推 jacky7987 :喔喔感謝你QAQ 09/10 14:23
※ 引述《jacky7987 (憶)》之銘言:
: Find∫ cos(ax+by+cz)dxdydz , where B is unit ball in |R^3 with center 0
: B
: a,b,c are constants
→
只要稍做變數變換就行了. (a,b,c)(≠ 0 )取成新的z'軸 , 以新的座標系
來看這個積分
d
let d = (a,b,c) , if ||d||≠0 , let v = ───── .
3 || d ||
3
Choose v , v so that { v , v , v } form a orthonomal basis in |R
1 2 1 2 3
┌ v ┐ ┌ ┐
│ 1 │ │ x │
│ │ │ │
let M = │ v │ , v =│ │
3x3 │ 2 │ │ y │
│ │ │ │
│ v │ │ │
│ 3 │ │ z │
└ ┘ └ ┘
3 3
Define a function g:|R -> |R by g(v) = Mv = v'
B : a Unit Ball at the origin . g(B) = B .
∫∫∫ cos (||d||*z') dx'dy'dz' = ∫∫∫cos(ax+by+bz)|Jg| dx dy dz
B B
|Jg(x,y,z)| = |detM | = 1 .
1 2
∫∫∫ cos (||d||*z') dx'dy'dz' = 2 ∫ π(1-z' ) cos (||d||*z') dz'
B 0
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◆ From: 61.224.202.55
※ 編輯: keroro321 來自: 61.224.202.55 (09/09 23:40)
※ 編輯: keroro321 來自: 61.224.202.55 (09/09 23:45)