1)\): $$ \frac{du}{dx}=f_x'+f_y'\cos x-f_z'\frac1{\psi_3'}(2x\psi_1'+e^y\cos x\psi_2') $$
法二 一阶微分不变性+微分四则运算
\[ \begin{align} u=f(x,y,z)&\Longrightarrow du=f_x'dx+f_y'dy+f_z'dz&(4)\\ \psi(x^2,e^y,z)=0&\Longrightarrow \psi_1'd(x^2)+\psi_2'd(e^y)+\psi_3'dz=0&(5)\\ &\Longrightarrow dz=-\frac1{\psi_3'}(2x\psi_1'+e^y\cos x\psi_2')dx&(6) \end{align} \]把\((5),(6)\)代入\((4)\): $$ du=[f_x'+f_y'\cos x-f_z'\frac1{\psi_3'}(2x\psi_1'+e^y\cos x\psi_2')]dx\ \Longrightarrow\frac{du}{dx}=f_x'+f_y'\cos x-f_z'\frac1{\psi_3'}(2x\psi_1'+e^y\cos x\psi_2') $$
4.3 反函数组的偏导数¶
隐函数组存在定理\(\large\Longrightarrow\frac{\partial (F,G)}{\partial (u,v)}\ne0\Longrightarrow\frac{\partial (x,y)}{\partial (u,v)}\ne0\Longrightarrow\)反函数组的存在性. $$ {\begin{cases} x=x(u,v)\ y=y(u,v) \end{cases} \iff \begin{cases} x-x(u,v)\equiv0\ y-y(u,v)\equiv0 \end{cases} \iff \begin{cases} x-x[u(x,y),v(x,y)]\equiv0\ y-y[u(x,y),v(x,y)]\equiv0 \end{cases}}\ {\Longrightarrow \begin{cases} 1-x_u'u_x'-x_v'v_x'=0\ 0-y_u'u_x'-y_v'v_x'=0\ 0-x_u'u_y'-x_v'v_y'=0\ 1-y_u'u_y'-y_v'v_y'=0 \end{cases} \Longrightarrow \begin{cases} x_u'u_x'+x_v'v_x'=1\ y_u'u_x'+y_v'v_x'=0\ x_u'u_y'+x_v'v_y'=0\ y_u'u_y'+y_v'v_y'=1 \end{cases}}\ \frac{\partial (u,v)}{\partial (x,y)}\cdot\frac{\partial (x,y)}{\partial (u,v)}=\left|\begin{array}{c}u_x' &v_x'\u_y' &v_y'\end{array}\right|\cdot\left|\begin{array}{c}x_u' &x_y'\y_u' &y_v'\end{array}\right|=\left|\begin{array}{c}x_u'u_x'+x_v'v_x' &y_u'u_x'+y_v'v_x'\x_u'u_y'+x_v'v_y' &y_u'u_y'+y_v'v_y'\end{array}\right|=\left|\begin{array}{c}1 &0\0 &1\end{array}\right|=1 $$ 注:这结果与一元函数的反函数的导数公式\({\large\frac{dy}{dx}\cdot\frac{dx}{dy}}=1\)一致.
5 场的方向导数和梯度¶
5.1 场的概念¶
场在数学上是指一个向量到另一个向量或数的映射.
5.2 场的方向导数¶
定义:设三元函数\(u\)在\(P_0(x_0,y_0,z_0)\)的某邻域\(u(P_0)\subset R^3\)内有定义,\(l\)为从点\(P_0\)出发的射线,\(P(x,y,z)\)为\(l\)上且含于\(u(P_0)\)内的任一点,以\(\rho\)表示\(P\)与\(P_0\)两点的距离,若极限 $$ \lim_{\rho\to0}\frac{u(P)-u(P_0)}{\rho}=\lim_{\rho\to0}\frac{\Delta_{\vec i}u}{\rho} $$ 存在,则称此极限为函数\(u\)在点\(P_0\)沿方向\(\vec l\)的方向导数,记作\({\large\frac{\partial u}{\partial \vec l}}\Big|_{P_0}\).
定理:若函数\(u\)在点\(P_0(x_0,y_0,z_0)\)处可微,则\(u\)在点\(P_0\)处沿任一方向\(\vec l\)的方向导数都存在,且 $$ \frac{\partial u}{\partial l}\Bigg|{P_0}=\frac{\partial u}{\partial x}\Bigg|\Bigg|}\cos\alpha+\frac{\partial u}{\partial y{P_0}\cos\beta+\frac{\partial u}{\partial z}\Bigg|\cos\gamma $$ 其中,\(\vec l=(\cos\alpha,\cos\beta,\cos\gamma)\).
例1:讨论\(f(x,y)=\sqrt{x^2+y^2}\)在\((0,0)\)处的任何方向\(\vec l\)的方向导数以及两个偏导数的存在性.
解:
\[ \frac{\partial f}{\partial \vec l}\Bigg|_{(0,0)}=\lim_{\rho\to0}\frac{f(\Delta x,\Delta y)-f(0,0)}\rho=\lim_{\rho\to0}\frac{\sqrt{(\Delta x)^2+(\Delta y)^2}}{\sqrt{(\Delta x)^2+(\Delta y)^2}}=1 \]方向导数存在. $$ \frac{\partial f}{\partial x}\Bigg|{(0,0)}=\lim\ \frac{\partial f}{\partial y}\Bigg|}\frac{f(\Delta x,0)-f(0,0)}{\Delta x}=\lim_{\Delta x\to 0}\frac{\sqrt{(\Delta x)^2}}{\Delta x{(0,0)}=\lim $$ 偏导数}\frac{f(0,\Delta y)-f(0,0)}{\Delta y}=\lim_{\Delta y\to 0}\frac{\sqrt{(\Delta y)^2}}{\Delta y不存在.
例2:讨论\(f(x,y)=\begin{cases}\large\frac{xy}{x^2+y^2},&x^2+y^2\ne0\\0,&x^2+y^2=0\end{cases}\) 在\((0,0)\)处的方向导数以及两个偏导数的存在性.
解:
\[ \frac{\partial f}{\partial \vec l}\Bigg|_{(0,0)}=\lim_{\rho\to0}\frac{f(\Delta x,\Delta y)-f(0,0)}\rho=\lim_{\rho\to0}\frac{\large\frac{\Delta x\Delta y}{\rho^2}}\rho=\lim_{\rho\to0}\frac{\Delta x}\rho\cdot\frac{\Delta y}\rho\cdot\frac1\rho \]方向导数不存在. $$ \frac{\partial f}{\partial x}\Bigg|{(0,0)}=\lim=0\ \frac{\partial f}{\partial y}\Bigg|}\frac{f(\Delta x,0)-f(0,0)}{\Delta x}=\lim_{\Delta x\to 0}\frac{0}{\Delta x{(0,0)}=\lim=0 $$ 偏导数}\frac{f(0,\Delta y)-f(0,0)}{\Delta y}=\lim_{\Delta y\to 0}\frac{0}{\Delta y存在.
结论:偏导数存在\(\not\Leftrightarrow\)方向导数存在.
5.3 梯度¶
定理:若函数\(u\)在点\(P(x,y,z)\)处可微,则\(u\)在点\(P\)处沿任一方向\(\vec l=(\cos\alpha,\cos\beta,\cos\gamma)\)的方向导数都存在,且 $$ \frac{\partial u}{\partial \vec l}=\frac{\partial u}{\partial x}\cdot\cos\alpha+\frac{\partial u}{\partial y}\cdot\cos\beta+\frac{\partial u}{\partial z}\cdot\cos\gamma=(\frac{\partial u}{\partial x},\frac{\partial u}{\partial y},\frac{\partial u}{\partial z})\cdot(\cos\alpha,\cos\beta,\cos\gamma) $$
定义:矢量\({\large\frac{\partial u}{\partial x}}\vec i+{\large\frac{\partial u}{\partial y}}\vec j+{\large\frac{\partial u}{\partial z}}\vec k=({\large\frac{\partial u}{\partial x},\frac{\partial u}{\partial y},\frac{\partial u}{\partial z}})\)称为函数\(u(P)\)在点\(P\)处的梯度,记为\(\pmb{grad}\ u\),即 $$ \pmb {grad} u\stackrel{\small\triangle}{=}\frac{\partial u}{\partial x}\vec i+\frac{\partial u}{\partial y}\vec j+\frac{\partial u}{\partial z}\vec k=(\frac{\partial u}{\partial x},\frac{\partial u}{\partial y},\frac{\partial u}{\partial z}) $$ 注:梯度这个矢量实质上是和坐标系的选择无关的.
梯度的运算法则:设\(u,v\)可微,\(\alpha,\beta\)为常数,则
(1)\(\pmb{grad}\ (\alpha u+\beta v)=\alpha\pmb{grad}\ u+\beta\pmb{grad}\ v\).
(2)\(\pmb{grad}\ (uv)=v\pmb{grad}\ u+u\pmb{grad}\ v\).
(3)\(\pmb{grad}\ f(u)=f'(u)\pmb{grad}\ u\).
6 多元函数的极值及应用¶
6.1 多元函数的泰勒公式¶
定理(泰勒定理):若函数\(f\)在点\(P_0(x_0,y_0)\)某邻域\(U(P_0)\)内有直到\(\pmb{n+1}\)阶的连续偏导数,则对\(U(P_0)\)内任一点\((x_0+h,y_0+k)\),存在相应\(\theta\in(0,1)\),使得 $$ \begin{align} &f(x_0+h,y_0+k)=f(x_0,y_0)+\Big(h\frac{\partial }{\partial x}+k\frac\partial {\partial y}\Big)f(x_0,y_0)\ &+\frac1{2!}\Big(h\frac{\partial }{\partial x}+k\frac\partial {\partial y}\Big)^2f(x_0,y_0)+···+\frac1{n!}\Big(h\frac{\partial }{\partial x}+k\frac\partial {\partial y}\Big)^nf(x_0,y_0)\ &+\frac1{(n+1)!}\Big(h\frac{\partial }{\partial x}+k\frac\partial {\partial y}\Big)^{n+1}f(x_0+\theta h,y_0+\theta k) \end{align} $$ 上式称为二元函数\(f\)在点\(P_0\)的\(\pmb n\)阶泰勒公式.
注:
(1)拉格朗日余项 $$ R_n\stackrel{\small\triangle}{=}\frac1{(n+1)!}\Big(h\frac{\partial }{\partial x}+k\frac\partial {\partial y}\Big)^{n+1}f(x_0+\theta h,y_0+\theta k),(0<\theta<1) $$ \(R_n\)称为泰勒公式的拉格朗日余项.
(2)当\((x_0,y_0)=(0,0)\)时,则称为二元函数的麦克劳林公式.
(3)二元函数的中值公式 $$ \begin{align} f(x_0+h,y_0+k)-f(x_0,y_0)&=\Big(h\frac{\partial }{\partial x}+k\frac\partial {\partial y}\Big)f(x_0+\theta h,y_0+\theta k)\ &=f_x'(x_0+\theta h,y_0+\theta k)h+f_y'(x_0+\theta h,y_0+\theta k)k \end{align} $$ 这就是二元函数的拉格朗日中值公式.
(4)推论:设\(f(x,y)\)在区域\(G\)上具有连续的一阶偏导.
(i)若\(f_x'(x,y)\equiv0,(x,y)\in G\),则\(f(x,y)\)在\(G\)上仅是\(\pmb y\)的函数.
(ii)若\(f_y'(x,y)\equiv0,(x,y)\in G\),则\(f(x,y)\)在\(G\)上仅是\(\pmb x\)的函数.
(iii)若\(f_x'(x,y)\equiv0,f_y'(x,y)\equiv0,(x,y)\in G\),则\(f(x,y)\)在\(G\)上是常值函数.
二元函数的中值定理:设函数\(f(x,y)\)在凸开域\(D\)上连续,在\(D\)的所有内点均可微,则对\(D\)内任意两点\(P(x_0,y_0),Q(x_0+h,y_0+h),\exists \theta\in(0,1)\),使得 $$ f(x_0+h,y_0+k)-f(x_0,y_0)=f_x'(x_0+\theta h,y_0+\theta k)h+f_y'(x_0+\theta h,y_0+\theta k)k $$
例1:计算\(\lim_{\substack{x\to0\\y\to0}}\frac{\ln{x^2+e^{y^2}}}{x^2+y^2}\).
解:
\[ f(x,y)\stackrel{\small\triangle}{=}\ln(x^2+e^{y^2})\Longrightarrow \begin{cases} f_x'(x,y)=\large\frac{2x}{x^2+e^{y^2}}\\ f_y'(x,y)=\large\frac{2ye^{y^2}}{x^2+e^{y^2}} \end{cases}\\ \Longrightarrow \begin{cases} f_{xx}''(x,y)=\large\frac{2e^{y^2}-2x^2}{(x^2+e^{y^2})^2}\\ f_{xy}''(x,y)=-\large\frac{4xye^{y^2}}{(x^2+e^{y^2})^2}\\ f_{yy}''(x,y)=\large\frac{(2+4y^2)e^{y^2}(x^2+e^{y^2})-4y^2e^{y^2}}{(x^2+e^{y^2})^2} \end{cases} \Longrightarrow \begin{cases} f_x'(0,0)=f_y'(0,0)=0\\ f_{xy}''(0,0)=0\\ f_{xx}''(0,0)=f_{yy}''(0,0)=2 \end{cases} \]由泰勒公式得 $$ f(x,y)=(x2+y2)+o(\rho2) (\rho\stackrel{\small\triangle}{=}\sqrt{x2+y2}\to0)\Longrightarrow\lim_{\substack{x\to0\y\to0}}\frac{\ln(x2+e{y2})}{x2+y2}=1 $$
6.2 多元函数的极值¶
多元函数的极值概念¶
极值定义:设函数\(z=f(x,y)\)在点\(P_0(x_0,y_0)\)的某邻域\(U(P_0)\)内有定义,\(\forall P(x,y)\in U(P_0)\),都有\(f(P)\le f(P_0)\)或\(f(P)\ge f(P_0)\),则称函数\(f\)在点\(P_0\)取到极大(极小)值,点\(P_0\)称为\(f\)的极大(极小)值点.
极值的必要条件:若函数\(f\)在点\(P_0(x_0,y_0)\)存在偏导数且在\(P_0\)取极值,则有 $$ f_x'(x_0,y_0)=0,f_y'(x_0,y_0)=0 $$ 反之,若函数\(f\)在点\(P_0\)满足上式,则称点\(P_0\)为\(f\)的稳定点或驻点.
注:
(1)上述定理指出,若\(f\)存在偏导数,则其极值点必是稳定点.
(2)稳定点不一定是极值点.
(3)函数\(f(x,y)\)在点\((x_0,y_0)\)取极值,\(f(x,y)\)的偏导数只有两种情形:
(i)\(f_x'(x_0,y_0),f_y'(x_0,y_0)\)都存在,则\(f_x'(x_0,y_0)=0,f_y'(x_0,y_0)=0\),即点\(P_0(x_0,y_0)\)为稳定点.
(ii)\(f_x'(x_0,y_0),f_y'(x_0,y_0)\)至少有一个不存在.
极值的充分条件:设函数\(z=f(x,y)\)在点\((x_0,y_0)\)的某邻域\(U(P_0)\)连续,且有一阶与二阶连续偏导数,若\(f_x'(x_0,y_0)=0,f_y'(x_0,y_0)=0\),设\(A=f_{xx}''(x_0,y_0),B=f_{xy}''(x_0,y_0),C=f_{yy}''(x_0,y_0)\),则
(1)当\(B^2-AC<0\)时,\(f(x_0,y_0)\)一定为极值,并且
当\(A(或C)>0\)时,\(f(x_0,y_0)\)为极小值;
当\(A(或C)<0\)时,\(f(x_0,y_0)\)为极大值.
(2)当\(B^2-AC>0\)时,\(f(x_0,y_0)\)不是极值.
(3)当\(B^2-AC=0\)时,无法判断,要进一步研究.
多元函数的最大值、最小值¶
\(f(P)\)在有界闭区域\(G\)上连续\(\Longrightarrow f(P)\)在\(G\)上一定能取到最大(小)值.
\(\iff \exists P_1,P_2\in G,f(P_1)=m,f(P_2)=M,\forall P\in G\),有\(m\le f(P)\le M\).
最值在
边界点取到(边界函数值最大值与最小值点).
内部取到:一定是极值点(稳定点或偏导数不存在点).
条件极值¶
条件极值的极值点的搜索范围受到条件的限制.
拉格朗日乘数法(拉格朗日乘子法):
条件极值问题:\(\max(\min)u=f(x,y,z)&(目标函数)\\s.t.\ G(x,y,z)=0&(约束函数)\).
设\(f\)和\(G\)具有连续的偏导数且\({\large\frac{\partial G}{\partial z}}\ne0\),隐函数存在定理得方程\(G(x,y,z)=0\)确定隐函数\(z=z(x,y)\).
将\(z=z(x,y)\)代入目标函数\(f\) $$ \begin{align} &条件极值问题\ u=&f(x,y,z)极值\ s.t. & G(x,y,z)=0 \end{align} \Longrightarrow \begin{aligned} &无条件极值问题\ u=&f(x,y,z(x,y))极值 \end{aligned} $$ 设\((x_0,y_0)\)为\(u=f(x,y,z(x,y))\)的极值点\((z_0=z(x_0,y_0))\),满足二元函数的极值必要条件 $$ \begin{cases} {\large\frac{\partial u}{\partial x}}=0\ {\large\frac{\partial u}{\partial y}}=0 \end{cases} \Longrightarrow \begin{cases} {\large\frac{\partial u}{\partial x}}={\large f_x'+f_z'\frac{\partial z}{\partial x}}={\large f_x'-\frac{G_x'}{G_z'}f_z'}=0\ {\large\frac{\partial u}{\partial y}}={\large f_y'+f_z'\frac{\partial z}{\partial y}}={\large f_y'-\frac{G_y'}{G_z'}f_z'}=0 \end{cases} $$ \(\iff\)所求问题的解\((x_0,y_0,z_0)\)必须满足关系式 $$ \frac{f_x'(x_0,y_0,z_0)}{G_x'(x_0,y_0,z_0)}=\frac{f_y'(x_0,y_0,z_0)}{G_y'(x_0,y_0,z_0)}=\frac{f_z'(x_0,y_0,z_0)}{G_z'(x_0,y_0,z_0)}\stackrel{\small\triangle}{=}-\lambda $$ 综上所述,\((x_0,y_0,z_0)\)是以下方程组的解 $$ \begin{cases} f_x'(x,y,z)+\lambda G_x'(x,y,z)=0\ f_y'(x,y,z)+\lambda G_y'(x,y,z)=0\ f_z'(x,y,z)+\lambda G_z'(x,y,z)=0\ G(x,y,z)=0 \end{cases} $$ 若函数\(L(x,y,z,\lambda)\stackrel{\small\triangle}{=}f(x,y,z)+\lambda G(x,y,z)\)存在极值点\((x^*,y^*,z^*,\lambda^*)\),则满足极值的必要条件 $$ \begin{cases} L_x'(x,y,z,\lambda)=f_x'(x,y,z)+\lambda G_x'(x,y,z)=0\ L_y'(x,y,z,\lambda)=f_y'(x,y,z)+\lambda G_y'(x,y,z)=0\ L_z'(x,y,z,\lambda)=f_z'(x,y,z)+\lambda G_z'(x,y,z)=0\ L_\lambda'(x,y,z,\lambda)=G(x,y,z)=0 \end{cases} $$ \(\Longrightarrow(x^*,y^*,z^*)=(x_0,y_0,z_0)\).
注:
(1)最后一定要根据实际情况验证怀疑点是否是极值点(最值点).
(2)上述函数\(L(x,y,z,\lambda)\)称为拉格朗日函数.
推广到多变量与多约束条件的情形:一般条件极值问题 $$ \max(\min) y=f(x_1,x_2,···,x_n)\ s.t. \psi_k(x_1,x_2,···,x_n)=0,k=1,2,···,m(m<n) $$ 此时拉格朗日函数为\(L(x_1,x_2,···,x_n,\lambda_1,\lambda_2,···,\lambda_m)=f(x_1,x_2,···,x_n)+\Sigma_{k=1}^{m}\lambda_k\psi_k(x_1,x_2,···,x_n)\)
其中\(\lambda_1,\lambda_2,···\lambda_m\)为拉格朗日乘数.
7 偏导数在几何上的应用¶
7.1 矢量函数的微分法¶
矢量函数的概念¶
如果把动点\(P\)与原点\(O\)连结起来就得到以原点为起点,动点为终点的矢量:\(\vec r=\vec {OP}\)(矢径),其坐标为\((x(t),y(t),z(t))\),则动点的运动方程为 $$ \vec r(t)=x(t)\vec i+y(t)\vec j+z(t)\vec k $$ 当\(t\)变动时,\(\vec r\)的模与方向一般都随之变动,即对每一个\(t\in[\alpha,\beta]\),按上式都有唯一的矢量\(\vec r\)与之对应,这个对应规律称为矢值函数,记为\(\vec r=\vec r(t)\),其动点\(P(x,y,z)\)画出的曲线,叫做矢值函数\(\vec r=\vec r(t)\)的矢端曲线,一般情况下\(t\)并不一定代表时间,而是在某一变化范围内的取值.
矢值函数的导数¶
设矢值函数\(\vec r=\vec r(t)=x(t)\vec i+y(t)\vec j+z(t)\vec k,\Delta\vec r=\vec r(t+\Delta t)-\vec r(t)=\Delta x\vec i+\Delta y\vec j+\Delta z\vec k\).
定义:若极限\(\lim_{\substack{\\\Delta t\to0}}\frac{\Delta x}{\Delta t},\lim_{\substack{\\\Delta t\to0}}\frac{\Delta y}{\Delta t},\lim_{\substack{\\\Delta t\to0}}\frac{\Delta z}{\Delta t}\)都存在,则极限 $$ \lim_{\Delta t\to0}\frac{\Delta\vec r}{\Delta t}=x'(t)\vec i+y'(t)\vec j+z'(t)\vec k $$ 称为矢值函数\(\vec r=\vec r(t)\)在\(t\)处的导数(称矢量导数),记作\(\vec r'(t)\)或\(\large\frac{d\vec r}{dt}\).
几何意义:\(\large\frac{d\vec r}{dt}\)为切线的方向向量\(\iff\)曲线在\(P\)点的切矢量.
7.2 空间曲线的切线与法平面¶
参数式方程的空间曲线的切线与法平面¶
设空间曲线的参数方程为\(\begin{cases}x=x(t)\\y=y(t)\\z=z(t)\end{cases}\) 且\(x'(t),y'(t),z'(t)\)连续不同时为零,曲线上一点\(P_0(x_0,y_0,z_0)\ \ (t=t_0)\)处切矢量\({\large\frac{d\vec r}{dt}}\bigg|_{t=t_0}=(x'(t_0),y'(t_0),z'(t_0))\),则曲线上点\(P_0(x_0,y_0,z_0)\)处的切线方程为 $$ \frac{x-x_0}{x'(t_0)}=\frac{y-y_0}{y'(t_0)}=\frac{z-z_0}{z'(t_0)} $$ 过点\(P_0\)且与该直线垂直的平面称为曲线在\(P_0\)点的法平面,则法平面方程为 $$ x'(t_0)(x-x_0)+y'(t_0)(y-y_0)+z'(t_0)(z-z_0)=0 $$
一般式方程的空间曲线的切线与法平面¶
曲线\(\Gamma\):\(\begin{cases}F(x,y,z)=0\\G(x,y,z)=0\end{cases}\) 在曲线上点\(P_0(x_0,y_0,z_0)\)处的切线方程.
曲线\(\Gamma\)在曲面\(S_1:F(x,y,z)=0\)上且经过曲面\(S_1\)上的点\(P_0\)
\(\Longrightarrow\)曲线\(\Gamma\)在\(P_0\)的切线必在曲面\(S_1\)在点\(P_0\)的切平面上
同理,曲线\(\Gamma\)在\(P_0\)的切线必在曲面\(S_2:G(x,y,z)=0\)在点\(P_0\)的切平面上.
故曲线\(\Gamma\)在点\(P_0\)处的切线方程为 $$ \begin{cases} F_x'(P_0)(x-x_0)+F_y'(P_0)(y-y_0)+F_z'(P_0)(z-z_0)=0\ G_x'(P_0)(x-x_0)+G_y'(P_0)(y-y_0)+G_z'(P_0)(z-z_0)=0 \end{cases} $$
7.3 空间曲面的切平面与法线¶
隐式方程的空间曲面的切平面与法向量¶
设已知曲面\(S:F(x,y,z)=0\),\(M_0(x_0,y_0,z_0)\)是曲面\(S\)上一点,过点\(M_0\)在曲面\(S\)上可以作无数条曲线,这些曲线在该点分别都有切线,则这无数条切线都在同一个平面上(切平面).
设过点\(M_0\)在曲面\(S\)上任作一曲线\(\Gamma:\begin{cases}x=x(t)\\y=y(t)\\z=z(t)\end{cases}\),且\(\begin{cases}x_0=x(t_0)\\y_0=y(t_0)\\z_0=z(t_0)\end{cases}\).
曲线\(\Gamma\)在曲面\(S\)上\(\Longrightarrow F\big(x(t),y(t),z(t)\big)\equiv0\),对\(t\)求导: $$ F_x'\big(x(t),y(t),z(t)\big)x'(t)+F_y'\big(x(t),y(t),z(t)\big)y'(t)+F_z'\big(x(t),y(t),z(t))z'(t\big) $$ 把\(t=t_0\)代入上式得 $$ F_x'(x_0,y_0,z_0)x'(t_0)+F_y'(x_0,y_0,z_0)y'(t_0)+F_z'(x_0,y_0,z_0)z'(t_0)=0\ \iff\big(F_x'(x_0,y_0,z_0),F_y'(x_0,y_0,z_0),F_z'(x_0,y_0,z_0)\big)\big(x'(t_0),y'(t_0),z'(t_0)\big)=0 $$ \(\Gamma\)在点\(M_0\)处的切矢量\(\vec v=\big(x'(t_0),y'(t_0),z'(t_0)\big)\),矢量\(\vec n\stackrel{\small\triangle}{=}\big(F_x'(x_0,y_0,z_0),F_y'(x_0,y_0,z_0),F_z'(x_0,y_0,z_0)\big)\)
\(\Longrightarrow\vec v\cdot\vec n=0\).
\(\Longrightarrow\)曲面上过点\(M_0\)的任意一条曲线的切线都与矢量\(\vec n\)垂直.
\(\Longrightarrow\)矢量\(\vec n\)为曲面在点\(M_0\)的法线矢量(法矢量),切平面方程为 $$ F_x'(x_0,y_0,z_0)(x-x_0)+F_y'(x_0,y_0,z_0)(y-y_0)+F_z'(x_0,y_0,z_0)(z-z_0)=0 $$ 曲面在点\(M_0\)处存在切平面的充分条件:\(F_x',F_y',F_z'\)在\(M_0\)点具有连续的偏导数且\(F_x'(x_0,y_0,z_0),F_y'(x_0,y_0,z_0),F_z'(x_0,y_0,z_0)\)不同时为零.
显式方程的空间曲面的切平面与法向量¶
若曲面方程\(z=f(x,y)\),则可将显式方程化为隐式方程求解.
参数式方程的空间曲面的切平面与法向量¶
设曲面参数方程\(\begin{cases}x=x(u,v)\\y=y(u,v)\\z=z(u,v)\end{cases}\) 在\((u_0,v_0)\)对应曲面上点\(P_0(x_0,y_0,z_0)\),则由\(\begin{cases}x=x(u,v)\\y=y(u,v)\end{cases}\) 可确定反函数组\(\begin{cases}u=u(x,y)\\v=v(x,y)\end{cases}\) 代入\(z=z(u,v)\Longrightarrow z\)是\(x,y\)的函数.
由复合函数求导法则得 $$ {\large \begin{cases} \frac{\partial z}{\partial u} = \frac{\partial z}{\partial x}\frac{\partial x}{\partial u} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial u} \ \frac{\partial z}{\partial v} = \frac{\partial z}{\partial x}\frac{\partial x}{\partial v} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial v} \end{cases}} \Longrightarrow \frac{\partial z}{\partial x} = {\large\frac{\frac{\partial (z,y)}{\partial (u,v)}}{\frac{\partial (x,y)}{\partial (u,v)}}}, \frac{\partial z}{\partial y} = {\large\frac{\frac{\partial (x,z)}{\partial (u,v)}}{\frac{\partial (x,y)}{\partial (u,v)}}} $$ 令\(F(x,y,z)=z[u(x,y),v(x,y)]-z\),则可转化为隐式方程求解.