Use the stroke theorem to evaluate
$$ \int_C{ \vec{F} \cdot \vec{dr}} $$ where C is oriented counterclockwise as viewed from above.
$$ \vec{F} = \langle x+y^2, y+z^2, z+x^2 \rangle $$
C is the triangle with vertices (3, 0, 0), (0, 3, 0), and (0, 0, 3).
Approach so far
I evaluated the curl of F, $$ curl \vec{F} = \langle -2z, -2x, -2y \rangle$$ Then I want to dot this with dS, but I'm not sure what dS is?
What is dS, or for that matter . What is S here.
Is it the triangular region (looked down toward XY plane) that would be bounded by line y= 3 -x and y = 0 ? If so, how do I describe S in order for it to be dotted?