[微積] 台大數研90高微考古題

看板Math作者 (Tasuku)時間14年前 (2011/12/11 14:28), 編輯推噓1(1推 0噓 21→)
留言22則, 3人參與, 最新討論串1/3 (看更多)
4.(b) Let a(n)>0, a(n+1)/a(n)<=(1-2/n) for n>=3. Show that the series of a(n) is convergent. 6. Prove that, for each integer n, there exists a C^2 function w=g(x,y) defined in some neighborhood of (0,0) such that x+2yw+cosw=0, g(0,0)=nπ+π/2. 請問有任何解題的方向嗎? 謝謝 -- ※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 134.208.26.13
推
12/11 15:44, , 1F
6 implicit function theorem
12/11 15:44, 1F
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12/11 16:00, , 2F
隱函數定理我有想過，不過不知道要怎麼使用
12/11 16:00, 2F
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12/11 16:01, , 3F
嚴格的寫下證明
12/11 16:01, 3F
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12/11 16:05, , 4F
4的不等號反了吧
12/11 16:05, 4F
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12/11 16:18, , 5F
F(x,y,w)=x+2yw+cosw=0 at (0,0,nπ+π/2)
12/11 16:18, 5F
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12/11 16:19, , 6F
所以可以把w寫成g(x,y)在一個neighborhood of (0,0)
12/11 16:19, 6F
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12/11 16:19, , 7F
然後對F(x,y,g(x,y))偏微 證明g是C^2
12/11 16:19, 7F
※ 編輯: tasukuchiyan 來自: 134.208.26.13 (12/11 17:40)
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12/11 19:17, , 8F
要怎麼對F(x,y,g(x,y))偏微，證明g是C^2呢?
12/11 19:17, 8F
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12/12 01:33, , 9F
F(x,y,g(x,y))在(0,0)的附近都是0 所以得到在0的附近
12/12 01:33, 9F
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12/12 01:34, , 10F
x+2yg(x,y)+cos(g(x,y))=0 對這個式子偏微後能夠把
12/12 01:34, 10F
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12/12 01:34, , 11F
g的一次微分提出來寫成x,y,g(x,y)的式子 然後再利用
12/12 01:34, 11F
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12/12 01:35, , 12F
g是C^1的性質
12/12 01:35, 12F
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12/12 12:09, , 13F
那g'不是從(0,0)的鄰域映射到L(R^2,R)的函數嗎?
12/12 12:09, 13F
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12/12 12:10, , 14F
怎麼用g是C^1的性質去證明g'可微分?
12/12 12:10, 14F
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12/12 13:30, , 15F
把w_x表成 w跟y,x的函數
12/12 13:30, 15F
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12/12 15:25, , 16F
If all mixed second order partial derivatives are
12/12 15:25, 16F
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12/12 15:26, , 17F
continuous at a point, f is termed a C^2 function
12/12 15:26, 17F
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12/12 15:26, , 18F
at that point.
12/12 15:26, 18F
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12/12 15:28, , 19F
是這一回事嗎?
12/12 15:28, 19F
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12/12 16:08, , 20F
yes
12/12 16:08, 20F
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12/12 16:27, , 21F
這句話要證明 雖然只是隱函數定理的一個推論
12/12 16:27, 21F
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12/13 11:05, , 22F
感謝所有回答的人
12/13 11:05, 22F
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