曲线积分求解,如下图
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曲线积分求解,如下图
![曲线积分求解,如下图](/uploads/image/z/19392090-42-0.jpg?t=%E6%9B%B2%E7%BA%BF%E7%A7%AF%E5%88%86%E6%B1%82%E8%A7%A3%2C%E5%A6%82%E4%B8%8B%E5%9B%BE)
{x = 1 + √2cost,dx/dt = - √2sint
{y = √2sint,dy/dt = √2cost
0 ≤ t ≤ 2π
∮_L (ydx - xdy)/[2(x² + y²)]
= (1/2)∫(0,2π) [(√2sint)(- √2sint) - (1 + √2cost)(√2cost)]/[(1 + √2cost)² + (√2sint)²] dt
= (1/2)∫(0,2π) [1/(- 4√2cost - 6) - 1/2] dt
= (1/2)(- π) - (1/4)(2π)
= - π
{y = √2sint,dy/dt = √2cost
0 ≤ t ≤ 2π
∮_L (ydx - xdy)/[2(x² + y²)]
= (1/2)∫(0,2π) [(√2sint)(- √2sint) - (1 + √2cost)(√2cost)]/[(1 + √2cost)² + (√2sint)²] dt
= (1/2)∫(0,2π) [1/(- 4√2cost - 6) - 1/2] dt
= (1/2)(- π) - (1/4)(2π)
= - π