Question

Consider the vector field Bold Upper Fequalsleft angle y minus x comma x right angle and curve​ C: Bold r (t )equalsleft angle 9 cosine t comma 9 sine t right angle for 0 less than or equals t less than or equals 2 pi. a. Based on the​ picture, make a conjecture about whether the circulation of Bold Upper F on C is​ positive, negative, or zero. b. Compute the circulation.

Answer 1

The circulation of F is 0.

Explanation:

Consider the provided vector field.

Some times F tends to clockwise and some time F tends to anti clock wise, Therefore the circulation of F is 0.

It is given that
`F=\left \langle y-x,x\right \rangle` and
`r(t)=\left \langle-9\text{cos} t , 9\text{sin}t\right \rangle`

`F(t)=\left \langle 9\sin t+9\cos t,-9\cos t\right \rangle`

`dr=\left \langle 9\sin t+9\cos t\right \rangle`

`F(t)\cdot dr=\left \langle 9\sin t+9\cos t,-9\cos t,\right \rangle \cdot \left \langle 9\sin t+9\cos t\right \rangle`

`F(t)\cdot dr=81\sin^2t+81\sin t\cos t-81\cos^2t`

`F(t)\cdot dr=-81\cos2t+(81)/(2)\sin2 t`

`\int\limits^a_b {F\cdot \, dr=\int\limits^(2\pi)_0 {-81\cos2t+(81)/(2)\sin2 t} \, dt`

`=[(-81\sin2t)/(2)-(81cos2t)/(4)]^(2\pi)_0\\=(-81)/(4)+(81)/(4)=0`

Hence the circulation of F is 0.