大师作文网曲线积分求解,如下图
来源：学生作业帮 编辑：大师作文网作业帮 分类：数学作业 时间：2024/11/12 10:08:39
曲线积分求解,如下图
{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