Question

高等数学题，第六题，求大神指点迷津

Answer 1

功 W = ∮<C> [y/(x^2+y^2)]dx+[-x/(x^2+y^2)]dy
被积函数在 O(0, 0) 不连续，不能用格林公式，为此
以顺时针方向作小圆 C0: x^2+y^2 =ε^2,
C 与 C0 围成的闭区域内
∮<C> [y/(x^2+y^2)]dx+[-x/(x^2+y^2)]dy
+∮<C0> [y/(x^2+y^2)]dx+[-x/(x^2+y^2)]dy
被积函数连续，用格林公式为 0，
则 W = -∮<C0> [y/(x^2+y^2)]dx+[-x/(x^2+y^2)]dy
= -(1/ε^2)∮<C0> ydx+(-x)dy
= -(1/ε^2)∫∫<x^2+y^2≤ε^2> (-2)dxdy
= -(1/ε^2)(-2πε^2) = 2π