计算曲线积分I=∫L ydx-xdy/x^2+y^2 其中L：(x-1)^2+(y-1)^2=1(逆

参考答案：

用格林公式：奇点(0,0)不在积分域内.

I = ∮L (ydx - xdy)/(x^2 + y^2)

= ∫∫D [(x^2 - y^2)/(x^2 + y^2)^2 - (x^2 - y^2)/(x^2 + y^2)^2 dxdy

= 0

用参数方程.

{ x = 1 + cost、dx = - sint dt

{ y = 1 + sint、dy = cost dt

0 ≤ t ≤ 2π

∮L (ydx - xdy)/(x^2 + y^2)

= ∫(0→2π) [(1 + sint)(- sint) - (1 + cost)(cost)/[(1 + cost)^2 + (1 + sint)^2 dt

= - ∫(0→2π) (sint + cost + 1)/(2sint + 2cost + 3) dt

令u = tan(t/2)、dt = 2/(1 + u^2) du,sint = 2u/(1 + u^2)、cost = (1 - u^2)/(1 + u^2)

∫ (sint + cost + 1)/(2sint + 2cost + 3) dt

= ∫ [2u/(1 + u^2) + (1 - u^2)/(1 + u^2) + 1/[2 * 2u/(1 + u^2) + 2 * (1 - u^2)/(1 + u^2) + 3 * 2/(1 + u^2) du

= 4∫ (u + 1)/[(u^2 + 1)(u^2 + 4u + 5) du

= ∫ du/(u^2 + 1) + ∫ du/(u^2 + 4u + 5)

= ∫ du/(u^2 + 1) + ∫ du/[(u + 2)^2 + 1

= arctan(u) + arctan(u + 2) + C

= arctan[tan(t/2) + arctan[2 + tan(t/2) + C

于是I = - arctan[tan(t/2) - arctan[2 + tan(t/2)：(0→2π)

将区间分为：0→π⁻,π⁺→2π

I = (- π/2 - π/2) - (- π/2 - π/2)

= 0