From cec41a289dae04500f18b05d04a4d7bfc4e6a493 Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Mon, 20 Sep 2021 23:22:37 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0=E5=BE=AE=E5=88=86?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- ...ial-calculus-of-multivariate-functions.pdf | Bin 52778 -> 72190 bytes ...ial-calculus-of-multivariate-functions.tex | 49 ++++++++++++++++++ .../differential-equation.pdf | Bin 95319 -> 95572 bytes .../differential-equation.tex | 6 ++- .../differential-equation.pdf | Bin 125449 -> 125552 bytes .../differential-equation.tex | 2 +- 6 files changed, 54 insertions(+), 3 deletions(-) diff --git a/advanced-math/exercise/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.pdf b/advanced-math/exercise/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.pdf index 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b/advanced-math/exercise/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.tex @@ -41,6 +41,55 @@ \newpage \pagestyle{plain} \setcounter{page}{1} + +\section{基本概念} + +\subsection{复合函数} + +函数以复合函数形式$f(g(x,y))$出现,函数的变量是一个整体。 + +\subsubsection{链式法则} + +若是给出相应的不等式可以通过链式法则求出对应的表达式。 + +\textbf{例题:}设$u=u(\sqrt{x^2+y^2})$($r=\sqrt{x^2+y^2}>0$)有二阶连续的偏导数,且满足$\dfrac{\partial^2u}{\partial x^2}+\dfrac{\partial^2u}{\partial y^2}-\dfrac{1}{x}\dfrac{\partial u}{\partial x}+u=x^2+y^2$,则求$u(\sqrt{x^2+y^2})$。 + +解:这个函数是复合函数$u=u(r)$和$r=\sqrt{x^2+y^2}$而成。根据复合函数求导法则: + +$\dfrac{\partial u}{\partial x}=\dfrac{\textrm{d}u}{\textrm{d}r}\dfrac{\partial r}{\partial x}=\dfrac{\textrm{d}u}{\textrm{d}r}\dfrac{x}{\sqrt{x^2+y^2}}=\dfrac{\textrm{d}u}{\textrm{d}r}\dfrac{x}{r}$。 + +$\dfrac{\partial^2u}{\partial x^2}=\dfrac{\partial}{\partial x}\left(\dfrac{\partial u}{\partial x}\right)=\dfrac{\partial}{\partial x}\left(\dfrac{\textrm{d}u}{\textrm{d}r}\dfrac{x}{r}\right)=\dfrac{x}{r}\cdot\dfrac{\partial}{\partial x}\left(\dfrac{\textrm{d}u}{\textrm{d}r}\right)+\dfrac{\textrm{d}u}{\textrm{d}r}\cdot\dfrac{\partial}{\partial x}\left(\dfrac{x}{r}\right)=$。 + +\subsubsection{特殊值反代} + +若是给出的不等式后还给出对应的特殊值,可以直接代入然后反代求出函数,而不用链式法则。 + +\textbf{例题:}设$z=e^x+y^2+f(x+y)$,且当$y=0$时,$z=x^3$,则求$\dfrac{\partial z}{\partial x}$。 + +解:已知$y=0$时,$z=e^x+f(x)=x^3$,$\therefore f(x)=x^3-e^x$,$f(x+y)=(x+y)^3-e^{x+y}$,$z=e^x+y^2+(x+y)^3-e^{x+y}$。 + +$\therefore\dfrac{\partial z}{\partial x}=e^x+3(x+y)^2-e^{x+y}$。 + +\section{二元函数} + +函数以$f(u,v)$的形式来出现,需要分别对其求偏导。 + +\subsection{链式法则} + +\textbf{例题:}设$z=e^{xy}+f(x+y,xy)$,$f(u,v)$有二阶连续偏导数,求$\dfrac{\partial^2z}{\partial x\partial y}$。 + +解:令$x+y$为$u$,$xy$为$v$,$f(u,v)$对$u$求导就是$f_1'$,对$v$求导就是$f_2'$,求$uv$依次求导就是$f_{12}''$,以此类推。 + +首先求一次偏导:$\dfrac{\partial z}{\partial x}=ye^{xy}+\dfrac{\partial f(u,v)}{\partial u}\dfrac{\partial u}{\partial x}+\dfrac{\partial f(u,v)}{\partial v}\dfrac{\partial v}{\partial x}=ye^{xy}+f_1'+f_2'y$。 + +接着对$y$求偏导:$\dfrac{\partial^2z}{\partial x\partial y}=e^{xy}+xye^{xy}+\dfrac{\partial f_1'}{\partial y}+\dfrac{\partial f_2'y}{\partial y}$。 + +$=e^{xy}+xye^{xy}+\dfrac{\partial f_1'}{\partial y}+\dfrac{\partial f_2'}{\partial y}y+f_2'\dfrac{\partial y}{\partial y}=e^{xy}+xye^{xy}+\dfrac{\partial f_1'}{\partial u}\dfrac{\partial u}{\partial y}+\dfrac{\partial f_1'}{\partial v}\dfrac{\partial v}{\partial y}+\dfrac{\partial f_2'}{\partial u}\dfrac{\partial u}{\partial y}y+\dfrac{\partial f_2'}{\partial v}\dfrac{\partial v}{\partial y}y+f_2'=e^{xy}+xye^{xy}+f_{11}''+f_{12}''x+f_{21}''y+f_{22}''xy+f_2'$。\medskip + +又$f(u,v)$具有两阶连续偏导数,所以$f_{12}''=f_{21}''$。 + +即$=e^{xy}+xye^{xy}+f_{11}''+(x+y)f_{12}''+xyf_{22}''+f_2'$。 + \section{多元函数微分应用} \subsection{空间曲线的切线与法平面} diff --git a/advanced-math/exercise/9-differential-equation/differential-equation.pdf b/advanced-math/exercise/9-differential-equation/differential-equation.pdf index 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b/advanced-math/exercise/9-differential-equation/differential-equation.tex index 549c44b..016cdfb 100644 --- a/advanced-math/exercise/9-differential-equation/differential-equation.tex +++ b/advanced-math/exercise/9-differential-equation/differential-equation.tex @@ -88,6 +88,8 @@ $\therefore\ln(u+\sqrt{1+u^2})=-\ln x+\ln C$,$u+\sqrt{1+u^2}=\dfrac{C}{x}$。 \subsection{一阶线性方程} +形如$\dfrac{\textrm{d}y}{\textrm{d}x}+P(x)y=Q(x)$。 + \textbf{例题:}求微分方程$y'+1=e^{-y}\sin x$的通解。 解:已知对$e^{-y}\sin x$无法处理,所以必然需要对其转换,$e^yy'+e^y=\sin x$。 @@ -98,6 +100,8 @@ $e^y=u=e^{-\int\textrm{d}x}(\int e^{\int\textrm{d}x}\sin x\,\textrm{d}x+C)=e^{-x \subsection{伯努利方程} +形如$\dfrac{\textrm{d}y}{\textrm{d}x}+P(x)y=Q(x)y^n$。 + \textbf{例题:}求$y\,\textrm{d}x=(1+x\ln y)x\,\textrm{d}y$($y>0$)的通解。 解:将导数放到一边:$\dfrac{\textrm{d}y}{\textrm{d}x}=\dfrac{y}{(1+x\ln y)x}$,这个算式无法处理。 @@ -132,8 +136,6 @@ $y'=C(1+x^2)$,$\therefore y=C_2\left(x+\dfrac{x^3}{3}+x\right)+C$。 \subsection{常系数齐次线性微分方程} - - \subsection{常系数非齐次线性微分方程} 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y=\pm e^{x^2}e^C=\pm C_1e^{x^2}=C_2e^{x^2}$。 也可能出现$\dfrac{\textrm{d}x}{\textrm{d}y}=\varphi\left(\dfrac{x}{y}\right)$。 -令$u=\dfrac{y}{x}$,则$y=ux$变为$\dfrac{\textrm{d}y}{\textrm{d}x}=u+x\dfrac{\textrm{d}u}{\textrm{d}x}$,从而原方程变为$x\dfrac{\textrm{d}u}{\textrm{d}x}+u=\varphi(u)$,即$\dfrac{\textrm{d}u}{\varphi(u)-u}=\dfrac{\textrm{d}x}{x}$。 +令$u=\dfrac{y}{x}$,则$y=ux$变为$\dfrac{\textrm{d}y}{\textrm{d}x}=u+x\dfrac{\textrm{d}u}{\textrm{d}x}$($u$不是一个常数而是一个关于$x$的函数,所以$\dfrac{\textrm{d}y}{\textrm{d}x}\neq u$),从而原方程变为$x\dfrac{\textrm{d}u}{\textrm{d}x}+u=\varphi(u)$,即$\dfrac{\textrm{d}u}{\varphi(u)-u}=\dfrac{\textrm{d}x}{x}$。 如$(xy-y^2)\textrm{d}x-(x^2-2xy)\textrm{d}y=0$可以化为$\dfrac{\textrm{d}y}{\textrm{d}x}=\dfrac{xy-y^2}{x^2-2xy}$,即$\dfrac{\textrm{d}y}{\textrm{d}x}=\dfrac{\dfrac{y}{x}-\left(\dfrac{y}{x}\right)^2}{1-2\left(\dfrac{y}{x}\right)}$。