VC4.12

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The given curve is r=\cos ti+\sin tj+tk\,

Hence its parametric equations are x=\cos t,y=\sin t,z=t\, --(1)

\int _{{C}}F\cdot ,dr=\int _{{C}}(zi+xj+yk)\cdot (dxi+dyj+dzk)=\int _{{C}}(zdx+xdy+ydz)\,

=\int _{{0}}^{{2\pi }}[z{\frac  {dx}{dt}}+x{\frac  {dy}{dt}}+y{\frac  {dz}{dt}}]\,dt\,

=\int _{{0}}^{{2\pi }}[t(-\sin t)+\cos t(\cos t)+\sin t]\,dt\,

=-\int _{{0}}^{{2\pi }}t\sin t\,dt+{\frac  {1}{2}}\int _{{0}}^{{2\pi }}(1+\cos 2t)\,dt+\int _{{0}}^{{2\pi }}\sin t\,dt\,

=-[t(-\cos t)_{{0}}^{{2\pi }}+(\sin t)_{{0}}^{{2\pi }}]+{\frac  {1}{2}}(2\pi )+[-\cos 2\pi +\cos 0]=2\pi +\pi =3\pi \,

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