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+$\dfrac{\partial^2z}{\partial y^2}=\dfrac{-(z-2)-(1-y)\dfrac{\partial z}{\partial y}}{(z-2)^2}$,同理得$C=-\dfrac{1}{4}$。 + +$\Delta<0$且$A>0$,所以得到极大值6。 + \subsection{有条件极值} 与无条件极值一样,在边界就是显函数可以直接求,在区域内就是隐函数需要求出可疑点再计算可疑点的二阶导数值判断。 diff --git a/advanced-math/exercise/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.pdf b/advanced-math/exercise/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.pdf index 5063fb1089927316f8727ef295b3dbc528666c24..272c6c573603a6234f0607e10406bf2608d9fccd 100644 GIT binary patch delta 25493 zcmY(pLwKNp(li=OY&)4uY}>YN+jidAwryJzOl;e>C&|RgJ>R*T^RIhX{d8AXRi6hy zRs=#eXo3UmtZe-Jur997W=3|fo*UP?GV%EBD7~i|k4_N9Jf>>*5Du=43D|dJ3B@8{ zo8h^cdImK=SB6*mW;L!nkk%JzYE8(XukQajzEG|lny58yOwY#J!GAT%6_CFy4KL;F z7Vg{l`s4E(F!VkHecqcN_Y8%Ng(R0HUhg5Ja}dC&oK2_9Qg@H9heJ&9szCQAq~3ND z%`1IogqJe~SF-4!?O5j6DevB_Lper?$$g4OL@tK4wnQ${BG}8&lkfv{8hxo%?qskW zRorDjhT-v5+vPevSGzWYvW0KA_MP{>7iIZOGl6;;9A5;}ylzo@yswdYHdO11TcGWt 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1f120c0..c22a257 100644 --- a/advanced-math/exercise/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.tex +++ b/advanced-math/exercise/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.tex @@ -54,31 +54,34 @@ 解:对于极坐标的积分次序交换需要利用直角坐标系来画图了解,特别是对于$r$的上下限。 -对$\theta=\dfrac{\pi}{2}$变为$y$轴,$y=-\dfrac{\pi}{4}$变为$y=-x$。 +对$\theta=\dfrac{\pi}{2}$变为$y$轴,$\theta=-\dfrac{\pi}{4}$变为$y=-x$。 对$r=2\cos\theta$变为$xy$的表达式,$r^2=2\cos\theta$,即$x^2+y^2=2x$,$(x-1)^2+y^2=1$。 +所以所得到的$\sigma$为一个圆割去一个扇形。 + \begin{minipage}{0.625\linewidth} - 所以所得到的$\sigma$为一个圆割去一个扇形。 交换积分次序后就需要以一个长度以极点为圆心做圆,切割$\sigma$。 由$\sigma$可知取长度$\sqrt{2}$可以切分。 + + 所以$\sigma$可以分为左边的$\sigma_1$和右边的$\sigma_2$。 \end{minipage} \hfill \begin{minipage}{0.25\linewidth} \begin{tikzpicture}[scale=1] + \draw [black, thick, smooth, domain=0:1,fill=gray!20] (1,-1) arc (-90:180:1); + \draw (0,0) -- (1,-1); + \draw (0,-1.4) arc (-90:90:1.4); \draw[-latex](-0.25,0) -- (2.5,0) node[below]{$x$}; \draw[-latex](0,-1.5) -- (0,1.5) node[above]{$y$}; \filldraw[black] (0,0) node[below]{$O$}; - \draw (1,-1) arc (-90:180:1); - \draw (0,0) -- (1,-1); - \draw (0,-1.4) arc (-90:90:1.4); + \filldraw (0.75,0) node{$\sigma_1$}; + \filldraw (1.65,0) node{$\sigma_2$}; \end{tikzpicture} \end{minipage} -所以$\sigma$可以分为左边的$\sigma_1$和右边的$\sigma_2$。 - $\sigma_1$的$r\in[0,\sqrt{2}]$,$\sigma_2$的$r\in[\sqrt{2},2]$。 $\sigma_1$的$\theta$下限是$y=-x$这条边,即$\theta=-\dfrac{\pi}{4}$,上限是$r=2\cos\theta$这个圆,则$\theta=\arccos\dfrac{r}{2}$。