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?