Trace
shows the Integrate
it computes:
Trace[
SurfaceIntegrate[Curl[vf, {x, y}], {x, y} \[Element] reg],
_Integrate
]
(* Integrate[-x + 2 x y,
{x, y} \[Element] Line[{{0, 0}, {1, 0}, {1, 1}, {0, 0}}],
GenerateConditions -> Automatic] *)
The docs say (in dimension
$n$):
For solid (of dimension n) and bounded RegionQ
objects R
, take the surface to be the region boundary (RegionBoundary[R]
) and the normal orientation to be pointed outward.
So it's integrating the scalar field Curl[vf, {x, y}]
around the boundary of the "solid" (dimension 2 region in
${\Bbb R}^2$)/
Embed the problem in 3D and it works fine:
SurfaceIntegrate[
Curl[vf, {x, y}] {0, 0, 1},
{x, y, z} \[Element] Polygon[{{0, 0, 0}, {1, 0, 0}, {1, 1, 0}}]]
(* -1/12 *)