Find the Jacobian of the transformation $x = u \sin v$ , $y = u \cos v$ .
A) $- u$

B)
$- u ^ { 2 }$
C)u
D) $u ^ { 2 }$
E) $- u ^ { 2 } \sin v \cos v$
F) $u ^ { 2 } \sin v \cos v$
G) $- 2 u ^ { 2 } \sin v \cos v$
H) $2 u ^ { 2 } \sin v \cos v$

Question

Find the Jacobian of the transformation $x = u \sin v$ , $y = u \cos v$ .
A) $- u$ 
 
B)
$- u ^ { 2 }$ 
C)u
D) $u ^ { 2 }$
E) $- u ^ { 2 } \sin v \cos v$
F) $u ^ { 2 } \sin v \cos v$ 
G) $- 2 u ^ { 2 } \sin v \cos v$  
H) $2 u ^ { 2 } \sin v \cos v$

Quizplus · Accepted Answer

The Jacobian of the transformation given by $x = u \sin v$ and $y = u \cos v$ is calculated as the determinant of the matrix formed by the partial derivatives of $x$ and $y$ with respect to $u$ and $v$. This gives:$$\begin{vmatrix}\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}\end{vmatrix}=\begin{vmatrix}\sin v & u \cos v \\cos v & -u \sin v\end{vmatrix}= (-u \sin^2 v) - (u \cos^2 v) = -u (\sin^2 v + \cos^2 v) = -u$$Hence, the correct answer is $-u$.

Find the Jacobian of the Transformation x=usin⁡vx = u \sin vx=usinv

Find the Jacobian of the Transformation $x = u \sin v$