Find the Jacobian of the transformation.
$$x = 5 \alpha \sin \beta , y = 4 \alpha \cos \beta$$
A)
$\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = 9 \alpha$
B)
$\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = 36 \alpha$
C)
$\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = - 20 \alpha$
D)
$\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = - 20 \alpha \sin \beta \cos \beta$
E)
$\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = - \alpha$

Question

Find the Jacobian of the transformation.
$$x = 5 \alpha \sin \beta , y = 4 \alpha \cos \beta$$
A) 
$\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = 9 \alpha$
B) 
$\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = 36 \alpha$
C)
$\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = - 20 \alpha$
D)
$\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = - 20 \alpha \sin \beta \cos \beta$
E)
$\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = - \alpha$

Quizplus · Accepted Answer

The Jacobian is calculated as the determinant of the matrix of partial derivatives: $\frac{\partial(x, y)}{\partial(\alpha, \beta)} = \begin{vmatrix} \frac{\partial x}{\partial \alpha} & \frac{\partial x}{\partial \beta} \ \frac{\partial y}{\partial \alpha} & \frac{\partial y}{\partial \beta} \end{vmatrix}$. Calculating these derivatives and the determinant gives $-20\alpha$.

Find the Jacobian of the Transformation x=5αsin⁡β,y=4αcos⁡βx = 5 \alpha \sin \beta , y = 4 \alpha \cos \betax=5αsinβ,y=4αcosβ

Find the Jacobian of the Transformation $x = 5 \alpha \sin \beta , y = 4 \alpha \cos \beta$