Question

Consider the transformation to a new set of coordinates $(\xi,\eta)$ from rectangular coordinates $(x,y)$, where $\xi=2x+3y$ and $\eta=3x-2y$. In the $(\xi,\eta)$ coordinate system, the area element $dx\,dy$ is

• A. $\dfrac{1}{13}\, d\xi\, d\eta$
• B. $\dfrac{2}{13}\, d\xi\, d\eta$
• C. $5\, d\xi\, d\eta$
• D. $\dfrac{3}{5}\, d\xi\, d\eta$

Hint:

Always take the absolute value of the Jacobian when converting area or volume elements.

Answer 1

The Correct Option is A

Step 1: Compute the Jacobian.
$\xi = 2x + 3y, \eta = 3x - 2y.$ Jacobian determinant:
$\displaystyle J = \begin{vmatrix} \frac{\partial \xi}{\partial x} & \frac{\partial \xi}{\partial y} \\ \frac{\partial \eta}{\partial x} & \frac{\partial \eta}{\partial y} \end{vmatrix} = \begin{vmatrix} 2 & 3 \\ 3 & -2 \end{vmatrix} = (2)(-2) - (3)(3) = -4 - 9 = -13.$ \\

Step 2: Area element relation.
$dx\,dy = \dfrac{1}{|J|}\, d\xi\, d\eta = \dfrac{1}{13} d\xi\, d\eta.$ \\

Step 3: Conclusion.
Thus $dx\,dy = \dfrac{1}{13} d\xi d\eta$.