From 4bb634c925ec33be852b360e27974aa039839146 Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Mon, 10 Oct 2022 23:12:35 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- ...ntegal-of-functions-of-single-variable.pdf | Bin 270525 -> 271181 bytes ...ntegal-of-functions-of-single-variable.tex | 82 +++++++------- ...ral-calculus-of-multivariate-functions.pdf | Bin 130805 -> 147813 bytes ...ral-calculus-of-multivariate-functions.tex | 104 ++++++++++++++++-- .../differential-equation.pdf | Bin 96830 -> 105745 bytes .../differential-equation.tex | 48 +++++++- ...efinite-integral-and-definite-integral.pdf | Bin 210510 -> 209922 bytes ...efinite-integral-and-definite-integral.tex | 14 ++- ...ral-calculus-of-multivariate-functions.pdf | Bin 201224 -> 202437 bytes ...ral-calculus-of-multivariate-functions.tex | 10 ++ 10 files changed, 198 insertions(+), 60 deletions(-) diff --git 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$\int_0^{\sqrt{2}}r\,\textrm{d}r\int_{-\frac{\pi}{4}}^{\arccos\frac{r}{2}}f(r\cos\theta,r\sin\theta)\textrm{d}\theta+\int_{\sqrt{2}}^2r\,\textrm{d}r\int_{-\arccos\frac{r}{2}}^{\arccos\frac{r}{2}}f(r\cos\theta,r\sin\theta)\textrm{d}\theta$。 -\subsection{极直互化} +\subsubsection{极直互化} \textbf{例题:}将$I=\int_0^{\frac{\sqrt{2}}{2}R}e^{-y^2}\textrm{d}y\int_0^ye^{-x^2}\,\textrm{d}x+\int_{\frac{\sqrt{2}}{2}R}^Re^{-y^2}\,\textrm{d}y\int_0^{\sqrt{R^2-y^2}}e^{-x^2}\,\textrm{d}x$转换为极坐标系并计算结果。 @@ -128,13 +130,21 @@ $\int_0^\pi\cos^2x\,\textrm{d}x\int_0^{\sin x}\textrm{d}y=\int_0^\pi\cos^2x\sin 直角坐标系和极坐标系都可以使用。 -若积分区域$\sigma$关于$x=k_1$或$y=k_2$对称,则当$f(x,y)$含有$x-k_1$或$y-k_2$因式时重积分值为0。 +\paragraph{直角坐标系} \leavevmode \medskip -\textbf{例题:}设$D:x^2+y^2\leqslant2x+2y$,求$\iint\limits_Dxy\,\textrm{d}x\textrm{d}y$。 +主要是一般对称性(积分区域关于$x$轴对称若$y$奇则为0,关于$y$轴对称若$x$奇则为0)和轮换对称性(积分区域关于$y=x$对称则$xy$可以互换)。 -解:本题目使用直角坐标系和极坐标系都不好做。所以需要利用积分性质,对$D$进行平移等操作。 +\paragraph{极坐标系} \leavevmode \medskip -利用平移,由于$D:(x-1)^2+(y-1)^2=2$,令$x=1+r\cos\theta$,$y=1+r\sin\theta$,则利用极坐标,$r\in[0,\sqrt{2}]$,$\theta\in[0,2\pi]$,$=\int_0^{2\pi}\textrm{d}\theta\int_0^{\sqrt{2}}((1+r\cos\theta)(1+r\sin\theta)r)\textrm{d}r=\int_0^{2\pi}\textrm{d}\theta\int_0^{\sqrt{2}}(1+r\sin\theta+r\cos\theta+r^2\sin\theta\cos\theta)r\,\textrm{d}r$,又将$\sin\theta$和$\cos\theta$对$\theta$在$[0,2\pi]$进行积分全部为0,所以直接把后面的全消掉,变为$\int_0^{2\pi}\textrm{d}\theta\int_0^{\sqrt{2}}r\,\textrm{d}r=2\pi$。 +主要就是指一般对称性(画图可知)。 + +\textbf{例题:}求曲线$r^2=2ax^2\cos2\theta$($a>0$)所围图形面积。 + +解:即求$\iint\limits_D\textrm{d}x\textrm{d}y$。重点就是求$D$,已知$r$的表达式,要求$\theta$的取值范围。 + +又$r^2>0$,$x^2>0$,$a>0$,所以$\cos2\theta\geqslant0$,$-\dfrac{\pi}{4}\leqslant\theta\leqslant\dfrac{\pi}{4}$,$\dfrac{3\pi}{4}\leqslant\theta\leqslant\dfrac{5\pi}{4}$。 + +由对称性,所以$=4\int_0^{\frac{\pi}{4}}\textrm{d}\theta\int_0^{a\sqrt{2\cos2\theta}}r\,\textrm{d}r=4a^2\int_0^{\frac{\pi}{4}}\cos2\theta\,\textrm{d}\theta=2a^2$。 \subsubsection{切分区域} @@ -178,6 +188,68 @@ $\iint\limits_{D'}(u^2+v^2)\textrm{d}u\textrm{d}v=\int_0^{2\pi}\textrm{d}\theta\ 所以原式$=2\pi+6\pi=8\pi$。 +若积分区域$\sigma$关于$x=k_1$或$y=k_2$对称,则当$f(x,y)$含有$x-k_1$或$y-k_2$因式时重积分值为0。 + +\textbf{例题:}设$D:x^2+y^2\leqslant2x+2y$,求$\iint\limits_Dxy\,\textrm{d}x\textrm{d}y$。 + +解:本题目使用直角坐标系和极坐标系都不好做。所以需要利用积分性质,对$D$进行平移等操作。 + +利用平移,由于$D:(x-1)^2+(y-1)^2=2$,令$x=1+r\cos\theta$,$y=1+r\sin\theta$,则利用极坐标,$r\in[0,\sqrt{2}]$,$\theta\in[0,2\pi]$,$=\int_0^{2\pi}\textrm{d}\theta\int_0^{\sqrt{2}}((1+r\cos\theta)(1+r\sin\theta)r)\textrm{d}r=\int_0^{2\pi}\textrm{d}\theta\int_0^{\sqrt{2}}(1+r\sin\theta+r\cos\theta+r^2\sin\theta\cos\theta)r\,\textrm{d}r$,又将$\sin\theta$和$\cos\theta$对$\theta$在$[0,2\pi]$进行积分全部为0,所以直接把后面的全消掉,变为$\int_0^{2\pi}\textrm{d}\theta\int_0^{\sqrt{2}}r\,\textrm{d}r=2\pi$。 + +\subsubsection{极限化为二重积分} + +类似极限转换为定积分,有$\lim\limits_{n\to\infty}\dfrac{1}{n^2}\sum\limits_{i=1}^n\sum\limits_{j=1}f\left(\dfrac{i}{n},\dfrac{j}{n}\right)=\int_0^1\textrm{d}x\int_0^1f(x,y)\,\textrm{d}y$: + +\begin{enumerate} + \item 先提出$\dfrac{1}{n^2}$。 + \item 凑出$\dfrac{i}{n}$和$\dfrac{j}{n}$。 + \item 写出$\int_0^1\textrm{d}x$、$\int_0^1f(x,y)\,\textrm{d}y$,其中$\dfrac{1}{n^2}$没有了,将所有$\dfrac{i}{n}$换为$x$,$\dfrac{j}{n}$换为$y$。 +\end{enumerate} + +\textbf{例题:}求$I=\lim\limits_{n\to\infty}\sum\limits_{i=1}^n\sum\limits_{j=1}^n\dfrac{1}{\left(1+\dfrac{i}{n}\right)(n^2+j^2)}$。 + +解:首先提出$\dfrac{1}{n^2}$,正好在分母右边等式中$=\lim\limits_{n\to\infty}\sum\limits_{i=1}^n\sum\limits_{j=1}^n\dfrac{1}{\left(1+\dfrac{i}{n}\right)(1+\dfrac{j^2}{n^2})}\dfrac{1}{n^2}$。 + +已完全化为$\dfrac{i}{n}$和$\dfrac{j}{n}$,所以$=\displaystyle{\int_0^1\dfrac{1}{1+x}\textrm{d}x\int_0^1\dfrac{1}{1+y^2}\textrm{d}y}$。 + +$=\ln(1+x)\vert_0^1\arctan y\vert_0^1=\dfrac{\pi}{4}\ln2$。 + +\subsubsection{二重积分极限} + +即对存在二重积分的式子求极限。当对二重积分求极限时基本上都需要使用洛必达法则对积分求导。 + +令中间的定积分为$g(x)$,并记住$(\int_{f(x)}^{g(x)}h(t)\,\textrm{d}t)'=(g'(x)-f'(x))h(x)$。上下限最好换为$x$和$0$。 + +\textbf{例题:}求$\lim\limits_{t\to0^+}\dfrac{1}{t^6}\int_0^t\textrm{d}x\int_x^t\sin(xy)^2\textrm{d}y$。 + +解:这个式子求极限必然使用洛必达法则,而$\int_0^t\textrm{d}x\int_x^t\sin(xy)^2\textrm{d}y$里面那层定积分的上下限不规整,所以更换积分次序变成$\int_0^t\textrm{d}y\int_0^x\sin(xy)^2\textrm{d}x$。 + +对其求导$\int_0^t[\int_0^x\sin(xy)^2\textrm{d}x]\textrm{d}y$,由于该层积分变量为$y$,令$\int_0^x\sin(xy)^2\textrm{d}x=g(y)$,所以$=\int_0^tg(y)\,\textrm{d}y$,对其求导得$g(t)$,里面的$y$全部用$t$替换$=\int_0^x\sin(xt)^2\textrm{d}x$。 + +$=\lim\limits_{t\to0^+}\dfrac{\int_0^t\textrm{d}x\int_x^t\sin(xy)^2\textrm{d}y}{t^6}=\lim\limits_{t\to0^+}\dfrac{\int_0^t\sin(xt)^2\textrm{d}x}{6t^5}=\lim\limits_{t\to0^+}\dfrac{\int_0^t\sin(xt)^2\textrm{d}(xt)}{6t^6}$。 + +由于$x$和$t$都是变量,所以令$xt=u$,$x=\dfrac{u}{t}$,$u\in(0,t^2)$,$=\lim\limits_{t\to0^+}\dfrac{\int_0^{t^2}\sin u^2\textrm{d}u}{6t^6}=\lim\limits_{t\to0^+}\dfrac{2t\sin t^4}{36t^5}=\lim\limits_{t\to0^+}\dfrac{t^5}{18t^5}=\dfrac{1}{18}$。 + +\subsubsection{广义极坐标系} + +广义极坐标系是对极坐标系的延伸,极坐标系是广义坐标系的特例。极坐标系是基于直线和圆周进行积分,而广义极坐标系可以对圆锥曲线进行积分。 + +当对面积为非圆形进行积分时可以使用广义极坐标系。 + +\textbf{例题:}设$D:\dfrac{x^2}{a}+\dfrac{y^2}{b}\leqslant1$,求$\iint\limits_Dy^2\,\textrm{d}x\textrm{d}y$。 + +解:令$x=ar\cos\theta$,$y=br\sin\theta$,所以$D'$为$0\leqslant r\leqslant 1$,$0\leqslant\theta\leqslant2\pi$。 + +变换雅可比行列式$J=\dfrac{\partial(x,y)}{\partial(r,\theta)}=\left\vert\begin{array}{cc} + \dfrac{\partial x}{\partial r} & \dfrac{\partial x}{\partial\theta} \\ + \dfrac{\partial y}{\partial r} & \dfrac{\partial y}{\partial\theta} +\end{array}\right\vert=\left\vert\begin{array}{cc} + a\cos\theta & -ar\sin\theta \\ + b\sin\theta & br\cos\theta +\end{array}\right\vert=abr$。 + +所以$I=\iint\limits_{D'}(br\sin\theta)^2\vert J\vert\,\textrm{d}r\textrm{d}\theta=\int_0^{2\pi}\textrm{d}\theta\int_0^1b^2r^2\sin^2\theta\cdot abr\,\textrm{d}r=\dfrac{\pi ab^3}{4}$。 + \subsection{二重积分等式} \subsubsection{函数} @@ -268,6 +340,22 @@ $=\displaystyle{\int_0^\frac{\pi}{2}\textrm{d}\theta\int_0^{+\infty}e^{-r^2}r\,\ $\therefore I=\dfrac{\sqrt{\pi}}{2}$。 +\subsection{二重积分应用} + +\subsubsection{体积} + +\subsubsection{形心公式} + +直角坐标系和极坐标系的形心公式都是一样的,使用极坐标系可以转换。 + +\textbf{例题:}求曲线$ay=x^2$与$x+y=2a$所围平面区域$D$的形心坐标。 + +解:根据表达式可知有两个交点$(-2a,4a)$、$(a,a)$,形心公式: + +$\overline{x}=\dfrac{\iint\limits_Dx\,\textrm{d}\sigma}{\iint\limits_D\textrm{d}\sigma}=\dfrac{\int_{-2a}^ax\,\textrm{d}x\int_{\frac{x^2}{a}}^{-x+2a}\textrm{d}y}{\int_{-2a}^a\,\textrm{d}x\int_{\frac{x^2}{a}}^{-x+2a}\textrm{d}y}=\dfrac{\int_{-2a}^a(-\dfrac{x^3}{a}-x^2+2ax)\textrm{d}x}{\int_{-2a}^a(-\dfrac{x^2}{a}-x+2a)\textrm{d}x}=-\dfrac{1}{2}a$ + +$\overline{y}=\dfrac{\iint\limits_Dy\,\textrm{d}\sigma}{\iint\limits_D\textrm{d}\sigma}=\dfrac{\int_{-2a}^a\,\textrm{d}x\int_{\frac{x^2}{a}}^{-x+2a}y\textrm{d}y}{\int_{-2a}^a\,\textrm{d}x\int_{\frac{x^2}{a}}^{-x+2a}\textrm{d}y}=\dfrac{\dfrac{1}{2}\int_{-2a}^a(-\dfrac{x^4}{4}+x^2-4ax+4a^2)}{\int_{-2a}^a(-\dfrac{x^2}{a}-x+2a)\textrm{d}x}=\dfrac{8}{5}a$ + \section{弧长曲线积分} \section{坐标曲线积分} diff --git a/advanced-math/exercise/9-differential-equation/differential-equation.pdf 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b/advanced-math/exercise/9-differential-equation/differential-equation.tex index 9913399..65bdf03 100644 --- a/advanced-math/exercise/9-differential-equation/differential-equation.tex +++ b/advanced-math/exercise/9-differential-equation/differential-equation.tex @@ -42,6 +42,8 @@ \subsection{可分离变量微分方程} +\subsubsection{交叉积分法} + \textbf{例题:}求$y\sin\dfrac{x}{2}\,\textrm{d}x-\cos\dfrac{x}{2}\,\textrm{d}y=0$的通解。 解:$\dfrac{\textrm{d}y}{\textrm{d}x}=y\tan\dfrac{x}{2}$,$\dfrac{\textrm{d}y}{y}=\tan\dfrac{x}{2}\,\textrm{d}x$,$\displaystyle{\int\dfrac{\textrm{d}y}{y}=2\int\tan\dfrac{x}{2}\,\textrm{d}\dfrac{x}{2}}$。 @@ -54,7 +56,9 @@ $y=\dfrac{\pm C_1}{\left(\cos\dfrac{x}{2}\right)^2}$,令$C=\pm C_1$,得$y=\d 此时解为全部解,为通解加上$y=0$的奇解。 -\subsection{多项式换元} +\subsubsection{多项式换元法} + +$x$和$y$是以和差作为一个整体形式。 \textbf{例题:}求微分方程$\textrm{d}y=\sin(x+y+100)\,\textrm{d}x$的通解。 @@ -68,7 +72,13 @@ $\int\sec^2u-\tan u\sec u\,\textrm{d}u=x$,即$\tan u-\sec u=x+C$。代回$u=x+ 所有解:$\tan(x+y+100)-\sec(x+y+100)=x+C$,$x+y+100=2k\pi-\dfrac{\pi}{2}$。 -\subsection{自然齐次方程} +\subsection{一阶线性方程} + +形如$\dfrac{\textrm{d}y}{\textrm{d}x}+P(x)y=Q(x)$。 + +可以直接求也可以使用公式求。 + +\subsubsection{交叉积分法} \textbf{例题:}设$L$是一条平面曲线,其上任意一点$P(x,y)$($x>0$)到坐标原点的距离恒等于该点处的切线在$y$轴上的截距,且$L$经过点$\left(\dfrac{1}{2},0\right)$,求$L$的方程。 @@ -86,9 +96,9 @@ $\therefore\ln(u+\sqrt{1+u^2})=-\ln x+\ln C$,$u+\sqrt{1+u^2}=\dfrac{C}{x}$。 代入$\dfrac{y}{x}+\sqrt{1+\dfrac{y^2}{x^2}}=\dfrac{C}{x}$,$y+\sqrt{x^2+y^2}=C$。 -\subsection{一阶线性方程} +\subsubsection{公式法} -形如$\dfrac{\textrm{d}y}{\textrm{d}x}+P(x)y=Q(x)$。 +即使用非齐次和非齐次的一阶线性微分方程公式。 \subsubsection{换元法} @@ -102,6 +112,28 @@ $\therefore(e^y)'+e^y=\sin x$,令$e^y=u$,$u'+u=\sin x$,$P(x)=1$,$Q(x)=\s $e^y=u=e^{-\int\textrm{d}x}(\int e^{\int\textrm{d}x}\sin x\,\textrm{d}x+C)=e^{-x}(\int e^x\sin x\,\textrm{d}x+C)$,积分再现表格解出$\int e^x\sin x\,\textrm{d}x$:$=e^{-x}\left(\dfrac{1}{2}e^x(\sin x-\cos x)+C\right)$。 +\textbf{例题:}求$y'=\dfrac{y^2-x}{2y(x+1)}$的通解。 + +解:这个式子首先分子分母等长,$xy$都合在一起,所以很难去分离出基本的微分方程。基本的微分方程式子为$y'+P(x)y=Q(x)$,对比可以看出里面$y^2$是不能化简的,所以很容易想到把这个当作一个整体。 + +$2y'y=\dfrac{y^2-x}{x+1}$,此时出现了$y^2$和$y^2$的导数,令$y^2=u$,$u'=\dfrac{u-x}{x+1}$。 + +即$u'-\dfrac{y}{x+1}=\dfrac{1}{x+1}-1$,此时就化为了一般非齐次方程。 + +根据公式算出$y=C(x+1)-(x+1)\ln\vert x+1\vert-1$。 + +\subsubsection{交换微分变量} + +当出现$y'=\dfrac{f(x)}{g(x)}$,$g(x)$多项式的次数远高于$f(x)$,此时就没办法分离变量了,可以用$\dfrac{\textrm{d}x}{\textrm{d}y}$颠倒求导顺序。 + +\textbf{例题:}求$y'=\dfrac{y}{x+(y+1)^2}$的通解。($y$不为常函数) + +解:由于$y'$对应的式子分母较复杂,而分子较简单,所以上下颠倒: + +$\dfrac{\textrm{d}x}{\textrm{d}y}=\dfrac{x+(y+1)^2}{y}=\dfrac{x}{y}+y+\dfrac{1}{y}+2$。$x'-\dfrac{1}{y}x=y+\dfrac{1}{y}+2$。 + +根据公式:$x=e^{\int\frac{1}{y}\,\textrm{d}y}\left[\displaystyle{\int\left(y+\dfrac{1}{y}+2\right)}e^{\int-\frac{1}{y}\,\textrm{d}y}\,\textrm{d}y+C\right]=y^2+2\ln\vert y\vert y-1+Cy$。 + \subsection{伯努利方程} 形如$\dfrac{\textrm{d}y}{\textrm{d}x}+P(x)y=Q(x)y^n$。 @@ -138,12 +170,16 @@ $y'=C(1+x^2)$,$\therefore y=C_2\left(x+\dfrac{x^3}{3}+x\right)+C$。 \section{高阶线性微分方程} -\subsection{常系数齐次线性微分方程} +% \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} +% \subsection{常系数非齐次线性微分方程} + +\subsection{基本解法} 先将常系数非齐次线性微分方程变为常系数齐次线性微分方程求解,然后加上非齐次方程的一个特解,就是非齐次方程的一个通解。 +特解只能拆为和的形式而不能拆为乘商的形式,如$Q(x)=\sin^2x$,则应该拆为$\dfrac{1-\cos2x}{2}$。 + \textbf{例题:}求$y''-4y'+4y=3xe^{2x}$的通解。 解:变为常系数齐次线性微分方程:$y''-4y'+4y$。 diff --git a/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.pdf b/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.pdf index fdce19be0085a59ec253cdca075cc10ae30a4670..9fe003368384c8231d1e6a7e60c9dbcd3b8e7e14 100644 GIT binary patch delta 46681 zcmY(qV|1WB*ezVUQ)_D5cBjVFwvDOX-nDJpwr$&<+P3C<-t&Gx&RJREpDQanSCYMx z%v>S_oFddkLjY_70&q^wj;4mTaPAvd8MAf=9LOU#K5*SaYi|;cFz8+y2Ax8GAiZfl z<4io{`5(U+i}5z6#}dRUU>nxq{0$}I*bm-(KDp4BKer9Q-;gpTWf8_Lt=@a@{cGQz zuI}I8_koY^x4F+xTFx*A5Bcf8OZN82#U7plOL#HBCjf@b15yi8mHW;2nbI4;*eg6Q 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zARG2Lk>E~Dhy%#*ql)l@B-jB7*db)hxM%=J>=8rW4o)$A#8FY%0Tk@M87|l)C;Zs{ z`r}87AMyh?T%anBr1HaKT#%(kH1eZtvJ$P88$SJZKCKHI{`x`ojqQBr9Mx;VGShYy zy1D@}_0SVDeDr1&`CpISbL{Rw3*R)0UHU1(KL6T>3Mqh(nS5ai>_R)ho8k?LHU&@t z$J&+XGbo*6y(yc?S%x|6Sz7E-ty&2z^(@{cqQz6I{O487FfmTK@T#6^ql!n?H3$4j zo2jOARH;%dRNLlM@>gE2U+CdFLztOAIS|iqD)on$(`}I<{*(>wazsKIC@Re}n z!Eg4du3tLowa~|Qv*t6HEmPs)Z#D(T(j(uI!k34(BV)^+|9Ky=`;HNBIy43Asja=I~yzog_wdk?Ee8^>)%`e diff --git a/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.tex b/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.tex index d3ab450..5590bee 100644 --- a/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.tex +++ b/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.tex @@ -759,7 +759,7 @@ $S_n=\sum\limits_{i=1}^n\Vert\overline{M_{i-1}M_{i}}\Vert$,$S=\lim\limits_{\de 如果是极坐标方程,则$S=\int_\alpha^\beta\sqrt{\rho^2+\rho'^2}\,\textrm{d}\theta$。 -\subsubsection{形心坐标公式} +\subsubsection{形心公式} \paragraph{曲线} \leavevmode \medskip @@ -781,15 +781,19 @@ $\overline{x}=\dfrac{\int_\alpha^\beta x\rho(x)\,\textrm{d}x}{\int_\alpha^\beta\ 设曲边梯形平面区域$D=\{(x,y)|0\leqslant y\leqslant f(x),a\leqslant x\leqslant b\}$,$f(x)$在$[a,b]$上连续,则平面$D$的形心坐标计算公式为:\medskip -$\overline{x}=\dfrac{\iint\limits_Dx\,\textrm{d}\sigma}{\iint\limits_D\textrm{d}\sigma}=\dfrac{\int_a^b\textrm{d}x\int_0^{f(x)}x\textrm{d}y}{\int_a^b\textrm{d}x\int_0^{f(x)}\,\textrm{d}y}=\dfrac{\int_a^bxf(x)\,\textrm{d}x}{\int_a^bf(x)\,\textrm{d}x}$。 +% $\overline{x}=\dfrac{\iint\limits_Dx\,\textrm{d}\sigma}{\iint\limits_D\textrm{d}\sigma}=\dfrac{\int_a^b\textrm{d}x\int_0^{f(x)}x\textrm{d}y}{\int_a^b\textrm{d}x\int_0^{f(x)}\,\textrm{d}y}=\dfrac{\int_a^bxf(x)\,\textrm{d}x}{\int_a^bf(x)\,\textrm{d}x}$。 -$\overline{y}=\dfrac{\iint\limits_Dy\,\textrm{d}\sigma}{\iint\limits_D\textrm{d}\sigma}=\dfrac{\int_a^b\textrm{d}x\int_0^{f(x)}y\textrm{d}y}{\int_a^b\textrm{d}x\int_0^{f(x)}\,\textrm{d}y}=\dfrac{\int_a^bf^2(x)\,\textrm{d}x}{2\int_a^bf(x)\,\textrm{d}x}$。 +% $\overline{y}=\dfrac{\iint\limits_Dy\,\textrm{d}\sigma}{\iint\limits_D\textrm{d}\sigma}=\dfrac{\int_a^b\textrm{d}x\int_0^{f(x)}y\textrm{d}y}{\int_a^b\textrm{d}x\int_0^{f(x)}\,\textrm{d}y}=\dfrac{\int_a^bf^2(x)\,\textrm{d}x}{2\int_a^bf(x)\,\textrm{d}x}$。 + +$\overline{x}=\dfrac{\iint\limits_Dx\,\textrm{d}\sigma}{\iint\limits_D\textrm{d}\sigma}=\dfrac{\int_a^bxf(x)\,\textrm{d}x}{\int_a^bf(x)\,\textrm{d}x}$,$\overline{y}=\dfrac{\iint\limits_Dy\,\textrm{d}\sigma}{\iint\limits_D\textrm{d}\sigma}=\dfrac{\int_a^bf^2(x)\,\textrm{d}x}{2\int_a^bf(x)\,\textrm{d}x}$。 设曲边梯形平面区域$D=\{(x,y)|g(x)\leqslant y\leqslant f(x),a\leqslant x\leqslant b\}$,$f(x)$、$g(x)$在$[a,b]$上连续,则平面$D$的形心坐标计算公式为:\medskip -$\overline{x}=\dfrac{\iint\limits_Dx\,\textrm{d}\sigma}{\iint\limits_D\textrm{d}\sigma}=\dfrac{\int_a^b\textrm{d}x\int_{g(x)}^{f(x)}x\textrm{d}y}{\int_a^b\textrm{d}x\int_{g(x)}^{f(x)}\,\textrm{d}y}=\dfrac{\int_a^bx[f(x)-g(x)]\,\textrm{d}x}{\int_a^b[f(x)-g(x)]\,\textrm{d}x}$。 +% $\overline{x}=\dfrac{\iint\limits_Dx\,\textrm{d}\sigma}{\iint\limits_D\textrm{d}\sigma}=\dfrac{\int_a^b\textrm{d}x\int_{g(x)}^{f(x)}x\textrm{d}y}{\int_a^b\textrm{d}x\int_{g(x)}^{f(x)}\,\textrm{d}y}=\dfrac{\int_a^bx[f(x)-g(x)]\,\textrm{d}x}{\int_a^b[f(x)-g(x)]\,\textrm{d}x}$。 -$\overline{y}=\dfrac{\iint\limits_Dy\,\textrm{d}\sigma}{\iint\limits_D\textrm{d}\sigma}=\dfrac{\int_a^b\textrm{d}x\int_{g(x)}^{f(x)}y\textrm{d}y}{\int_a^b\textrm{d}x\int_{g(x)}^{f(x)}\,\textrm{d}y}=\dfrac{\int_a^b[f^2(x)-g^2(x)]\,\textrm{d}x}{2\int_a^b[f(x)-g(x)]\,\textrm{d}x}$。 +% $\overline{y}=\dfrac{\iint\limits_Dy\,\textrm{d}\sigma}{\iint\limits_D\textrm{d}\sigma}=\dfrac{\int_a^b\textrm{d}x\int_{g(x)}^{f(x)}y\textrm{d}y}{\int_a^b\textrm{d}x\int_{g(x)}^{f(x)}\,\textrm{d}y}=\dfrac{\int_a^b[f^2(x)-g^2(x)]\,\textrm{d}x}{2\int_a^b[f(x)-g(x)]\,\textrm{d}x}$。 + +$\overline{x}=\dfrac{\iint\limits_Dx\,\textrm{d}\sigma}{\iint\limits_D\textrm{d}\sigma}=\dfrac{\int_a^bx[f(x)-g(x)]\,\textrm{d}x}{\int_a^b[f(x)-g(x)]\,\textrm{d}x}$,$\overline{y}=\dfrac{\iint\limits_Dy\,\textrm{d}\sigma}{\iint\limits_D\textrm{d}\sigma}=\dfrac{\int_a^b[f^2(x)-g^2(x)]\,\textrm{d}x}{2\int_a^b[f(x)-g(x)]\,\textrm{d}x}$。 \subsection{物理应用} diff --git a/advanced-math/knowledge/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.pdf b/advanced-math/knowledge/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.pdf index a5817c1e80fb60e8da5cd09dd583e7ed06a30f80..fddb5f60541d1cf496489244b4098765970f10bd 100644 GIT binary patch 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z(`e+|Jv~pY=kOGr{q<{+t)GsH+Trbec;FX z4+^&}@dk-8Oa3>u5pZ7(RcnT+Gq_)$ptv3IF_(0K;Q)Q5L7 z;~|xNr2h*(!bU!dMt;omclquuo7AhJ=!UThY+5R zxbMoVpOTZl4-Ln+5H}HqAFl3ISy`IZa$-jsqgrK$vkM)y>aKA8tW?-qi?Q9EM&*&y zTrPZJ(lm9YN$6h$Y`lWsY2myfB(VAY9-2#z VN{i1gDk>rX#CiQ%K}!+me*lSHhA034 diff --git a/advanced-math/knowledge/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.tex b/advanced-math/knowledge/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.tex index 9f68e9f..d24f052 100644 --- a/advanced-math/knowledge/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.tex +++ b/advanced-math/knowledge/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.tex @@ -244,6 +244,16 @@ $\textrm{d}\sigma$是第一象限,可以看作一个广义的圆,半径无 $=\int_0^\frac{\pi}{2}\textrm{d}\theta\int_0^{+\infty}e^{-r^2}r\,\textrm{d}r=\displaystyle{\int_0^\frac{\pi}{2}\dfrac{1}{2}\,\textrm{d}\theta}=\dfrac{\pi}{2}$。$\therefore I=\dfrac{\sqrt{\pi}}{2}$。 +\subsection{二重积分应用} + +\subsubsection{体积} + +\subsubsection{形心公式} + +对于直角坐标系和参数方程: + +$\overline{x}=\dfrac{\iint\limits_Dx\,\textrm{d}\sigma}{\iint\limits_D\textrm{d}\sigma}=\dfrac{\int x(t)y(t)x'(t)\,\textrm{d}t}{\int y(t)x'(t)\,\textrm{d}t}$,$\overline{y}=\dfrac{\iint\limits_Dy\,\textrm{d}\sigma}{\iint\limits_D\textrm{d}\sigma}=\dfrac{\int y(t)^tx'(t)\,\textrm{d}t}{\int y(t)x'(t)\,\textrm{d}t}$。 + \section{三重积分} \subsection{概念}

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