VC4.6

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Let the given curve be C and let A(0,1,1) and B(1,0,1)be two points on curve C.

Therefore,\int _{{C}}F\cdot \,dr=\int _{{A}}^{{B}}[(2x+yz)i+xzj+(xy+2z)k]\cdot (dxi+dyj+dzk)\,

=\int _{{A}}^{{B}}[(2x+yz)dx+xzdy+(xy+2z)dz]\, --(1)

While moving from A to B on the curve C,x varies from 0 to 1,y varies from 1 to 0 and z remains constant.Since z=1,so dz=0.Hence (1) reduces to

\int _{{C}}F\cdot \,dr=\int _{{0}}^{{1}}(2x+y)dx+\int _{{1}}^{{0}}xdy+0\,

=\int _{{0}}^{{1}}[2x+{\sqrt  {1-x^{2}}}]dx-\int _{{0}}^{{1}}{\sqrt  {1-y^{2}}}dy\,

=2\int _{{0}}^{{1}}xdx+\int _{{0}}^{{1}}{\sqrt  {(1-x^{2})}}dx-\int _{{0}}^{{1}}{\sqrt  {(1-x^{2})}}dx=2[{\frac  {x^{2}}{2}}]_{{0}}^{{1}}=1\,

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