[微積] Green Theorem
Let x = 6cost, y = 6sint, 0 ≦ t ≦ 2pi
x 3 x 3
試求線積分 ∮( e y - y ) dx + ( e + x )dy
我有疑問的是, 如果以 Green Theorem 下去求解,
令 M = (e^x)y - y^3
N = e^x + x^3
∫c [ 2y - e^(cosx) ] dx + ∫c { 5x + (y^4 + 2)^(1/2)} dy
partial (N) partial (M)
= ∫∫{ ------------- - ------------- } dA
Partial x partial y
( R: x^2 + y^2 < = 36)
= ∫∫( 3x^2 + 3y^2 ) dA
2π 6
= ∫ ∫ (3r^2)rdrdθ
0 0
6
= 2π(3/4)(r^4) |
0
= 1944π
這樣是得到正確的解答,
但是為什麼不能夠將( 3x^2 + 3y^2 )<= 3*36 = 108
代入逕行積分呢？
（這樣得出來答案會是兩倍…）
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◆ From: 114.46.115.17
推
03/23 00:42, , 1F
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03/23 00:42, , 2F
03/23 00:42, 2F
恩... 但是另外有一題
2 2 2
令 C 為平面 x - 2y + 2z = 9 與球面 x + y + z = 25 相交之圓
2 2 2
若 Vector F = ( -2z )i + ( x )j - ( 4y )k, 依順時針方向旋轉
求線積分 ∮F dot dr
curlF = ( -8 )i + ( -4z )j + ( 2x )k
unit vector n = (1/3)i + (-2/3)j + (2/3)k
ds = (3/2)dA
Then ∮F dot dr = ∫∫(curlF dot n) ds
= ∫∫(1/3)(4x-8y+8z) ds
= 12 ∫∫ds = 192pi
這邊卻又是直接代入?
※ 編輯: EggAche 來自: 114.46.115.17 (03/23 00:54)
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