∫L (x^2+y^2)dx+(x^2-y^2)dy L为从A(0,0) 至点B(1,1) 到点C(2,0)折线段
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∫L (x^2+y^2)dx+(x^2-y^2)dy L为从A(0,0) 至点B(1,1) 到点C(2,0)折线段
![∫L (x^2+y^2)dx+(x^2-y^2)dy L为从A(0,0) 至点B(1,1) 到点C(2,0)折线段](/uploads/image/z/4253347-19-7.jpg?t=%E2%88%ABL+%28x%5E2%2By%5E2%29dx%2B%28x%5E2-y%5E2%29dy+L%E4%B8%BA%E4%BB%8EA%280%2C0%29+%E8%87%B3%E7%82%B9B%281%2C1%29+%E5%88%B0%E7%82%B9C%282%2C0%29%E6%8A%98%E7%BA%BF%E6%AE%B5)
普通方法:
L1:y = x、dy = dx
L2:y = 2 - x、dy = - dx
∫L (x² + y²) dx + (x² - y²) dy
= ∫(0→1) 2x² dx + ∫(1→2) [x² + (2 - x)² + (x² - (2 - x)²)(- 1)] dx
= ∫(0→1) 2x² dx + ∫(1→2) 2(x - 2)² dx
= 2/3 + 2/3
= 4/3
格林公式:
补上线段N:y = 0、dy = 0、逆时针、使L围成闭区域D
P = x² + y²、P'y = 2y
Q = x² - y²、Q'x = 2x
∮L (x² + y²) dx + (x² - y²) dy
= ∫∫D (2x - 2y) dxdy
= 2∫(0→1) dy ∫(y→2 - y) (y - x) dx
= 4/3
∫N (x² + y²) dx + (x² - y²) dy = ∫(0→2) x² dx = 8/3
- I(L) + I(N) = ∮(L)
L1:y = x、dy = dx
L2:y = 2 - x、dy = - dx
∫L (x² + y²) dx + (x² - y²) dy
= ∫(0→1) 2x² dx + ∫(1→2) [x² + (2 - x)² + (x² - (2 - x)²)(- 1)] dx
= ∫(0→1) 2x² dx + ∫(1→2) 2(x - 2)² dx
= 2/3 + 2/3
= 4/3
格林公式:
补上线段N:y = 0、dy = 0、逆时针、使L围成闭区域D
P = x² + y²、P'y = 2y
Q = x² - y²、Q'x = 2x
∮L (x² + y²) dx + (x² - y²) dy
= ∫∫D (2x - 2y) dxdy
= 2∫(0→1) dy ∫(y→2 - y) (y - x) dx
= 4/3
∫N (x² + y²) dx + (x² - y²) dy = ∫(0→2) x² dx = 8/3
- I(L) + I(N) = ∮(L)
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