Use the transformation $x = \sqrt { 2 } u - \sqrt { \frac { 2 } { 3 } } v , y = \sqrt { 2 } u + \sqrt { \frac { 2 } { 3 } } v$ to evaluate the integral $\iint _ { R } \left( x ^ { 2 } - x y + y ^ { 2 } \right) d A$, where $R$ is the region bounded by the ellipse $x ^ { 2 } - x y + y ^ { 2 } = 2$.

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The Answer of Use the transformation $x = \sqrt { 2 } u - \sqrt { \frac { 2 } { 3 } } v , y = \sqrt { 2 } u + \sqrt { \frac { 2 } { 3 } } v$ to evaluate the integral $\iint _ { R } \left( x ^ { 2 } - x y + y ^ { 2 } \right) d A$, where $R$ is the region bounded by the ellipse $x ^ { 2 } - x y + y ^ { 2 } = 2$.

Use the Transformation x=2u−23v,y=2u+23vx = \sqrt { 2 } u - \sqrt { \frac { 2 } { 3 } } v , y = \sqrt { 2 } u + \sqrt { \frac { 2 } { 3 } } vx=2​u−32​​v,y=2​u+32​​v

Use the Transformation $x = \sqrt { 2 } u - \sqrt { \frac { 2 } { 3 } } v , y = \sqrt { 2 } u + \sqrt { \frac { 2 } { 3 } } v$