Discussion
Consider the transformation to a new set of coordinates \((\xi, \eta)\) from rectangular Cartesian coordinates \((x, y)\), where \(\xi = 2x + 3y\) and \(\eta = 3x - 2y\). In the \((\xi, \eta)\) coordinate system, the area element dxdy is
Answer: A
The area element is \(\frac{1}{13}\,d\xi\,d\eta\).
Areas transform with the Jacobian of the change of variables:
\[ \begin{aligned} \frac{\partial(\xi, \eta)}{\partial(x, y)} &= \begin{vmatrix} 2 & 3 \\ 3 & -2 \end{vmatrix} = -13 \\ d\xi\,d\eta &= 13\,dx\,dy \\ dx\,dy &= \tfrac{1}{13}\,d\xi\,d\eta \end{aligned} \]
Only the absolute value of the Jacobian matters for an area. The trap is to use 13 directly instead of its inverse; the options 5 and 3/5 come from adding coefficients instead of taking the determinant.
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