VC5.4

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As per the Green's theorm,\oint _{C}[(3x^{2}-8y^{2})dx+(4y-6xy)dy]\,

=\iint _{R}[{\frac  {\partial }{\partial x}}(4y-6xy)-{\frac  {\partial }{\partial }}(3x^{2}-8y^{2})]dxdy\, --(1)

RHS of (1)=\iint _{R}(-6y+16y)dxdy=10\int _{{x=0}}^{{1}}\int _{{y=0}}^{{1-x}}ydxdy\,

=10\int _{0}^{1}[{\frac  {y^{2}}{2}}]_{{0}}^{{1-x}}dx=5\int _{0}^{1}(1-x)^{2}dx\,

=5[{\frac  {(x-1)^{3}}{3}}]_{0}^{1}={\frac  {5}{3}}[0-(-1)^{3}]={\frac  {5}{3}}\, --(2)

To evaluate LHS of (1),along O(0,0),A(1,0),y=0,dy=0,x varies from 0 to 1.

Along A(1,0),B(0,1),x=1-y,dx=-dy and y varies from 0 to 1.

Along B(0,1),O(0,0),x=0,dx=0 and y varies from 1 to 0.

Therefore,LHS =\int _{{OA}}[(3x^{2}-8y^{2})dx+(4y-6xy)dy]+\int _{{AB}}[(3x^{2}-8y^{2})dx+(4y-6xy)dy]+\int _{{BO}}[(3x^{2}-8y^{2})dx+(4y-6xy)dy]\,

=\int _{0}^{1}3x^{2}dx+\int _{0}^{1}[(3(1-y)^{2}-8y^{2})(-dy)+(4y-6y(1-y))]dy+2[y^{2}]_{0}^{1}\,

=1+\int _{0}^{1}(11y^{2}+4y-3)dy-2=-1+[{\frac  {11y^{3}}{3}}+2y^{2}-3y]_{0}^{1}\,

=-1+{\frac  {11}{3}}+2-3={\frac  {5}{3}}\, --(3)

From (2) and (3),LHS = RHS,Green's theorm is verified.

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