Find the Jacobian for the change of variables given below.
$x = - 4 u - 4 v ^ { 2 } , y = 4 u + 5 v$
A) $- 20 + 32 v$
B) $- 20 v + 36$
C) $32 - 19 v$
D) $- 20 + 32 v ^ { 2 }$
E) $36 + 19 v$

Question

Find the Jacobian for the change of variables given below. 
$x = - 4 u - 4 v ^ { 2 } , y = 4 u + 5 v$
A) $- 20 + 32 v$
B) $- 20 v + 36$
C) $32 - 19 v$
D) $- 20 + 32 v ^ { 2 }$
E) $36 + 19 v$

Quizplus · Accepted Answer

The Jacobian of the transformation from variables $u, v$ to $x, y$ is given by the determinant of the matrix formed by the partial derivatives of $x$ and $y$ with respect to $u$ and $v$. Thus, we have:$$J = \begin{vmatrix}\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}\end{vmatrix}= \begin{vmatrix}-4 & -8v \4 & 5\end{vmatrix}= (-4)(5) - (-8v)(4) = -20 + 32v$$

Find the Jacobian for the Change of Variables Given Below x=−4u−4v2,y=4u+5vx = - 4 u - 4 v ^ { 2 } , y = 4 u + 5 vx=−4u−4v2,y=4u+5v

Find the Jacobian for the Change of Variables Given Below $x = - 4 u - 4 v ^ { 2 } , y = 4 u + 5 v$