VC5.1

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The coordinates of the four vertices of the rectangle are (0,0),(a,0),(a,b) and (0,b). Let R be the plane region bounded by C.

Therefore,\int _{C}F\cdot dr=\int _{C}[(x^{2}-y^{2})i+2xyj]\cdot (dxi+dyj)\,

=\int _{C}[(x^{2}-y^{2})dx+2xydy]\,

=\iint _{R}[{\frac  {\partial }{\partial x}}(2xy)-{\frac  {\partial }{\partial y}}(x^{2}-y^{2})]dxdy\,,By using Green's theorm.

=\iint _{R}(4y)dxdy=4\int _{{x=0}}^{{a}}\int _{{y=0}}^{{b}}ydxdy\,

=4\int _{o}^{a}[{\frac  {y^{2}}{2}}]_{{0}}^{{b}}dx=2b^{2}\int _{o}^{a}dx=2ab^{2}\,

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