Question

Find the circulation and flux of the field Bold Upper F equals negative 3 x squared Bold i minus 3 y squared Bold j around and across the closed semicircular path that consists of the semicircular arch Bold Bold r 1 (t )equals (a cosine t )iplus (a sine t )j​, 0 less than or equals t less than or equals pi​, followed by the line segment Bold r 2 (t )equals t Bold i​, negative a less than or equals t less than or equals a.

Answer 1

Answer:0

Explanation:

CPdx+Qdy=C1Pdx+Qdy+C2Pdx+Qdy=circulation

C1:x=acost,dx=−asintdty=asint,dy=acostdt.C2:x=t,dx=dt,y=0,dy=0.

P=−y=−asint.Q=x=acost

circulation=π0[−asint(−asint)+acost(acost)]dt+a−atdt=a2π0dt+t22a−a=πa*2

Flux=CPdy−Qdx=C1Pdy−Qdx+C2Pdy−Qdx=π0[−asint(acost)−acost(−asint)]dt+a−a0dt=0