又是这问题哦哦
设闭曲线L由三条线段围成
L1：y = 0,dy = 0,x由0变化到a
L2：y = x,dy = dx,2x^2 = a^2 ==> x = a/√2,x由0变化到a/√2
L3：x^2 + y^2 = a^2,用参数方程化简,ds = √(x'²(t) + y'²(t)) dt = a dt,t由0变化到π/4
∮L e^(x^2 + y^2) ds
= ∫L1 e^(x^2 + y^2) ds + ∫L2 e^(x^2 + y^2) ds + ∫L3 e^(x^2 + y^2) ds
= ∫(0→a) e^x √(1 + 0) dx + ∫(0→a/√2) e^√(2x^2) √2 dx + ∫(0→π/4) e^a * a dt
= ∫(0→a) e^x dx + √2∫(0→a/√2) e^(√2 * x) dx + ae^a * π/4
= e^a - 1 + e^(√2 * x)：(0→a/√2) + (1/4)e^a * πa
= e^a - 1 + e^a - 1 + (πa/4)e^a
= (2 + πa/4)e^a - 2