9. (10 pts) Let C be the curve in defined by the parametric equation
x(t)=cos(t),y(t)=sin(t),z(t)=t
for 0 ≤ t ≤ a. Suppose that the arc length of C is . Evaluate the line integal of the vector field F=

詳解 (共 1 筆)

助人為本

詳解 #7406382

2026/06/13

所以第一步先算弧長

弧長會等於x(t)^2+y(t)^2+z(t)^2等於r(t)^2

所以會等於根號2

然後根號2a會等於根號2pi/4

所以F(r(t))會等於(tcost,tsint,cos^3t)

微分後會等於

再把兩者做內積

前兩項抵銷

所以

ㅤㅤ