Vector Calculus · Superchallenge problem

The map $(u, v)\mapsto(x, y) = (u^2 - v^2,\; 2uv)$ carries the boundary of the unit square $0\le u\le 1$, $0\le v\le 1$, traversed counterclockwise, to a closed curve $C$ in the $xy$-plane. Evaluate $\displaystyle\oint_C xy\,dy$.