From 8e4d76da144aacd7de3d5aca83a92cc84e1e67bf Mon Sep 17 00:00:00 2001 From: kuriv Date: Sun, 10 Nov 2024 18:45:20 +0800 Subject: [PATCH] New update --- .../6.\346\200\273\347\272\277.emmx" | Bin 0 -> 19005 bytes .../6.\346\200\273\347\272\277.png" | Bin 0 -> 107816 bytes .../6.\346\200\273\347\272\277/README.md" | 6 ++ ...\345\207\272\347\263\273\347\273\237.emmx" | Bin 0 -> 24638 bytes ...3\345\207\272\347\263\273\347\273\237.png" | Bin 0 -> 140286 bytes .../README.md" | 6 ++ .../README.md" | 20 +++++ .../README.md" | 82 ++++++++++++++++++ .../README.md" | 54 ++++++++++++ 9 files changed, 168 insertions(+) create mode 100644 "\350\200\203\347\240\224\344\270\223\344\270\232\350\257\276/4.\350\256\241\347\256\227\346\234\272\347\273\204\346\210\220\345\216\237\347\220\206/6.\346\200\273\347\272\277/6.\346\200\273\347\272\277.emmx" create mode 100644 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z=4VFplQ=-lu>BoZNI0d6Xt<>xYjiy*-Af&#a@~0hIgP5WmX_7H@~V68(8q3doY~{4 zydkyyOJ4Ft-YCr_gR(5*-Dv72gbEY1kG-&*s1gty#^e6+mYy{8Yz(4T^L^QChD`HQ zFdd1=wAQjq_0`W#N#Ag68Wuu=_;L8m(TD7kZbCdw`jn&h+9Yl?I|-AAPAeqJF+53m z`n)y#MNf|+F-8q5qXRLG1JC*PLltRgmL5{4k*vDE${&)`{IaqrR9mp)_4{x_QObok zUfjxv`c|w~Ru*r#I5#24Nm1%@RJ%#PAm_54x61r-lqRJd&wOkiOki);1BxVAT9ExZp^Ih=l!KQ@ z9SSb>%&pP;au+TldEKrCseouCvNQlsHlm{$Hze#*l5(CTkg4UF^7cs{L)p|r_aSLN zf$xj4mFl>+;76T#$cOlf&#vv?KD=Z~d+|2?yRVdfa`c;&%M~lzsPx5RM|C>+!Oj8~ zrlxU0x@OI}{hGw<8q7$k?oik20eB)qXjRmHjpX6z+w=QlU`HQwcK*lCn#t2cNE^j^ zkU0YE7k2}IGp78ysiut}BCM^g3z67NvJVMSfTSVm{cERVfq`5vQz>+;ppk&&X$Rwn zuy^QN{DA(FKA{j7)_DKw0S@{k6Ff|s%jp&)aq)hYTJk=D;rEjxDRY1e9An}@@DoG~ zEJs}8tG5~2eU`27JuD)4(@1oMKw*-NH(L>p*{uU_0)-HPB$7oJl5|354%Ea+U zME9M`R6~Q&*}CjP>>S**CMtH#cEx)PrAcYX?e2u6z?kD8{_78`YA)#fxx|M22j^)~ zyFu)}4Lb7>?P};T=H)jS69rdW5Tg&KndRb zJl1i&(7HtzG>eh?n(1y!YhQY{O)1PX`I?t0E{fsapGe_!6*z%m2FK0lD>Jhiwqnj@ z*>0wpnOfM7JPJwS<$VRJ+f=kRZ`OLw!>k>1IK;5>C^L!sxyWW2EHp2!Tf}&X4hYjm z)kD#Wiy^PUD#4vfu;8}@z;|5 zJ&`7#sSkGdqo2XzsYgTxF#gkWA5=zs*{<8lcOZ1m2 f{eMSR)y^%bbSRiM8%~TQ;bXMh#31+Y_80yMvq$~e literal 0 HcmV?d00001 diff --git "a/\350\200\203\347\240\224\344\270\223\344\270\232\350\257\276/4.\350\256\241\347\256\227\346\234\272\347\273\204\346\210\220\345\216\237\347\220\206/7.\350\276\223\345\205\245\350\276\223\345\207\272\347\263\273\347\273\237/README.md" "b/\350\200\203\347\240\224\344\270\223\344\270\232\350\257\276/4.\350\256\241\347\256\227\346\234\272\347\273\204\346\210\220\345\216\237\347\220\206/7.\350\276\223\345\205\245\350\276\223\345\207\272\347\263\273\347\273\237/README.md" new file mode 100644 index 0000000..11a934d --- /dev/null +++ "b/\350\200\203\347\240\224\344\270\223\344\270\232\350\257\276/4.\350\256\241\347\256\227\346\234\272\347\273\204\346\210\220\345\216\237\347\220\206/7.\350\276\223\345\205\245\350\276\223\345\207\272\347\263\273\347\273\237/README.md" @@ -0,0 +1,6 @@ +# 输入输出系统 + +![输入输出系统](7.输入输出系统.png) + + + diff --git "a/\350\200\203\347\240\224\346\225\260\345\255\246/7.\344\272\214\351\207\215\347\247\257\345\210\206/1.\345\237\272\346\234\254\346\200\247\350\264\250/README.md" "b/\350\200\203\347\240\224\346\225\260\345\255\246/7.\344\272\214\351\207\215\347\247\257\345\210\206/1.\345\237\272\346\234\254\346\200\247\350\264\250/README.md" index 9c0b907..7789039 100644 --- "a/\350\200\203\347\240\224\346\225\260\345\255\246/7.\344\272\214\351\207\215\347\247\257\345\210\206/1.\345\237\272\346\234\254\346\200\247\350\264\250/README.md" +++ "b/\350\200\203\347\240\224\346\225\260\345\255\246/7.\344\272\214\351\207\215\347\247\257\345\210\206/1.\345\237\272\346\234\254\346\200\247\350\264\250/README.md" @@ -35,3 +35,23 @@ $$ ## 经典例题 +$$ +如:设 D_t = \{(x , y|2{x ^ 2} + 3{y ^ 2} \le 6t)\} (t \ge 0) , f(x , y) = +\begin{cases} +\frac{\sqrt[3]{1 - (x ^ 2 + y ^ 2)} - 1}{e ^ {x ^ 2 + y ^ 2}} , (x , y) \neq (0 , 0) , +\\ +a , (x , y) = (0 , 0) , +\end{cases} +为连续函数,令 F(t) = \iint_{D_t}{f(x , y)}dxdy ,求 F_{+} ^ {\prime} (0) . +\\ +\because f(x , y) 为连续函数, +\\ +\therefore \lim_{(x , y) \to (0 , 0)} f(x , y) = \lim_{(x , y) \to (0 , 0)} \frac{\sqrt[3]{1 - (x ^ 2 + y ^ 2)} - 1}{e ^ {x ^ 2 + y ^ 2}} = \lim_{(x , y) \to (0 , 0)} \frac{-\frac{1}{3}(x ^ 2 + y ^ 2)}{x ^ 2 + y ^ 2} = -\frac{1}{3} = a , +\\ +\therefore F_{+} ^ {\prime} (0) = \lim_{t \to 0} \frac{F(t) - F(0)}{t - 0} = \lim_{t \to 0} \frac{F(t)}{t} = \lim_{t \to 0} \frac{\iint_{D_t}{f(x , y)}dxdy}{t} = \lim_{t \to 0} \frac{\sqrt{6}\pi{t}f(\xi , \eta)}{t} , 其中 (\xi , \eta) \in D_t , +\\ +\therefore \lim_{t \to 0} \frac{\sqrt{6}\pi{t}f(\xi , \eta)}{t} = -\frac{1}{3}\sqrt{6}\pi . +$$ + + + diff --git "a/\350\200\203\347\240\224\346\225\260\345\255\246/7.\344\272\214\351\207\215\347\247\257\345\210\206/2.\346\231\256\351\200\232\345\257\271\347\247\260\346\200\247\344\270\216\350\275\256\346\215\242\345\257\271\347\247\260\346\200\247/README.md" "b/\350\200\203\347\240\224\346\225\260\345\255\246/7.\344\272\214\351\207\215\347\247\257\345\210\206/2.\346\231\256\351\200\232\345\257\271\347\247\260\346\200\247\344\270\216\350\275\256\346\215\242\345\257\271\347\247\260\346\200\247/README.md" new file mode 100644 index 0000000..42789ed --- /dev/null +++ "b/\350\200\203\347\240\224\346\225\260\345\255\246/7.\344\272\214\351\207\215\347\247\257\345\210\206/2.\346\231\256\351\200\232\345\257\271\347\247\260\346\200\247\344\270\216\350\275\256\346\215\242\345\257\271\347\247\260\346\200\247/README.md" @@ -0,0 +1,82 @@ +# 普通对称性与轮换对称性 + +* [普通对称性](#普通对称性) +* [轮换对称性](#轮换对称性) +* [经典例题](#经典例题) + +## 普通对称性 + +$$ +若 D 关于 y 轴对称,则 \iint_D{f(x , y)}d\sigma = +\begin{cases} +2\iint_{D_1}{f(x , y)}d\sigma , f(x , y) = f(-x , y) , +\\ +0 , f(x , y) = -f(-x , y) . +\end{cases} +$$ + +$$ +若 D 关于 x = a (a \neq 0) 对称,则 \iint_D{f(x , y)}d\sigma = +\begin{cases} +2\iint_{D_1}{f(x , y)}d\sigma , f(x , y) = f(2a - x , y) , +\\ +0 , f(x , y) = -f(2a - x , y) . +\end{cases} +$$ + +$$ +若 D 关于 x 轴对称,则 \iint_D{f(x , y)}d\sigma = +\begin{cases} +2\iint_{D_1}{f(x , y)}d\sigma , f(x , y) = f(x , -y) , +\\ +0 , f(x , y) = -f(x , -y) . +\end{cases} +$$ + +$$ +若 D 关于 y = a (a \neq 0) 对称,则 \iint_D{f(x , y)}d\sigma = +\begin{cases} +2\iint_{D_1}{f(x , y)}d\sigma , f(x , y) = f(x , 2a - y) , +\\ +0 , f(x , y) = -f(x , 2a - y) . +\end{cases} +$$ + +$$ +若 D 关于原点对称,则 \iint_D{f(x , y)}d\sigma = +\begin{cases} +2\iint_{D_1}{f(x , y)}d\sigma , f(x , y) = f(-x , -y) , +\\ +0 , f(x , y) = -f(-x , -y) . +\end{cases} +$$ + +$$ +若 D 关于 y = x 对称,则 \iint_D{f(x , y)}d\sigma = +\begin{cases} +2\iint_{D_1}{f(x , y)}d\sigma , f(x , y) = f(y , x) , +\\ +0 , f(x , y) = -f(y , x) . +\end{cases} +$$ + +## 轮换对称性 + +$$ +在直角坐标系下,若把 x 和 y 对调后,区域 D 不变(或区域 D 关于 y = x 对称),则 \iint_D{f(x , y)}d\sigma = \iint_D{f(y , x)}d\sigma . +\\ +若 f(x , y) + f(y , x) = a ,则 I = \frac{1}{2}\iint_D{[f(x , y) + f(y , x)]}dxdy = \frac{1}{2}\iint_D{a}dxdy = \frac{a}{2}S_D . +$$ + +## 经典例题 + +$$ +如:设 f(x) = \iint_{D(x)}{\frac{v\ln{\sqrt{u ^ 2 + v ^ 2}}}{u + v}}dudv , D(x) = \{(u , v)|\frac{1}{4} \le u ^ 2 + v ^ 2 \le x ^ 2 , u \gt 0 , v \gt 0\} ,求 f(x) . +\\ +由轮换对称性可得, f(x) = \iint_{D(x)}{\frac{v\ln{\sqrt{u ^ 2 + v ^ 2}}}{u + v}}dudv = \iint_{D(x)}{\frac{u\ln{\sqrt{u ^ 2 + v ^ 2}}}{u + v}}dudv = \frac{1}{2}\iint_{D(x)}{\ln{\sqrt{u ^ 2 + v ^ 2}}}dudv , +\\ +\therefore f(x) = \frac{1}{4}\iint_{D(x)}{\ln{(u ^ 2 + v ^ 2)}}dudv . +$$ + + + diff --git "a/\350\200\203\347\240\224\346\225\260\345\255\246/7.\344\272\214\351\207\215\347\247\257\345\210\206/3.\344\272\214\351\207\215\347\247\257\345\210\206\350\256\241\347\256\227/README.md" "b/\350\200\203\347\240\224\346\225\260\345\255\246/7.\344\272\214\351\207\215\347\247\257\345\210\206/3.\344\272\214\351\207\215\347\247\257\345\210\206\350\256\241\347\256\227/README.md" new file mode 100644 index 0000000..87fb640 --- /dev/null +++ "b/\350\200\203\347\240\224\346\225\260\345\255\246/7.\344\272\214\351\207\215\347\247\257\345\210\206/3.\344\272\214\351\207\215\347\247\257\345\210\206\350\256\241\347\256\227/README.md" @@ -0,0 +1,54 @@ +# 二重积分计算 + +* [直角坐标系下计算](#直角坐标系下计算) +* [极坐标系下计算](#极坐标系下计算) +* [经典例题](#经典例题) + +## 直角坐标系下计算 + +$$ +\ +\ +\iint_D{f(x , y)}d\sigma = \int_{a} ^ {b}{dx}\int_{\phi_1(x)} ^ {\phi_2(x)}{f(x , y)}dy ,其中 D 为 X 型区域: \phi_1(x) \le y \le \phi_2(x) , a \le x \le b . +$$ + +$$ +\ +\ +\iint_D{f(x , y)}d\sigma = \int_{c} ^ {d}{dy}\int_{\psi_1(y)} ^ {\psi_2(y)}{f(x , y)}dx ,其中 D 为 Y 型区域: \psi_1(y) \le y \le \psi_2(y) , c \le x \le d . +$$ + +## 极坐标系下计算 + +$$ +\iint_D{f(x , y)}d\sigma = \int_{\alpha} ^ {\beta}d\theta\int_{r_1(\theta)} ^ {r_2(\theta)}{f(r\cos{\theta} , r\sin{\theta})}rdr (极点 O 在区域 D 外部) . +\\ +\iint_D{f(x , y)}d\sigma = \int_{\alpha} ^ {\beta}d\theta\int_{0} ^ {r(\theta)}{f(r\cos{\theta} , r\sin{\theta})}rdr (极点 O 在区域 D 边界上) . +\\ +\iint_D{f(x , y)}d\sigma = \int_{0} ^ {2\pi}d\theta\int_{0} ^ {r(\theta)}{f(r\cos{\theta} , r\sin{\theta})}rdr (极点 O 在区域 D 内部) . +$$ + +## 经典例题 + +$$ +如:当 x \to 0 ^ {+} 时, f(x) = \int_{0} ^ {x ^ 2}{dy}\int_{x} ^ {\sqrt{y}}{\sin{\frac{y}{t}}}dt 与 g(x) = a{x ^ b} 是等价无穷小量,求 ab . +\\ +\because \int_{0} ^ {x ^ 2}{dy}\int_{x} ^ {\sqrt{y}}{\sin{\frac{y}{t}}}dt = -\int_{0} ^ {x ^ 2}{dy}\int_{\sqrt{y}} ^ {x}{\sin{\frac{y}{t}}}dt = -\int_{0} ^ {x}{dt}\int_{0} ^ {t ^ 2}{\sin{\frac{y}{t}}}dy = \int_{0} ^ {x}{t \cdot \cos{\frac{y}{t}}|_{y = 0} ^ {y = t ^ 2}}dt = \int_{0} ^ {x}{t \cdot (\cos{t} - 1)}dt , +\\ +\because f(x) 与 g(x) 是等价无穷小量 , +\\ +\therefore \lim_{x \to 0 ^ {+}} \frac{\int_{0} ^ {x}{t \cdot (\cos{t} - 1)}dt}{a{x ^ b}} = \lim_{x \to 0 ^ {+}} \frac{x \cdot (\cos{x} - 1)}{ab{x ^ {b - 1}}} = \lim_{x \to 0 ^ {+}} \frac{-\frac{1}{2}x ^ 3}{ab{x ^ {b - 1}}} = 1 , +\\ +\therefore ab = -\frac{1}{2} . +$$ + +$$ +如:设区域 D = \{(x , y)|x ^ 2 + y ^ 2 \le \sqrt{2}\} ,求 \iint_D{(x ^ 2 + \frac{y ^ 2}{2})}dxdy . +\\ +由轮换对称性可得, I = \frac{1}{2}[\iint_D{(x ^ 2 + \frac{y ^ 2}{2})}dxdy + \iint_D{(y ^ 2 + \frac{x ^ 2}{2})}dxdy] = \frac{3}{4}\iint_D{(x ^ 2 + y ^ 2)}dxdy , +\\ +\therefore \frac{3}{4}\iint_D{(x ^ 2 + y ^ 2)}dxdy = \frac{3}{4}\int_{0} ^ {2\pi}d\theta\int_{0} ^ {\sqrt[4]{2}}{r ^ 3}dr = \frac{3}{4}\pi . +$$ + + +