Question

高等数学～计算有向曲线积分

Answer 1

答：h³/3

 

这个有点复杂，我用软件化简过来的

x = acost，dx/dt = - asint

y = asint，dy/dt = acost

z = ht/2π，dz/dt = h/2π

t：0->2π

x²-yz = a²cos²t-ahtsint/2π

y²-xz = a²sin²t-ahtcost/2π

z²-xy = h²t²/4π²-a²sintcost

(x²-yz)dx + (y²-xz)dy + (z²-xy)dz

= h³t²/8π³ - a²htcos²t/2π - a²hsintcost/2π - a³sintcos²t + a²htsin²t/2π + a³sin²tcost

取不定积分得

h³t³/24π³ + a³cost/4 + a²hcos²t/4π - a²hcos2t/8π + a³cos3t/12 + a³sint/4 - a²htsin2t/4π - a³sin3t/12

再取0到2π的积分得

h³/3

所以∮_(Γ) (x²-yz)dx + (y²-xz)dy + (z²-xy)dz = h³/3