VC4.20

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Since the integration is performed in xy-plane,z=0,so given value o f F in xy-plane becomes F=yi\,

Also,r=xi+yj,dr=dxi+dyj\,

The parametric equation of circle C,x^{2}+y^{2}=1\, are x=\cos \theta ,y=\sin \theta \,

By definition,the circulation of F along the circle C =\oint _{C}F\cdot \,dr=\oint _{C}(yi)\cdot (dxi+dyj)\,

=\oint _{C}ydx=\int _{{0}}^{{2\pi }}\sin \theta (-\sin \theta )\,d\theta \, [Since for the complete circle,theta varies from 0 to 2pi]

=-{\frac  {1}{2}}\int _{{0}}^{{2\pi }}(1-\cos 2\theta )\,d\theta =-{\frac  {1}{2}}[\theta -{\frac  {1}{2}}\sin 2\theta ]_{{0}}^{{2\pi }}=-\pi \,

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