Our vector field is
Clear[vf, x, y, reg, bnd];
vf = {x y - x^2, x^2 y};
Our region and its boundary are:
reg = Polygon[{{0, 0}, {1, 0}, {1, 1}}];
bnd = RegionBoundary[r]
(*Line[{{0, 0}, {1, 0}, {1, 1}, {0, 0}}]*)
Left hand side of Green theorem is line integral over boundary.
We need to break boundary in separate Lines, find integrals and Total:
Total[Table[
Integrate[
vf . (#2 - #1) & @@seg
, {x, y} \[Element] Line@seg]
, {seg, Partition[First@bnd, 2, 1]}]]
(*-(1/12)*)
Right hand side of Green theorem is 2D integral of Curl over region:
Integrate[Curl[vf, {x, y}], {x, y} \[Element] reg]
(*-(1/12)*)
