∫Le^xcosydy+e^xsinydx,L从O(0,0)沿摆线X=a(t-sint),y=a(1-cost)到A(πa,2a)

Question

∫Le^xcosydy+e^xsinydx,L从O(0,0)沿摆线X=a(t-sint),y=a(1-cost)到A(πa,2a)

Answer 1

∂P/∂y = ∂/∂y (e^x siny) = e^x cosy
∂Q/∂x = ∂/∂x (e^x cosy) = e^x cosy
即∂P/∂y = ∂Q/∂x,积分结果与路径L无关.
选择最简单的积分折线：
L1：y = 0,0 ≤ x ≤ πa
L2：x = πa,0 ≤ y ≤ 2a
∫L e^x cosy dy + e^x siny dx
= ∫(0→πa) 0 dx + ∫(0→2a) e^(πa) cosy dy
= e^(πa) * [siny]：(0,2a)
= e^(πa) sin(2a)