Discussion
Under a change of variables from \( (x,y) \) to \( (u,v) \), the area element transforms with a factor of
Answer: A
The rule is
\[ dx\,dy \;=\; \left|\frac{\partial(x,y)}{\partial(u,v)}\right| du\,dv, \]
the Jacobian of the *old* variables with respect to the new ones, and its modulus because an area is positive whichever way the map orients the plane. Using the reciprocal Jacobian scales the integral by the wrong factor
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