Past exam of the mathematical methods II course of the University of Oxford / 2016 resit / 9 / a / Solution

For a change of variables in a double integral, first solve for $x=x(u,v)$ and $y=y(u,v)$ and transform every boundary inequality to identify the new integration region $S$. Compute the Jacobian determinant, replace the integrand by its expression in $u,v$, and useA sufficient standard hypothesis is a continuously differentiable one-to-one transformation with nonzero Jacobian determinant in the interior, with harmless measure-zero boundaries. If one coordinate chart does not cover the region injectively, split into suitable pieces rather than silently counting points multiple times. The absolute value is essential for an ordinary area integral.