From bcc283ebedeb4a6054d6e302087c47003b6163e0 Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Thu, 23 Sep 2021 23:34:22 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0=E5=90=91=E9=87=8F?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- ...ial-calculus-of-multivariate-functions.pdf | Bin 81930 -> 95852 bytes ...ial-calculus-of-multivariate-functions.tex | 74 +++++++++++++++++- linear-algebra/exercise/2-matrix/matrix.pdf | Bin 151054 -> 102147 bytes linear-algebra/exercise/3-vector/vector.pdf | Bin 68563 -> 75471 bytes linear-algebra/exercise/3-vector/vector.tex | 32 +++++++- 5 files changed, 104 insertions(+), 2 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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zwsBCTxoS;#O`CM(*=e5S19{t--p+~hind{d_=-^c!{EaPTceRxZM(;EWEJhpduPdb zf*VSq&bSx(gCgP8v`bXCBty-zPXCKI@N)R?_Rl}S)-NE~&(R5AOi&0ch|k8Rq^*qq EA6&Bd*Z=?k diff --git a/advanced-math/exercise/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.tex b/advanced-math/exercise/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.tex index 31fd1b5..a3dc635 100644 --- a/advanced-math/exercise/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.tex +++ b/advanced-math/exercise/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.tex @@ -120,7 +120,79 @@ $z_{xx}''=y^2f(xy)+y^2f(xy)=2y^2f(xy)$,同理根据变量对称性$z_{yy}''=2x \textbf{例题:}设$z=f(x,y)$满足$\dfrac{\partial^2z}{\partial x\partial y}=x+y$,且$f(x,0)=x$,$f(0,y)=y^2$,求$f(x,y)$。 -解: +解:根据$\partial x\partial y$的求导顺序反向积分: + +$\dfrac{\partial z}{\partial x}=\int(x+y)\,\textrm{d}y=xy+\dfrac{1}{2}y^2+C_1(x)$。($x$看作常数) + +再次积分$z=\displaystyle{\int\left(xy+\dfrac{1}{2}y^2+C_1(x)\right)\,\textrm{d}x}=\dfrac{1}{2}x^2y+\dfrac{1}{2}xy^2+\int C_1(x)\,\textrm{d}x+C_2(y)$。($y$看作常数) + +又$f(x,0)=x$,代入$\int C_1(x)\,\textrm{d}x+C_2(0)=x$,两边求导$C_1(x)=1$,即$\int C_1(x)\,\textrm{d}x=\int\textrm{d}x=x$,$z=\dfrac{1}{2}x^2y+\dfrac{1}{2}xy^2+x+C_2(y)$。 + +又$f(0,y)=y^2$,代入$C_2(y)=y^2$。 + +$\therefore z=\dfrac{1}{2}x^2y+\dfrac{1}{2}xy^2+x+y^2$。 + +% \subsection{极限} + +\subsection{全微分} + +\subsubsection{含参数} + +基本上是用含参数的全微分来求参数。有多种方法。 + +\textbf{例题:}设$(ax^2y^2-2xy^2)\textrm{d}x+(2x^3y+bx^2y+1)\textrm{d}y$是函数$f(x,y)$的全微分,求参数。 + +解:由全微分定义可知,$f_x'=ax^2y^2-2xy^2$,$f_y'=2x^3y+bx^2y+1$。 + +分别对其积分:$f(x,y)=\int(ax^2y^2-2xy^2)\textrm{d}x=\int(2x^3y+bx^2y+1)\textrm{d}y$。 + +从而$\dfrac{a}{3}x^3y^2-x^2y^2+C(y)=x^3y^2+\dfrac{b}{2}x^2y^2+y+C(x)$,解得$a=3$,$b=-2$,$f(x)=x^3y^2-x^2y^2+y$。 + +\subsubsection{极限定义} + +全微分形式:$\lim\limits_{\substack{\Delta x\to0\\\Delta y\to0}}\dfrac{\Delta z-(A\Delta x+B\Delta y)}{\sqrt{(\Delta x)^2+(\Delta y)^2}}$。 + +要求$\textrm{d}z|_{(a,b)}$,就要求$\lim\limits_{(x,y)\to(a,b)}f(x,y)-f(a,b)=cx+dy+o(\rho)$,$c$和$d$就是$\textrm{d}x\textrm{d}y$的参数。 + +\textbf{例题:}连续函数$z=f(x,y)$满足$\lim\limits_{\substack{x\to0\\ y\to1}}\dfrac{f(x,y)-2x+y-2}{\sqrt{x^2+(y-1)^2}}=0$,求$\textrm{d}z|_{(0,1)}$。 + +解:当$x\to0$,$y\to1$时$\sqrt{x^2+(y-1)^2}\to0$,又$\lim\limits_{\substack{x\to0\\ y\to1}}\dfrac{f(x,y)-2x+y-2}{\sqrt{x^2+(y-1)^2}}=0$,$\therefore\lim\limits_{\substack{x\to0\\ y\to1}}f(x,y)-2x+y-2=0$。 + +又$f(x,y)$连续,则$f(0,1)+1-2=0$,$f(0,1)=1$。将值代入,并按分子配方: + +$\lim\limits_{\substack{x\to0\\ y\to1}}\dfrac{f(x,y)-f(0,1)-2x+(y-1)}{\sqrt{x^2+(y-1)^2}}=0$,即$f(x,y)-f(0,1)=2x-(y-1)+o(\rho)$。 + +根据全微分的定义偏导数就是其系数,$f_x'(0,1)=2$,$f_y'(0,1)=-1$。 + +$\therefore\textrm{d}z|_{(0,1)}=2\textrm{d}x-\textrm{d}y$。 + +\textbf{例题:}设$f(x,y)$在$(0,0)$处连续,且$\lim\limits_{(x,y)\to(0,0)}\dfrac{f(x,y)-a-bx-cy}{\ln(1+x^2+y^2)}=1$,其中$a,b,c$为常数,求$\textrm{d}f(x,y)|_{(0,0)}$。 + +解:根据全微分的定义,分母应该是根号的形式,所以对于极限使用等价无穷小替换$\ln(x+1)\sim x$,$\ln(1+x^2+y^2)=x^2+y^2$,$\lim\limits_{(x,y)\to(0,0)}\dfrac{f(x,y)-a-bx-cy}{x^2+y^2}=1$。 + +又$(x,y)\to0$时$x^2+y^2\to0$,$\therefore f(x,y)-a-bx-cy\to0$。 + +又$f(x,y)$在$(0,0)$处连续,$f(0,0)=a$。根据极限和无穷小的关系将其代回: + +$\lim\limits_{(x,y)\to(0,0)}\dfrac{f(x,y)-f(0,0)-bx-cy}{x^2+y^2}=1+o(1)$。 + +$\therefore\lim\limits_{(x,y)\to(0,0)}f(x,y)-f(0,0)-bx-cy=x^2+y^2+o(1)\cdot(x^2+y^2)=o(\rho)$。 + +$\therefore\lim\limits_{(x,y)\to(0,0)}f(x,y)-f(0,0)=bx+cy+o(\rho)$。 + +即$f_x'(0,0)=b$,$f_y'(0,0)=c$。$\textrm{d}f(x,y)|_{(0,0)}=b\textrm{d}x+c\textrm{d}y$。 + +\subsubsection{隐函数} + +二元隐函数求导公式:$\dfrac{\textrm{d}y}{\textrm{d}x}=-\dfrac{F_x'}{F_y'}$。 + +三元隐函数求导公式:$\dfrac{\partial z}{\partial x}=-\dfrac{F_x'}{F_z'}$,$\dfrac{\partial z}{\partial y}=-\dfrac{F_y'}{F_z'}$。 + +\textbf{例题:}设$f(x,y,z)=e^x+y^2z$,其中$z=z(x,y)$由$x+y+z+xyz=0$确定,求$f_x'(0,1,-1)$。 + +解:$f_x'(x,y,z)=e^x+y^2z_x'$。 + 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A^{k-1}\alpha\neq0$,$\therefore\lambda_2=0$,消去$\lambda_2$:$\ 综上当$s>n$时线性相关,$s\leqslant n$时线性无关。 -\section{极大线性无关组与向量组秩} +\subsection{矩阵秩} + +当向量的个数与维数不同时就不能使用行列式去分析,而只能用矩阵的秩来分析。当矩阵满秩则线性无关,当矩阵降秩则线性相关。 + +\subsubsection{线性相关性} + +当谈到多个向量是否线性相关时可以将向量组组成矩阵,判断其秩。 + +\subsubsection{线性表出} + +当谈到一个向量是否能被其他向量线性表出时,要将这些向量全部组成一起,判断能否被其他向量表出的向量放在最右边,然后判断增广矩阵的秩。 + +\textbf{例题:}已知$\alpha_1=(1,2,1)^T$,$\alpha_2(2,3,a)^T$,$\alpha_3=(1,a+2,-2)^T$,$\beta=(1,3,0)^T$,若$\beta$可以由$\alpha_1$、$\alpha_2$,$\alpha_3$线性表示,且表示法不唯一,求$a$。 + +解:设$x_1\alpha_1+x_2\alpha_2+x_2\alpha_3=\beta$,由$\beta$可以由$\alpha_1$、$\alpha_2$,$\alpha_3$线性表示,且表示法不唯一可知$Ax=\beta$有无穷解,即$r(A)=r(A|B)<3$。 + +$=[\alpha_1,\alpha_2,\alpha_3|\beta]=\left[\begin{array}{cccc} + 1 & 2 & 1 & 1 \\ + 2 & 3 & a+2 & 3 \\ + 1 & a & -2 & 0 +\end{array}\right]=\left[\begin{array}{cccc} + 1 & 2 & 1 & 1 \\ + 0 & -1 & a & 1 \\ + 0 & 0 & a^2-2a-3 & a-3 +\end{array}\right]$。 + +$\therefore a=3$。 + +\subsection{极大线性无关组} 极大线性无关组一般与向量组秩在一起使用。一般解出极大线性无关组与秩,还要用极大线性无关组表示出其余的向量,基本步骤: