作者air11 (拿出一张白纸...)
看板Math
标题Re: [微积] Stokes' Theorem
时间Mon May 23 21:14:52 2011
※ 引述《alex800813 (L)》之铭言：
: use Stokes' Theorem to calculate ∫c [(x-z)dx+(x+y)dy+(y+z)dz]
: where C is the ellipse where plane z=y intersects the cylinder
: x^2+y^2=1 , oriented counterclockwise as viewed from above.
: 知道题目大概的意思 应该是要用curl F 下去解的吧..
: 但是我解出来有点复杂 希望有高手愿意帮忙
∫ Fdr = ∫∫ curlFdS
C S
S:h(r,θ) = (rcosθ,rsinθ,rsinθ) 0≦r≦1 and 0≦θ≦2pi
h_r X h_θ = (0,-r,r) (Note that "X" is cross )
By calculating, we get curlF = (1,-1,1)
by Stokes' thm, we have
2pi 1
∫ ∫ 2rdrdθ = 2pi*1=2pi
0 0
BTY, we check ans by line integral as below :
C:(cosθ,sinθ,sinθ) 0≦θ≦2pi
so ∫(x-z)dx+(x+y)dy+(y+z)dz
C
2pi
= ∫ (cosθ-sinθ)*(-sinθ)+(cosθ+sinθ)*(cosθ)+(2sinθ)*(cosθ)dθ
0
=2pi
#
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1F：推 alex800813 :非常谢谢你的帮忙! 我会了! 05/23 21:31
2F：推 peicachu :刚刚眼花，现在手残，curlF算成(1,1,1)XD 05/23 21:38
3F：推 alex800813 :突然有个地方想不通 他的h_r X h_θ是怎麽算出来的@@ 05/23 21:41
4F：推 peicachu :cylinder被z=y截面的法向量，取逆时钟方向 05/23 21:46
5F：推 peicachu :h_r的意思是对r偏微分，直接外积就好 05/23 22:06