Let $x = u ^ { 3 } - 3 u v ^ { 2 }$ and $y = 3 u ^ { 2 } v - v ^ { 3 }$ Then $\frac { \partial ( x , y ) } { \partial ( u , v ) }$ is
A) $9 \left( u ^ { 2 } + v ^ { 2 } \right) ^ { 2 }$
B) $9 \left( u ^ { 2 } - v ^ { 2 } \right) ^ { 2 }$
C) $- 9 \left( u ^ { 2 } + v ^ { 2 } \right) ^ { 2 }$
D) $- 9 \left( u ^ { 2 } - v ^ { 2 } \right) ^ { 2 }$
E) $9 \left( u ^ { 2 } + v ^ { 2 } \right)$

Question

Quizplus · Accepted Answer

The Jacobian determinant $\frac { \partial ( x , y ) } { \partial ( u , v ) }$ is calculated by finding the determinant of the matrix formed by the partial derivatives of $x$ and $y$ with respect to $u$ and $v$. Calculating these derivatives and their determinant yields $9(u^2 + v^2)^2$.

Let x=u3−3uv2x = u ^ { 3 } - 3 u v ^ { 2 }x=u3−3uv2

Let $x = u ^ { 3 } - 3 u v ^ { 2 }$