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更新多元微分
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Didnelpsun
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@@ -599,9 +599,23 @@ $=\displaystyle{\int_0^1\dfrac{1+x}{1+x^2}\textrm{d}x}=\displaystyle{\int_0^1\df
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$=\left[\arctan x+\dfrac{1}{2}\ln(1+x^2)\right]_0^1=\dfrac{\pi}{4}+\dfrac{1}{2}\ln2$。
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\subsection{变限积分}
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\subsection{变限积分与极限}
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\subsection{牛莱公式}
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变限积分也常与极限共同出现。
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\textbf{例题：}若$f(x)$连续，$f(0)=0$，$f'(0)=\pi$，求$\lim\limits_{x\to0}\dfrac{\int_0^xf(t)\,\textrm{d}t}{x^2}$。
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解：$=\lim\limits_{x\to0}\dfrac{f(x)}{2x}=\dfrac{1}{2}\lim\limits_{x\to0}\dfrac{f(x)-f(0)}{x-0}=\dfrac{1}{2}f'(0)=\dfrac{\pi}{2}$。
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\textbf{例题：}若$f(x)$连续，$F(x)=\int_0^x(x-t)f(t)\,\textrm{d}t$，求$F''(x)$。
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解：因为$x$与$t$混合在一起很麻烦，$x$为上限是常数，$t$为积分变量。
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$F(x)=\int_0^xxf(t)\,\textrm{d}t-\int_0^xtf(t)\,\textrm{d}t=x\int_0^xf(t)\,\textrm{d}t-\int_0^xtf(t)\,\textrm{d}t$
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$\therefore F'(x)=\int_0^xf(t)\,\textrm{d}t+xf(x)-xf(x)=\int_0^xf(t)\,\textrm{d}t$。
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$\therefore F''(x)=f(x)$。
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\subsection{换元积分}
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@@ -830,24 +830,30 @@ $
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\subsection{\texorpdfstring{$\lim\limits_{x\to 0}\dfrac{\sin x}{x}=1$}{}}
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证明：当$x\to 0$时$x\in[0,\dfrac{\pi}{2}]$。
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证明：
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\begin{tikzpicture}[scale=1.5]
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\draw (0,0) circle (1);
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\filldraw[black] (0,0) node[below]{$O$};
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\draw[black](0,0) -- (1,0) node[right]{$A$};
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\draw[black](0,0) -- (1/2,{sqrt(3)/2}) node[above]{$B$};
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\draw[black](1/2,{sqrt(3)/2}) -- (1/2,0) node[below]{$D$};
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\draw[black](1,0) -- (1,{sqrt(3)}) node[above]{$C$};
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\draw[black](1,0) -- (1/2,{sqrt(3)/2});
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\draw[black](1,{sqrt(3)}) -- (1/2,{sqrt(3)/2});
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\end{tikzpicture}
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\begin{minipage}{0.7\linewidth}
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当$x\to 0$时$x\in[0,\dfrac{\pi}{2}]$。
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设$\angle AOB$的弧度为$x$，圆$O$的半径为$1$，则$OD=\sin x$。
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设$\angle AOB$的弧度为$x$，圆$O$的半径为$1$，则$OD=\sin x$。
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则$S_\vartriangle AOB=\dfrac{\sin x}{2}$。根据扇形面积公式：$S_{\text{扇形}}AOB=\dfrac{x}{2}$。
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则$S_\vartriangle AOB=\dfrac{\sin x}{2}$。根据扇形面积公式：$S_{\text{扇形}}AOB=\dfrac{x}{2}$。
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又$\because CA=\tan x$，则$S_\vartriangle AOC=\dfrac{\tan x}{2}$。
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又$\because CA=\tan x$，则$S_\vartriangle AOC=\dfrac{\tan x}{2}$。
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\end{minipage}
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\hfill
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\begin{minipage}{0.2\linewidth}
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\begin{tikzpicture}[scale=1.125]
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\draw (0,0) circle (1);
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\filldraw[black] (0,0) node[below]{$O$};
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\draw[black](0,0) -- (1,0) node[right]{$A$};
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\draw[black](0,0) -- (1/2,{sqrt(3)/2}) node[above]{$B$};
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\draw[black](1/2,{sqrt(3)/2}) -- (1/2,0) node[below]{$D$};
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\draw[black](1,0) -- (1,{sqrt(3)}) node[above]{$C$};
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\draw[black](1,0) -- (1/2,{sqrt(3)/2});
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\draw[black](1,{sqrt(3)}) -- (1/2,{sqrt(3)/2});
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\end{tikzpicture}
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\end{minipage}
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根据图，在$x\in[0,\dfrac{\pi}{2}]$，$\sin x<x<\tan x$。
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@@ -957,64 +963,80 @@ $=e^{\lim\limits_{x\to\infty}\frac{\left(\frac{1}{1+\frac{1}{x}}\right)\cdot\lef
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\subsubsection{可去间断点（可补间断点）}
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\textcolor{violet}{\textbf{定义：}}若$\lim\limits_{x\to x_0}f(x)=A\neq f(x_0)$（甚至可以没有定义）。
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\begin{tikzpicture}
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\draw[-latex](-0.5,0) -- (3,0) node[below]{$x$};
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\draw[-latex](0,-0.5) -- (0,3) node[above]{$y$};
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\filldraw[black] (0,0) node[below]{$O$};
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\draw[black, thick, domain=-0.5:2] plot (\x,{pow(e,\x-1)});
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\filldraw[white, draw=black, line width=1pt] (1,1) circle (2pt);
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\draw[black, densely dashed](1,2) -- (0,2) node[left]{$B$};
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\draw[black, densely dashed](1,1) -- (0,1) node[left]{$A$};
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\draw[black, densely dashed](1,2) -- (1,0) node[below]{$x_0$};
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\filldraw[black] (1,2) circle (2pt) node[above]{$f(x_0)=B$};
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\filldraw[black] (1,1) node[right]{$\lim\limits_{x\to x_0}=A$};
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\end{tikzpicture}
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\begin{minipage}{0.6\linewidth}
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\textcolor{violet}{\textbf{定义：}}若$\lim\limits_{x\to x_0}f(x)=A\neq f(x_0)$（甚至可以没有定义）。
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\end{minipage}
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\hfill
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\begin{minipage}{0.3\linewidth}
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\begin{tikzpicture}
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\draw[-latex](-0.5,0) -- (3,0) node[below]{$x$};
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\draw[-latex](0,-0.5) -- (0,3) node[above]{$y$};
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\filldraw[black] (0,0) node[below]{$O$};
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\draw[black, thick, domain=-0.5:2] plot (\x,{pow(e,\x-1)});
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\filldraw[white, draw=black, line width=1pt] (1,1) circle (2pt);
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\draw[black, densely dashed](1,2) -- (0,2) node[left]{$B$};
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\draw[black, densely dashed](1,1) -- (0,1) node[left]{$A$};
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\draw[black, densely dashed](1,2) -- (1,0) node[below]{$x_0$};
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\filldraw[black] (1,2) circle (2pt) node[above]{$f(x_0)=B$};
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\filldraw[black] (1,1) node[right]{$\lim\limits_{x\to x_0}=A$};
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\end{tikzpicture}
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\end{minipage}
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\subsubsection{跳跃间断点}
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\textcolor{violet}{\textbf{定义：}}若$\lim\limits_{x\to x_0^-}f(x)$与$\lim\limits_{x\to x_0^+}f(x)$都存在，但是$\lim\limits_{x\to x_0^-}f(x)\neq\lim\limits_{x\to x_0^+}f(x)$。
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\begin{tikzpicture}
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\draw[-latex](-0.5,0) -- (3,0) node[below]{$x$};
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\draw[-latex](0,-0.5) -- (0,3) node[above]{$y$};
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\filldraw[black] (0,0) node[below]{$O$};
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\draw[black, thick, domain=-0.5:1] plot (\x,{pow(e,\x-1)});
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\draw[black, thick, domain=1:1.5] plot (\x,{pow(e,\x-1)+1});
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\filldraw[white, draw=black, line width=1pt] (1,1) circle (2pt);
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\draw[black, densely dashed](1,2) -- (0,2) node[left]{$B$};
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\draw[black, densely dashed](1,1) -- (0,1) node[left]{$A$};
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\draw[black, densely dashed](1,2) -- (1,0) node[below]{$x_0$};
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\filldraw[black] (1,2) circle (2pt) node[above]{$f(x_0)=B$};
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\filldraw[black] (1,1) node[right]{$\lim\limits_{x\to x_0}=A$};
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\end{tikzpicture}
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\begin{minipage}{0.6\linewidth}
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\textcolor{violet}{\textbf{定义：}}若$\lim\limits_{x\to x_0^-}f(x)$与$\lim\limits_{x\to x_0^+}f(x)$都存在，但是$\lim\limits_{x\to x_0^-}f(x)\neq\lim\limits_{x\to x_0^+}f(x)$。
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\end{minipage}
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\hfill
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\begin{minipage}{0.3\linewidth}
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\begin{tikzpicture}
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\draw[-latex](-0.5,0) -- (3,0) node[below]{$x$};
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\draw[-latex](0,-0.5) -- (0,3) node[above]{$y$};
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\filldraw[black] (0,0) node[below]{$O$};
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\draw[black, thick, domain=-0.5:1] plot (\x,{pow(e,\x-1)});
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\draw[black, thick, domain=1:1.5] plot (\x,{pow(e,\x-1)+1});
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\filldraw[white, draw=black, line width=1pt] (1,1) circle (2pt);
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\draw[black, densely dashed](1,2) -- (0,2) node[left]{$B$};
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\draw[black, densely dashed](1,1) -- (0,1) node[left]{$A$};
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\draw[black, densely dashed](1,2) -- (1,0) node[below]{$x_0$};
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\filldraw[black] (1,2) circle (2pt) node[above]{$f(x_0)=B$};
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\filldraw[black] (1,1) node[right]{$\lim\limits_{x\to x_0}=A$};
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\end{tikzpicture}
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\end{minipage} \medskip
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可去间断点与跳跃间断点的左右极限都存在的间断点都称为第一类间断点。
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\subsubsection{无穷间断点}
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\textcolor{violet}{\textbf{定义：}}若$\lim\limits_{x\to x_0}f(x)=\infty$，或至少一个方向为无穷大（定义分歧）。如$y=\dfrac{1}{x}$在$x=0$处为无穷间断点。
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\begin{tikzpicture}
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\draw[-latex](-2,0) -- (2,0) node[below]{$x$};
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\draw[-latex](0,-2) -- (0,2) node[above]{$y$};
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\filldraw[black] (0,0) node[below]{$O$};
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\draw[black, thick, domain=0.5:2] plot (\x,{pow(\x,-1)});
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\draw[black, thick, domain=-2:-0.5] plot (\x,{pow(\x,-1)});
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\end{tikzpicture}
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\begin{minipage}{0.6\linewidth}
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\textcolor{violet}{\textbf{定义：}}若$\lim\limits_{x\to x_0}f(x)=\infty$，或至少一个方向为无穷大（定义分歧）。如$y=\dfrac{1}{x}$在$x=0$处为无穷间断点。
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\end{minipage}
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\hfill
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\begin{minipage}{0.3\linewidth}
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\begin{tikzpicture}
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\draw[-latex](-2,0) -- (2,0) node[below]{$x$};
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\draw[-latex](0,-2) -- (0,2) node[above]{$y$};
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\filldraw[black] (0,0) node[below]{$O$};
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\draw[black, thick, domain=0.5:2] plot (\x,{pow(\x,-1)});
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\draw[black, thick, domain=-2:-0.5] plot (\x,{pow(\x,-1)});
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\end{tikzpicture}
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\end{minipage}
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\subsubsection{振荡间断点}
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\textcolor{violet}{\textbf{定义：}}若$\lim\limits_{x\to x_0}f(x)$为振荡不存在。如$\lim\limits_{x\to 0}\sin\dfrac{1}{x}$的$x=0$就是振荡间断点。
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\begin{tikzpicture}
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\draw[-latex](-2,0) -- (2,0) node[below]{$x$};
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\draw[-latex](0,-1.5) -- (0,1.5) node[above]{$y$};
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\filldraw[black] (0,0) node[below]{$O$};
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\draw[black, thick, domain=0.01:2] plot (\x,{sin(pow(\x,-1) r)});
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\draw[black, thick, domain=-2:-0.01] plot (\x,{sin(pow(\x,-1) r)});
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\end{tikzpicture}
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\begin{minipage}{0.6\linewidth}
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\textcolor{violet}{\textbf{定义：}}若$\lim\limits_{x\to x_0}f(x)$为振荡不存在。如$\lim\limits_{x\to 0}\sin\dfrac{1}{x}$的$x=0$就是振荡间断点。
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\end{minipage}
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\hfill
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\begin{minipage}{0.3\linewidth}
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\begin{tikzpicture}
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\draw[-latex](-2,0) -- (2,0) node[below]{$x$};
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\draw[-latex](0,-1.5) -- (0,1.5) node[above]{$y$};
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\filldraw[black] (0,0) node[below]{$O$};
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\draw[black, thick, domain=0.01:2] plot (\x,{sin(pow(\x,-1) r)});
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\draw[black, thick, domain=-2:-0.01] plot (\x,{sin(pow(\x,-1) r)});
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\end{tikzpicture}
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\end{minipage}
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无穷间断点与振荡间断点的左右极限都不存在的点都是第二类间断点。
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Binary file not shown.
@@ -487,36 +487,40 @@ $\therefore \sqrt{2}$。
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有一个边长为$x$的正方形，变化了$\Delta x$，其面积$\Delta S=(x+\Delta x)^2-x^2=2x\Delta x+(\Delta x)^2$。
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当$\Delta x\to 0$时，将这个变化定义为$2x\cdot\Delta x+o(\Delta x)$，前项为线性主部，后面为误差。这个就是$S$的微分。
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\begin{minipage}{0.45\linewidth}
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当$\Delta x\to 0$时，将这个变化定义为$2x\cdot\Delta x+o(\Delta x)$，前项为线性主部，后面为误差。这个就是$S$的微分。
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增量$\Delta y=f(x_0+\Delta)-f(x_0)=A\Delta x+o(\Delta x)$，这个$A\Delta x$定义为$\textrm{d}y$，叫做$y$的微分。
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增量$\Delta y=f(x_0+\Delta)-f(x_0)=A\Delta x+o(\Delta x)$，这个$A\Delta x$定义为$\textrm{d}y$，叫做$y$的微分。
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$\therefore \textrm{d}y\vert_{x=x_0}=A\Delta x=y'(x_0)\cdot\Delta x=y'(x_0)\cdot\textrm{d}x$
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$\therefore \textrm{d}y\vert_{x=x_0}=A\Delta x=y'(x_0)\cdot\Delta x=y'(x_0)\cdot\textrm{d}x$。
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由此，可导必可微，可微必可导。
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\begin{tikzpicture}[scale=0.9]
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\draw[-latex](-0.5,0) -- (4.5,0) node[below]{$x$};
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\draw[-latex](0,-0.5) -- (0,4) node[above]{$y$};
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\draw[black, thick, domain=-0.5:3] plot (\x,{pow(\x-1,2)/2+1}) node[above]{$y(x)$};
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\filldraw[black] (0,0) node[below]{$O$};
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\draw[black, densely dashed](1.5,1.125) -- (1.5,0) node[below]{$x_0$};
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\draw[black, densely dashed](1.5,1.125) -- (0,1.125) node[left]{$y_0$};
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\draw[black, densely dashed](3,3) -- (3,0) node[below]{$x_0+\Delta x$};
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\draw[black, densely dashed](3,3) -- (0,3) node[left]{$y_0+\Delta x$};
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\draw[black, densely dashed](3,1.875) -- (0,0.375) node[left]{$\textrm{d}y\cdot x+b$};
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\draw[<->, black](1.5,1.125) -- (3,1.125);
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\draw[<->, black](4,1.125) -- (4,3);
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\draw[<->, black](3.25,1.125) -- (3.25,1.875);
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\draw[<->, black](3.25,3) -- (3.25,1.875);
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\draw[black](3,3) -- (4.5,3);
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\draw[black](3,1.125) -- (4.5,1.125);
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\draw[black](3,1.875) -- (3.75,1.875);
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\filldraw[black] (2.25,0.75) node{$\Delta x$};
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\filldraw[black] (4.3,2) node{$\Delta y$};
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\filldraw[black] (3.5,1.5) node{\scriptsize{$\textrm{d}y$}};
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\filldraw[black] (3.5,2.5) node{\scriptsize{$o(\Delta x)$}};
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\end{tikzpicture}
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由此，可导必可微，可微必可导。
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\end{minipage}
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\hfill
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\begin{minipage}{0.45\linewidth}
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\begin{tikzpicture}[scale=0.9]
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\draw[-latex](-0.5,0) -- (4.5,0) node[below]{$x$};
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\draw[-latex](0,-0.5) -- (0,4) node[above]{$y$};
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\draw[black, thick, domain=-0.5:3] plot (\x,{pow(\x-1,2)/2+1}) node[above]{$y(x)$};
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\filldraw[black] (0,0) node[below]{$O$};
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\draw[black, densely dashed](1.5,1.125) -- (1.5,0) node[below]{$x_0$};
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\draw[black, densely dashed](1.5,1.125) -- (0,1.125) node[left]{$y_0$};
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\draw[black, densely dashed](3,3) -- (3,0) node[below]{$x_0+\Delta x$};
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\draw[black, densely dashed](3,3) -- (0,3) node[left]{$y_0+\Delta x$};
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\draw[black, densely dashed](3,1.875) -- (0,0.375) node[left]{$\textrm{d}y\cdot x+b$};
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\draw[<->, black](1.5,1.125) -- (3,1.125);
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\draw[<->, black](4,1.125) -- (4,3);
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||||
\draw[<->, black](3.25,1.125) -- (3.25,1.875);
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\draw[<->, black](3.25,3) -- (3.25,1.875);
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\draw[black](3,3) -- (4.5,3);
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\draw[black](3,1.125) -- (4.5,1.125);
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\draw[black](3,1.875) -- (3.75,1.875);
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\filldraw[black] (2.25,0.75) node{$\Delta x$};
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\filldraw[black] (4.3,2) node{$\Delta y$};
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\filldraw[black] (3.5,1.5) node{\scriptsize{$\textrm{d}y$}};
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\filldraw[black] (3.5,2.5) node{\scriptsize{$o(\Delta x)$}};
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\end{tikzpicture}
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\end{minipage} \medskip
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所以可微就是用简单线性取代复杂线性，如图用直线取替代曲线。微分就是瞬时改变量，而导数就是瞬时改变速率。
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Binary file not shown.
@@ -31,8 +31,6 @@
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% 圆圈序号
|
||||
\usepackage{mathtools}
|
||||
% 有字的长箭头
|
||||
\usepackage{yhmath}
|
||||
% 弧线标识
|
||||
\usetikzlibrary{decorations.pathreplacing}
|
||||
% tikz的大括号
|
||||
\usepackage{yhmath}
|
||||
@@ -64,28 +62,32 @@ $\text{罗尔定理}\xrightleftharpoons[\text{特例：}f(a)=f(b)]{\text{泛化
|
||||
|
||||
罗尔定理\textcolor{violet}{\textbf{定义：}}
|
||||
|
||||
\begin{enumerate}
|
||||
\item $f(x)$在$[a,b]$上连续。
|
||||
\item $f(x)$在$(a,b)$内可导。
|
||||
\item $f(a)=f(b)$。
|
||||
\end{enumerate}
|
||||
|
||||
则$\exists\,\xi\in(a,b)$，使得$f'(\xi)=0$。
|
||||
|
||||
\begin{tikzpicture}[scale=0.7]
|
||||
\draw[-latex](-0.5,0) -- (8,0) node[below]{$x$};
|
||||
\draw[-latex](0,-0.5) -- (0, 4) node[above]{$y$};
|
||||
\filldraw[black] (0,0) node[below]{$O$};
|
||||
\draw[black, thick,domain=-0.5:8] plot (\x, {sin((\x-0.5) r)+2});
|
||||
\filldraw[black] (6,3.5) node {$y=f(x)$};
|
||||
\draw[densely dashed](0.5,2) -- (0.5+2*pi, 2);
|
||||
\draw[densely dashed](0.5,2) -- (0.5, 0) node[below]{$a$};
|
||||
\draw[densely dashed](0.5+2*pi,2) -- (0.5+2*pi, 0) node[below]{$b$};
|
||||
\draw[densely dashed](0.5+pi/2,3) -- (0.5+pi/2, 0) node[below]{$\xi_1$};
|
||||
\draw[densely dashed](0.5+pi/2*3,1) -- (0.5+pi/2*3, 0) node[below]{$\xi_2$};
|
||||
\draw[black](pi/2-0.5,3) -- (1.5+pi/2,3);
|
||||
\draw[black](pi/2*3-0.5,1) -- (1.5+pi/2*3,1);
|
||||
\end{tikzpicture}
|
||||
\begin{minipage}{0.45\linewidth}
|
||||
\begin{enumerate}
|
||||
\item $f(x)$在$[a,b]$上连续。
|
||||
\item $f(x)$在$(a,b)$内可导。
|
||||
\item $f(a)=f(b)$。
|
||||
\end{enumerate}
|
||||
|
||||
则$\exists\,\xi\in(a,b)$，使得$f'(\xi)=0$。
|
||||
\end{minipage}
|
||||
\hfill
|
||||
\begin{minipage}{0.45\linewidth}
|
||||
\begin{tikzpicture}[scale=0.7]
|
||||
\draw[-latex](-0.5,0) -- (8,0) node[below]{$x$};
|
||||
\draw[-latex](0,-0.5) -- (0, 4) node[above]{$y$};
|
||||
\filldraw[black] (0,0) node[below]{$O$};
|
||||
\draw[black, thick,domain=-0.5:8] plot (\x, {sin((\x-0.5) r)+2});
|
||||
\filldraw[black] (6,3.5) node {$y=f(x)$};
|
||||
\draw[densely dashed](0.5,2) -- (0.5+2*pi, 2);
|
||||
\draw[densely dashed](0.5,2) -- (0.5, 0) node[below]{$a$};
|
||||
\draw[densely dashed](0.5+2*pi,2) -- (0.5+2*pi, 0) node[below]{$b$};
|
||||
\draw[densely dashed](0.5+pi/2,3) -- (0.5+pi/2, 0) node[below]{$\xi_1$};
|
||||
\draw[densely dashed](0.5+pi/2*3,1) -- (0.5+pi/2*3, 0) node[below]{$\xi_2$};
|
||||
\draw[black](pi/2-0.5,3) -- (1.5+pi/2,3);
|
||||
\draw[black](pi/2*3-0.5,1) -- (1.5+pi/2*3,1);
|
||||
\end{tikzpicture}
|
||||
\end{minipage}
|
||||
|
||||
\subsection{拉格朗日中值定理}
|
||||
|
||||
|
||||
Binary file not shown.
@@ -315,6 +315,10 @@ $\dfrac{P}{x^n}=\dfrac{A}{x}+\dfrac{B_0x+B_1}{x^2}+\cdots+\dfrac{N_0x^{n-1}+\cdo
|
||||
|
||||
\subsection{性质}
|
||||
|
||||
\textcolor{aqua}{\textbf{定理：}}定积分由积分上下限与函数关系确定，与积分变量无关。$\int_a^xf(x)\,\textrm{d}x=\int_a^xf(t)\,\textrm{d}t=\varPhi(x)$。
|
||||
|
||||
\subsubsection{计算性质}
|
||||
|
||||
设函数$f(x)$在区间$[a,b]$上连续，则得到定积分基本计算性质：
|
||||
|
||||
\begin{enumerate}
|
||||
@@ -329,8 +333,15 @@ $\dfrac{P}{x^n}=\dfrac{A}{x}+\dfrac{B_0x+B_1}{x^2}+\cdots+\dfrac{N_0x^{n-1}+\cdo
|
||||
\item 已知$f(x)\in[m,M]$在$[a,b]$上成立，则$m(b-a)\leqslant\int_a^bf(x)\,\textrm{d}x\leqslant M(a-b)$。
|
||||
\item 估值定理：当$M$，$m$分别为$f(x)$在$[a,b]$上的最大值和最小值，则$m(b-a)\leqslant\int_a^bf(x)\,\textrm{d}x\leqslant M(b-a)$。
|
||||
\item 积分中值定理：$\exists\,\xi\in[a,b]$，使得$\int_a^bf(x)\,\textrm{d}x=f(\xi)(b-a)$。
|
||||
\item 积分中值定理推广：设$f(x)\in[a,b]$，$\exists\,\xi\in(a,b)$，使得$\int_a^bf(x)\,\textrm{d}x=f(\xi)(b-a)$。
|
||||
\end{enumerate}
|
||||
|
||||
证明第八条：已知$-\vert f(x)\vert\leqslant f(x)\leqslant\vert f(x)\vert$。
|
||||
|
||||
即得到$-\int_a^b\vert f(x)\vert\,\textrm{d}x\leqslant\int_a^bf(x)\,\textrm{d}x\leqslant\int_a^b\vert f(x)\vert\,\textrm{d}x$。
|
||||
|
||||
从而得证。
|
||||
|
||||
证明第十一条积分中值定理：
|
||||
|
||||
设函数$f(x)$在区间$[a,b]$上连续，因为闭区间上连续函数必然有最大最小值，所以设最大值为$M$，最小值为$m$，$M\geqslant m$。
|
||||
@@ -349,7 +360,11 @@ $\dfrac{P}{x^n}=\dfrac{A}{x}+\dfrac{B_0x+B_1}{x^2}+\cdots+\dfrac{N_0x^{n-1}+\cdo
|
||||
|
||||
对$F(x)$使用拉格朗日中值定理：$F(b)-F(a)=F'(\xi)(b-a)$，即$\int_a^bf(x)\,\textrm{d}x=f(\xi)(b-a)$，其中$\xi\in(a,b)\subset[a,b]$。
|
||||
|
||||
定积分的存在性性质：
|
||||
证明第十二条积分中值定理的推广。令$F(x)=\int_a^xf(t)\,\textrm{d}t$，$F'(x)=f(x)$。
|
||||
|
||||
$\int_a^bf(x)\,\textrm{d}x=F(b)-F(a)=F'(\xi)(b-a)=f(\xi)(b-a)$（$a<\xi<b$）。
|
||||
|
||||
\subsubsection{存在性性质}
|
||||
|
||||
\begin{enumerate}
|
||||
\item 设函数$f(x)$在区间$[a,b]$上连续，则$f(x)$在该区间上可积。
|
||||
@@ -357,7 +372,7 @@ $\dfrac{P}{x^n}=\dfrac{A}{x}+\dfrac{B_0x+B_1}{x^2}+\cdots+\dfrac{N_0x^{n-1}+\cdo
|
||||
\item 设函数$f(x)$在区间$[a,b]$上有界，且只有有限个间断点，则$f(x)$在该区间上可积。
|
||||
\end{enumerate}
|
||||
|
||||
定积分与函数性质：
|
||||
\subsubsection{定积分与函数性质}
|
||||
|
||||
\begin{enumerate}
|
||||
\item 若函数$f(x)$是周期函数且周期为$T$，$\int_a^{a+T}f(x)\,\textrm{d}x=\int_0^Tf(x)\,\textrm{d}x$对于$\forall a$成立。
|
||||
@@ -436,8 +451,6 @@ $\therefore\int_0^\pi x\sin^9x\,\textrm{d}x=\dfrac{\pi}{2}\int_0^\pi\sin^9x\,\te
|
||||
|
||||
设$f(x)$在$[a,b]$上连续，且$\Phi(x)=\int_a^xf(t)\,\textrm{d}t(x\in[a,b])$，这个函数就是积分上限函数或叫积分变限函数（如果$\int_x^af(t)\,\textrm{d}t$就是变下限积分或积分下限函数）。
|
||||
|
||||
积分值与积分变量无关，即$\int_a^xf(x)\,\textrm{d}x=\int_a^xf(t)\,\textrm{d}t$。
|
||||
|
||||
对变限积分$\int_{a}^xf(t)\,\textrm{d}t$求导得到$f(t)$，再求导就得到$f'(t)$。
|
||||
|
||||
定限积分是一个数值，而变限积分是一个函数。
|
||||
|
||||
Binary file not shown.
@@ -1,7 +1,10 @@
|
||||
\documentclass[UTF8, 12pt]{ctexart}
|
||||
% UTF8编码，ctexart现实中文
|
||||
\usepackage{color}
|
||||
\usepackage{xcolor}
|
||||
% 使用颜色
|
||||
\definecolor{orange}{RGB}{255,127,0}
|
||||
\definecolor{violet}{RGB}{192,0,255}
|
||||
\definecolor{aqua}{RGB}{0,255,255}
|
||||
\usepackage{geometry}
|
||||
\setcounter{tocdepth}{4}
|
||||
\setcounter{secnumdepth}{4}
|
||||
@@ -22,6 +25,10 @@
|
||||
% 数学公式
|
||||
\usepackage[colorlinks,linkcolor=black,urlcolor=blue]{hyperref}
|
||||
% 超链接
|
||||
\usepackage{pifont}
|
||||
% 圆圈序号
|
||||
\usepackage{tikz}
|
||||
% 绘图
|
||||
\author{Didnelpsun}
|
||||
\title{多元函数微分学}
|
||||
\date{}
|
||||
@@ -34,6 +41,223 @@
|
||||
\newpage
|
||||
\pagestyle{plain}
|
||||
\setcounter{page}{1}
|
||||
\section{}
|
||||
\section{基本概念}
|
||||
\subsection{平面与点}
|
||||
\subsubsection{平面点集}
|
||||
|
||||
\textcolor{violet}{\textbf{定义：}}在平面上建立直角坐标系$xOy$，则平面上的点可用两个实数组的有序数组$(x,y)$表示，而二元函数$f(x,y)$的定义域是$(x,y)$为元素的几何，所以$f(x,y)$的定义域就是\textbf{平面上的点集}。
|
||||
|
||||
\subsubsection{距离}
|
||||
|
||||
\textcolor{aqua}{\textbf{定理：}}平面上任意两点$M_1(x_1,y_1)$与$M_2(x_2,y_2)$之间距离定义为$\rho(M_1,M_2)$\\$=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$。
|
||||
|
||||
$\rho(M_1,M_2)$满足：
|
||||
|
||||
\begin{itemize}
|
||||
\item 非负性：$\rho(M_1,M_2)\geqslant0$。
|
||||
\item 对称性：$\rho(M_1,M_2)=\rho(M_2,M_1)$。
|
||||
\item 三角不等式：$\rho(M_1,M_3)\leqslant\rho(M_1,M_2)+\rho(M_2,M_3)$。
|
||||
\end{itemize}
|
||||
|
||||
\subsubsection{邻域}
|
||||
|
||||
设$M_0$为平面上一点，$\delta>0$，则平面上以$M_0$为圆心，$\delta$为半径的圆的内部称为$M_0$的\textbf{$\delta$领域}，记为$U(M_0,\delta)$。
|
||||
|
||||
若领域中去掉圆心$M_0$，称为$M_0$的\textbf{$\delta$去心邻域}，记为$\mathring{U}(M_0,\delta)$。
|
||||
|
||||
\subsubsection{点的分类}
|
||||
|
||||
\textcolor{violet}{\textbf{定义：}}设$M$为平面上一个点，若存在$\delta>0$，使得$U(M,\delta)\subset E$，则$M$为点集$E$的\textbf{内点}。
|
||||
|
||||
\textcolor{violet}{\textbf{定义：}}若存在$\delta>0$，使得$U(M,\delta)\cap E=\varnothing$，则$M$为点集$E$的的\textbf{外点}。
|
||||
|
||||
\textcolor{violet}{\textbf{定义：}}若对任意$\delta>0$，$U(M,\delta)$即有$E$内的点也有外的点，则$M$为点集$E$的\textbf{边界点}。
|
||||
|
||||
\textcolor{violet}{\textbf{定义：}}$E$所有边界点的集合称为$E$的\textbf{边界}，记为$\partial E$。对于任意一个点集$E$与其余集$E^C$有公共边界，即$\partial E=\partial E^C$。
|
||||
|
||||
\subsubsection{集合}
|
||||
|
||||
\textcolor{violet}{\textbf{定义：}}设$E$为一个平面点集，若存在常数$\delta>0$，使得$E\subset U(O,\delta)$，则$E$为\textbf{有界集}，否则为\textbf{无界集}。
|
||||
|
||||
\textcolor{violet}{\textbf{定义：}}若$E$中的每个点都是$E$的内点，则$E$为\textbf{开集}，若$E$的边界点都是$E$的点，则$E$为\textbf{闭集}。若一个点集是开集，则其余集为闭集，若一个点集为闭集，则其余集为开集。
|
||||
|
||||
\textcolor{violet}{\textbf{定义：}}若$E$中任意两点，都可用一条完全属于$E$的曲线将其两点连接，则$E$为\textbf{（道路）连通集}，连通的开集为\textbf{开区域}，一个开区域和其边界点的并集为\textbf{闭区域}，统称\textbf{区域}。
|
||||
|
||||
\textcolor{violet}{\textbf{定义：}}若$E$内任意一条\textbf{简单闭曲线}的内部还在$E$内，则$E$为\textbf{单连通区域}，否则为\textbf{多连通区域}。
|
||||
|
||||
\subsubsection{聚点}
|
||||
|
||||
\textcolor{violet}{\textbf{定义：}}对一个平面点集$E$，$M_0$为平面上一点，若对任意$\delta>0$，总有$\mathring{U}(M_0,\delta)\cap E\neq\varnothing$，即$M_0$的任意邻域中都含有异于$M_0$的$E$中的点，则$M_0$为$E$的\textbf{聚点}。
|
||||
|
||||
\textcolor{aqua}{\textbf{定理：}}非空开集的内点余边界点都是这个点集的聚点，闭区域的任意一点都是其聚点。
|
||||
|
||||
\textcolor{violet}{\textbf{定义：}}若存在$\delta>0$，使得$U(M_0,\delta)\cap E=\{M_0\}$，即如果$M_0$的某一邻域与点集$E$的交集是一个孤立的点$M_0$，则称$M_0$为$E$的\textbf{孤立点}。边界点要么是聚点要么是孤立点。
|
||||
|
||||
\subsection{极限}
|
||||
|
||||
对于一元函数的极限可用列举法，从两端逼近该点取极限，但是对于多元函数所处的邻域，逼近方向为无穷，所以不可能再通过取两个方向逼近的方式求极限。
|
||||
|
||||
从点集来看\textcolor{violet}{\textbf{定义：}}设二元函数$f(P)=f(x,y)$的定义域为$D$，$P_0(x_0,y_0)$为$D$聚点。若存在常数$A$，对于任意给定正数$\epsilon$，总存在正数$\delta$，使得当$P(x,y)\in D\cap\mathring{U}(P_0,\delta)$时，都有$\vert f(x,y)-A\vert<\epsilon$成立，则常数$A$为$f(x,y)$当$(x,y)\rightarrow(x_0,y_0)$时的极限，记为$\lim\limits_{(x,y)\to(x_0,y_0)}f(x,y)=A$或$f(x,y)\to A((x,y)\to(x_0,y_0))$。
|
||||
|
||||
如$\because xy\neq0$排除$xy$轴：$\lim\limits_{(x,y)\to(0,0)}\dfrac{\sqrt{xy+1}-1}{xy}=\lim\limits_{(x,y)\to(0,0)}\dfrac{xy+1-1}{xy(\sqrt{xy+1}+1)}$\\$=\lim\limits_{(x,y)\to(0,0)}\dfrac{1}{\sqrt{xy+1}+1}=\dfrac{1}{2}$。\medskip
|
||||
|
||||
从邻域来看\textcolor{violet}{\textbf{定义：}}若二元函数$f(x,y)$在$(x_0,y_0)$的去心邻域内有定义，且$(x,y)$以任意方式趋向$(x_0,y_0)$时，$f(x,y)$均趋向于$A$，则$\lim\limits_{\substack{x\to x_0\\y\to y_0}}f(x,y)=A$。
|
||||
|
||||
根据邻域的定义，由于函数$\lim\limits_{(x,y)\to(0,0)}\dfrac{\sqrt{xy+1}-1}{xy}$在坐标轴上无定义，则极限不存在。
|
||||
|
||||
此时两种定义就会有两种结论，所以为了避免这种定义不同的矛盾，就只会出现哪种定义下极限存在或都不存在的函数，如$\lim\limits_{\substack{x\to0\\y\to0}}(x^2+y^2)\sin\dfrac{1}{x^2+y^2}=0$。
|
||||
|
||||
从现实角度来看，点集定义是更合理的，若要求一根弯曲铁丝在某点的导数，第二种定义无法求，所以不合理。而第二种定义是从一元极限定义直接升级过来，所以有一定局限性。
|
||||
|
||||
\subsection{连续}
|
||||
|
||||
\textcolor{violet}{\textbf{定义：}}若$\lim\limits_{\substack{x\to x_0\\y\to y_0}}f(x,y)=f(x_0,y_0)$则称$f(x,y)$在点$(x_0,y_0)$处连续。
|
||||
|
||||
若不连续，则不讨论间断类型。
|
||||
|
||||
\subsection{偏导数}
|
||||
|
||||
当含有两个以及三个变量时，若求一个极限，则有多个变量同时趋向，所以多个变量同时在变。为了运算简单，就假定只有一个变量在变，其他变量固定，从而直接降低成一元变量，只对一个变量求导，从而就是偏导数。
|
||||
|
||||
\textcolor{violet}{\textbf{定义：}}设函数$z=f(x,y)$在点$(x_0,y_0)$的某邻域内有定义，若极限\\$\lim\limits_{\Delta x\to0}\dfrac{f(x_0+\Delta x,y_0)-f(x_0,y_0)}{\Delta x}$存在，则称此极限为函数$z=f(x,y)$在点$(x_0,y_0)$处对$x$的\textbf{偏导数}，记为$\dfrac{\partial z}{\partial x}\bigg|_{\substack{x=x_0\\y=y_0}}$，$\dfrac{\partial f}{\partial x}\bigg|_{\substack{x=x_0\\y=y_0}}$，$z'\bigg|_{\substack{x=x_0\\y=y_0}}$或$f'_x(x_0,y_0)$。\medskip
|
||||
|
||||
$f'_x(x_0,y_0)=\lim\limits_{\Delta x\to0}\dfrac{f(x_0+\Delta x,y_0)-f(x_0,y_0)}{\Delta x}=\lim\limits_{x\to x_0}\dfrac{f(x,y_0)-f(x_0,y_0)}{x-x_0}$。
|
||||
|
||||
$f'_y(x_0,y_0)=\lim\limits_{\Delta y\to0}\dfrac{f(x_0,y_0+\Delta y)-f(x_0,y_0)}{\Delta y}=\lim\limits_{y\to y_0}\dfrac{f(x_0,y)-f(x_0,y_0)}{y-y_0}$。
|
||||
|
||||
\textcolor{violet}{\textbf{定义：}}若函数$z=f(x,y)$在区域$D$内的偏导数$f_x'(x,y)$、$f_y'(x,y)$仍具有偏导数，则其偏导数为函数$z=f(x,y)$的\textbf{二阶偏导数}。按照求导次序不同，有如下四个二阶偏导数。
|
||||
|
||||
$\dfrac{\partial}{\partial x}\left(\dfrac{\partial z}{\partial x}\right)=\dfrac{\partial^2z}{\partial x^2}=f''_{xx}(x,y)$，$\dfrac{\partial}{\partial y}\left(\dfrac{\partial z}{\partial y}\right)=\dfrac{\partial^2z}{\partial y^2}=f''_{yy}(x,y)$，
|
||||
|
||||
$\dfrac{\partial}{\partial y}\left(\dfrac{\partial z}{\partial x}\right)=\dfrac{\partial^2z}{\partial x\partial y}=f''_{xy}(x,y)$，$\dfrac{\partial}{\partial x}\left(\dfrac{\partial z}{\partial y}\right)=\dfrac{\partial^2z}{\partial y\partial x}=f''_{yx}(x,y)$。
|
||||
|
||||
其中$f''_{xy}(x,y)$和$f''_{yx}(x,y)$为\textbf{混合偏导数}。二阶以及以上的偏导数均为\textbf{高阶偏导数}。
|
||||
|
||||
\subsection{全微分}
|
||||
|
||||
\textcolor{violet}{\textbf{定义：}}若函数$z=f(x,y)$在点$(x,y)$的全增量$\Delta z=f(x+\Delta x,y+\Delta y)-f(x,y)$可表示为$\Delta z=A\Delta x+B\Delta y+o(\rho)$，其中$\rho=\sqrt{(\Delta x)^2+(\Delta y)^2}$，$AB$不依赖$\Delta x$，$\Delta y$而仅与$x,y$相关，则称函数$z=f(x,y)$在点$(x,y)$可微，而称$A\Delta x+B\Delta y$为函数$z=f(x,y)$在点$(x,y)$的\textbf{全微分}，记为$\textrm{d}z$。
|
||||
|
||||
$\textrm{d}z=A\Delta x+B\Delta y=\dfrac{\partial z}{\partial x}\Delta x+\dfrac{\partial z}{\partial y}\Delta y=\dfrac{\partial z}{\partial x}\textrm{d}x+\dfrac{\partial z}{\partial y}\textrm{d}y$。
|
||||
|
||||
判断可微的步骤：
|
||||
|
||||
\begin{enumerate}
|
||||
\item 写出全增量$\Delta z=f(x_0+\Delta x,y_0+\Delta y)-f(x_0,y_0)$。
|
||||
\item 写出线性增量$A\Delta x+B\Delta y$，$A=f_x'(x_0,y_0)$，$B=f_y'(x_0,y_0)$。
|
||||
\item 写出极限$\lim\limits_{\substack{\Delta x\to0\\\Delta y\to0}}\dfrac{\Delta z-(A\Delta x+B\Delta y)}{\sqrt{(\Delta x)^2+(\Delta y)^2}}$，若极限等于0，则$z=f(x,y)$在点$(x_0,y_0)$可微，否则不可微。
|
||||
\end{enumerate}
|
||||
|
||||
% \textbf{例题：}已知函数$z=f(x,y)$的全微分$\textrm{d}z=2x\,\textrm{d}x+\sin y\,\textrm{d}y$且$f(1,0)=2$，求$f(x,y)$。
|
||||
|
||||
% 解：
|
||||
|
||||
\subsection{偏导数连续性}
|
||||
|
||||
对$z=f(x,y)$，讨论其在某特殊点$(x_0,y_0)$处偏导数是否连续的步骤：
|
||||
|
||||
\begin{enumerate}
|
||||
\item 用定义法求$f_x'(x_0,y_0)$，$f_y'(x_0,y_0)$。（求某点偏导数）
|
||||
\item 用公式法求$f_x'(x,y)$，$f_y'(x,y)$。（求偏导函数）
|
||||
\item 计算$\lim\limits_{\substack{x\to x_0\\y\to y_0}}f_x'(x,y)$，$\lim\limits_{\substack{x\to x_0\\y\to y_0}}f_y'(x,y)$。（偏导函数求极限）
|
||||
\item 若$\lim\limits_{\substack{x\to x_0\\y\to y_0}}f_x'(x,y)=f_x'(x_0,y_0)$，$\lim\limits_{\substack{x\to x_0\\y\to y_0}}f_y'(x,y)=f_y'(x_0,y_0)$若成立则连续，否则不连续。
|
||||
\end{enumerate}
|
||||
|
||||
\textbf{例题：}设$z=f(x,y)=\left\{\begin{array}{ll}
|
||||
(x^2+y^2)\sin\dfrac{1}{\sqrt{x^2+y^2}}, & x^2+y^2\neq0 \\
|
||||
0, & x^2+y^2=0
|
||||
\end{array}\right.$，则四个结论中正确的个数为()。
|
||||
|
||||
\ding{172}$f(x,y)$在$(0,0)$处连续。\qquad\ding{173}$f'_x(0,0)$，$f'_y(0,0)$存在。
|
||||
|
||||
\ding{174}$f_x'(x,y)$，$f_y'(x,y)$在$(0,0)$处连续。\qquad\ding{174}$f(x,y)$在$(0,0)$可微。
|
||||
|
||||
$A.1$\qquad$B.2$\qquad$C.3$\qquad$D.4$
|
||||
|
||||
解：$\lim\limits_{\substack{x\to0\\y\to0}}(x^2+y^2)\sin\dfrac{1}{\sqrt{x^2+y^2}}=0=f(0,0)$。所以$A$正确。
|
||||
|
||||
$f_x'(0,0)=\lim\limits_{\Delta x\to0}\dfrac{f(0+\Delta x,0)-f(0,0)}{\Delta x}=\lim\limits_{\Delta x\to0}\dfrac{(\Delta x)^2\sin\dfrac{1}{\sqrt{(\Delta x)^2}}-0}{\Delta x}=$\\$\lim\limits_{\Delta x\to0}(\Delta x)\sin\dfrac{1}{\vert\Delta x\vert}=0$。同理$f'_y(0,0)=0$。
|
||||
|
||||
判断连续性，首先计算偏导数值，之前计算过：$f_x'(0,0)=f_y'(0,0)=0$；然后求偏导函数$f_x'(x,y)=2x\sin\dfrac{1}{\sqrt{x^2+y^2}}+(x^2+y^2)\cos\dfrac{1}{\sqrt{x^2+y^2}}\left(-\dfrac{1}{2}\right)\dfrac{2x}{\sqrt{(x^2+y^2)^3}}$\\$=2x\sin\dfrac{1}{\sqrt{x^2+y^2}}-\dfrac{x}{\sqrt{x^2+y^2}}\cos\dfrac{1}{\sqrt{x^2+y^2}}$，同理得$f_y'(x,y)=2y\sin\dfrac{1}{\sqrt{x^2+y^2}}$\\$-\dfrac{y}{\sqrt{x^2+y^2}}\cos\dfrac{1}{\sqrt{x^2+y^2}}$；最后一步查看偏导函数值与偏导数值是否相等，$\because\lim\limits_{\substack{x\to0\\y\to0}}2x\sin\dfrac{1}{\sqrt{x^2+y^2}}=0$，且$\lim\limits_{\substack{x\to0\\y\to0}}\dfrac{x}{\sqrt{x^2+y^2}}\cos\dfrac{1}{\sqrt{x^2+y^2}}$震荡，所以总的来说极限值不存在，就不会等于偏导数值，同理可得函数的偏导数在该点不连续。
|
||||
|
||||
要求一个函数在某点可微，首先$\Delta z=f(0+\Delta x,0+\Delta y)-f(0,0)=[(\Delta x)^2+(\Delta y)^2]\sin\dfrac{1}{\sqrt{(\Delta x)^2+(\Delta y)^2}}$。然后$A\Delta x+B\Delta y=f_x'(0,0)\Delta x+f_y'(0,0)\Delta y=0$。最后求极限$\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}}=\lim\limits_{\substack{\Delta x\to0\\\Delta y\to0}}\sqrt{(\Delta x)^2+(\Delta y)^2}\sin\dfrac{1}{\sqrt{(\Delta x)^2+(\Delta y)^2}}$\\$=0$，所以在此点可微。
|
||||
|
||||
综上正确的结论有\ding{172}\ding{173}\ding{175}三个，所以选$C$。
|
||||
|
||||
\section{多元函数微分法则}
|
||||
|
||||
\subsection{链式求导法则}
|
||||
|
||||
主要对显函数的微分。
|
||||
|
||||
多元函数链式求导法则与一元函数的求导法则类似。都是从因变量从中间变量走到自变量。一条路径是一个加项，多少条从因变量到所有自变量的路就有多少个加项。每条路上由不同的路段组成，若有$n$层中间变量，则有$n+1$路段，路段之间项是乘积形式，若变量只与一个变量有一条路，则是导数$\textrm{d}$，若一个变量到多个变量有多条路，则是偏导数$\partial$。
|
||||
|
||||
\begin{minipage}{0.65\linewidth}
|
||||
因变量$z$到$x$一共有两条路，所以两个和项。每条路都有两端，所以和项中有两个乘项。$z$到$uv$两个中间变量，所以是两个偏导$\dfrac{\partial z}{\partial u}$和$\dfrac{\partial z}{\partial v}$。$uv$都只有一条路直接连通$x$，所以都是导数$\dfrac{\textrm{d}u}{\textrm{d}x}$和$\dfrac{\textrm{d}v}{\textrm{d}x}$。一条路的每个路段的项相乘：$\dfrac{\partial z}{\partial u}\dfrac{\textrm{d}u}{\textrm{d}x}$和$\dfrac{\partial z}{\partial v}\dfrac{\textrm{d}v}{\textrm{d}x}$。最后将每条路段相加：$\dfrac{\textrm{d}z}{\textrm{d}x}=\dfrac{\partial z}{\partial v}\dfrac{\textrm{d}v}{\textrm{d}x}$。
|
||||
\end{minipage}
|
||||
\hfill
|
||||
\begin{minipage}{0.25\linewidth}
|
||||
\begin{tikzpicture}[scale=1]
|
||||
\filldraw[black] (-0.25,0) node{$z$};
|
||||
\draw[black](0,0) -- (1,1) node[right]{$u$};
|
||||
\draw[black](0,0) -- (1,-1) node[right]{$v$};
|
||||
\draw[black](1.5,1) -- (2.5,0) node[right]{$x$};
|
||||
\draw[black](1.5,-1) -- (2.5,0);
|
||||
\end{tikzpicture}
|
||||
\end{minipage} \medskip
|
||||
|
||||
\begin{minipage}{0.25\linewidth}
|
||||
\begin{tikzpicture}[scale=1]
|
||||
\filldraw[black] (-0.25,0) node{$z$};
|
||||
\draw[black](0,0) -- (1,1) node[right]{$u$};
|
||||
\draw[black](0,0) -- (1,0) node[right]{$v$};
|
||||
\draw[black](0,0) -- (1,-1) node[right]{$w$};
|
||||
\draw[black](1.5,1) -- (2.5,1) node[right]{$x$};
|
||||
\draw[black](1.5,0) -- (2.5,1);
|
||||
\draw[black](1.5,-1) -- (2.5,-1) node[right]{$y$};
|
||||
\draw[black](1.5,1) -- (2.5,-1);
|
||||
\end{tikzpicture}
|
||||
\end{minipage}
|
||||
\hfill
|
||||
\begin{minipage}{0.65\linewidth}
|
||||
因为因变量$z$到自变量$x,y$有较多条路径，所以分开分析。
|
||||
|
||||
对于$x$，有$z-u-x$，所以这条路为$\dfrac{\partial z}{\partial u}\dfrac{\partial u}{\partial x}$，还有一条$z-v-x$，由于$v$只与$x$连通，所以是导数，该路为$\dfrac{\partial z}{\partial v}\dfrac{\textrm{d}v}{\textrm{d}x}$，所以$\dfrac{\partial z}{\partial x}=\dfrac{\partial z}{\partial u}\dfrac{\partial u}{\partial x}+\dfrac{\partial z}{\partial v}\dfrac{\textrm{d}v}{\textrm{d}x}$。、
|
||||
|
||||
同理对于$y$，有$z-u-y$和$z-w-y$，且$u$有两条出路，$w$只有一条，所以$u$偏导，$v$导数，$\dfrac{\partial z}{\partial y}=\dfrac{\partial z}{\partial u}\dfrac{\partial u}{\partial y}+\dfrac{\partial z}{\partial w}\dfrac{\textrm{d}w}{\textrm{d}y}$。
|
||||
\end{minipage} \medskip
|
||||
|
||||
无论$z$对谁求导也无论求了几阶到，求导过后的新函数仍具有与原函数完全相同的复合结构。
|
||||
|
||||
\textbf{例题：}设$z=f(e^x\sin y,x^2+y^2)$，其中$f$具有二阶连续偏导数，求$\dfrac{\partial^2z}{\partial x\partial y}$。
|
||||
|
||||
解：$\because\dfrac{\partial^2z}{\partial x\partial y}=\dfrac{\partial}{\partial y}\left(\dfrac{\partial z}{\partial x}\right)$，$\therefore\dfrac{\partial z}{\partial x}=f_1'\cdot e^x\sin y+f_2'\cdot2x$。
|
||||
|
||||
在求偏导时，将第一个中间变量记为$f_1$即之前的$u$，第二个中间变量记为$f_2$即之前的$v$。记$f$对$u$求偏导为$f_1'$，对$v$求偏导为$f_2'$同理二阶导也如此，下标为求导顺序。
|
||||
|
||||
$\dfrac{\partial^2z}{\partial x\partial y}=\dfrac{\partial(f_1'\cdot e^x\sin y)}{\partial y}+\dfrac{\partial(f_2'\cdot2x)}{\partial y}$。
|
||||
|
||||
其中$\dfrac{\partial(f_1'\cdot e^x\sin y)}{\partial y}=\dfrac{\partial f_1'}{\partial y}\cdot e^x\sin y+f_1'\cdot e^x\cos y$。所以难点就是$\dfrac{\partial f_1'}{\partial y}$。
|
||||
|
||||
求导路径$f_1'-1-y$和$f_1'-2-y$：$=(f_{11}''e^x\cos y+f_{12}''2y)\cdot e^x\sin y+f_1'\cdot e^x\cos y$。
|
||||
|
||||
\subsection{隐函数存在定理}
|
||||
|
||||
主要对隐函数的微分。
|
||||
|
||||
\section{多元函数极值最值}
|
||||
|
||||
\subsection{概念}
|
||||
|
||||
\subsection{无条件极值}
|
||||
|
||||
\subsubsection{隐函数}
|
||||
|
||||
\subsubsection{显函数}
|
||||
|
||||
\subsection{条件极值与拉格朗日乘数法}
|
||||
|
||||
\subsubsection{闭区域边界最值}
|
||||
|
||||
\subsubsection{闭区域上最值}
|
||||
|
||||
\end{document}
|
||||
|
||||
Reference in New Issue
Block a user