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?
Best Answer
$S$ here is any surface that has $C$ as its boundary, therefore it could be the triangular region made by the three points.
In order to make the dot product think about the geometrical interpretation of the surface integral.
It is just like a normal integral, but it is not on the usual axis/planes, but is on a given surface.
From that you can see that $d\vec S=\nabla dS=\vec n dS$ where $\vec n$ is the normal to the surface. Therefore, you last paragraph is wrong, what you need is to choose a proper surface and dot its normal.
In a general surface integral of a general vector field this process is as follows:$$\Phi=\int_S \vec V \cdot d\vec S=\int_S (\vec V \cdot \vec n)dS=\int_S V_x dydz+V_ydzdx+V_z dxdy$$
I believe you can go on from here.