∫ (e^xsiny-my)dx+(e^xcosy-m)dy其中L是按逆时针方向从圆周(x-1)^2+y^2=1上点A(2,0)到点(0,0)的曲线积分πm/2

Question

∫ (e^xsiny-my)dx+(e^xcosy-m)dy其中L是按逆时针方向从圆周(x-1)^2+y^2=1上点A(2,0)到点(0,0)的曲线积分
πm/2

Answer 1

补上直线N：y = 0、使得半圆y = √[1 - (x - 1)²]与直线N围成闭区域.
P = e^xsiny - my、Q = e^xcosy - m
∂P/∂y = e^xcosy - m、∂Q/∂x = e^xcosy
∫_(L) (e^xsiny - my) dx + (e^xcosy - m) dy + ∫_(N) y dx
= ∫_(L) (e^xsiny - my) dx + (e^xcosy - m) dy + ∫_(N) (0) dx
= ∫∫_(D) (∂Q/∂x - ∂P/∂y) dxdy、D是y = √[1 - (x - 1)²]的面积
= ∫∫_(D) m dxdy
= m · D
= m · (1/2)π(1)²
= mπ/2