For the following problem, I'm suppose to Stokes Theorem and then line integral but I'm getting different results.
Let S be the triangular region with vertices A(0,0,0), B(0,1,2), C(2,2,6). Let C = oriented counterclockwise as seen from above. Let F= (y,-x,y). Find
First, I find the equation of the plane and that comes out to z=x+2y.
Then when I parametrize the surface, I get S(x,y) = (x,y,x+2y).
Because it's counterclockwise, I need Sx X Sy = (-1,-2,1).
The curl of F is (1,0,-2)
By Stokes Theorem:
= -3 Area of D
By Line Integral:
Vector AC: for
Vector CB: for
Vector BA: for
Keeping in mind that F= (y,-x,y):
With Stokes theorem, I get -3. With line integral, I get 3. I should be getting the same results though.