Line integral in conservative vector field


c
x
2
ydx+(y+x
y
2
)dy

, where C is the region enclosed by
y=
x
2
y
and
x=
y
2
x
By
df
dy
=
dg
dx
d
test the function
(
x
2
y)i+(y+x
y
2
)
(
is not conservative vector field.
we know if
f(x,y)
f
and
g(x,y)
g
are continuous on some open connected region D, then
i)F(x,y)=f(x,y) i+ g(x,y) j is a conservative vector field on the region D.
ii)

c
F⋅dr=0

for every piecewise smooth curve C in D.
iii)

c
F⋅dr

is independent of path from any point P to Q in D for every piecewise smooth curve C in D.
These statements are all true or all false. Here it just goes to hell. Here the vector field is not independent of path and not conservative. But

c
F⋅dr=0

!!. So that means the vector field is conservative and independent of path.It gets contradictory. What is wrong with my understanding?

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