What is the result of applying Greens theorem to these vector fields?

[Larrytsai]

Homework Statement

Let C be the counter-clockwise planar circle with center at the origin and radius r > 0. Without computing them, determine F for the following vector fields whether the line integrals int(Fdr)
are positive negative or zero

F = xi + yj
F = -yi + xj
F = yi -xj
F= i + j

The Attempt at a Solution

I applies greens theorem

for the first force vector F = xi + yj
so I take the partial of Q wrt x and get 1 and p wrt y and get 1
and i get double zero

but when i do this for 2nd one F = -yi + xj and 3rd one, my answers seem to be opposite of what i got.

F = -yi + xj
partial Q / partial x = -1
partial P/ partial y = 1
so i would get a negative number. but answer is positive.

greens theorem = double integral (-2)dA

 

[Dick]

Homework Statement

Let C be the counter-clockwise planar circle with center at the origin and radius r > 0. Without computing them, determine F for the following vector fields whether the line integrals int(Fdr)
are positive negative or zero

F = xi + yj
F = -yi + xj
F = yi -xj
F= i + j

The Attempt at a Solution

I applies greens theorem

for the first force vector F = xi + yj
so I take the partial of Q wrt x and get 1 and p wrt y and get 1
and i get double zero

but when i do this for 2nd one F = -yi + xj and 3rd one, my answers seem to be opposite of what i got.

F = -yi + xj
partial Q / partial x = -1
partial P/ partial y = 1
so i would get a negative number. but answer is positive.

greens theorem = double integral (-2)dA

For F=(-yi)+xj, isn't P=(-y) and Q=x? How did you then get that the x derivative of Q is -1? You seem to be mixing up your P's and Q's.