For the vector field ,F= iy+jx a) Evaluate the line integral F.dl along a closed

Question

For the vector field ,F= iy+jx a) Evaluate the line integral F.dl along a closed contour of a half-circle of radius a b)Verify Stoke's theorem for the surface bounded by the contour c)Can F be expressed as the gradient of a scalar?Explain why.

Explanation / Answer

c(t): x=sin(t), y=cos(t), z=t ==> dx=cos(t)*dt, dy=-sin(t)*dt, dz=1*dt=dt dl=(dx, dy, dz)=(cos(t)*dt, -sin(t)*dt, dt) F=(Fx, Fy, Fz)=(-2x, -2y, 2z)=(-2sin(t), -2cos(t), 2t) ?[c] F•dl = ?[c] Fx*dx+Fy*dy+Fz*dz = ?[0, 3p/2] [-2sin(t)*cos(t)+2cos(t)*sin(t)+2t] dt = ?[0, 3p/2] 2t dt = t² [t from 0 to 3p/2] = (3p/2)² =9p²/4

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