Let $x = r \cos \theta$ and $y = r \sin \theta$ Then $\frac { \partial ( x , y ) } { \partial ( r , \theta ) }$ is

A) $2$
B) $r ^ { 2 }$
C) $r$
D) $2 r ^ { 2 }$
E) $\frac { r ^ { 2 } } { 2 }$

Question

Let $x = r \cos \theta$ and $y = r \sin \theta$ Then $\frac { \partial ( x , y ) } { \partial ( r , \theta ) }$ is

A) $2$
B) $r ^ { 2 }$
C) $r$
D) $2 r ^ { 2 }$
E) $\frac { r ^ { 2 } } { 2 }$

Quizplus · Accepted Answer

The Jacobian determinant $\frac { \partial ( x , y ) } { \partial ( r , \theta ) }$ is calculated as $\begin{vmatrix} \frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta} \ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta} \end{vmatrix}$. Substituting $x = r \cos \theta$ and $y = r \sin \theta$, we get $\begin{vmatrix} \cos \theta & -r \sin \theta \ \sin \theta & r \cos \theta \end{vmatrix}$, which equals $r(\cos^2 \theta + \sin^2 \theta) = r$.

Let x=rcos⁡θx = r \cos \thetax=rcosθ And y=rsin⁡θy = r \sin \thetay=rsinθ

Let $x = r \cos \theta$ And $y = r \sin \theta$