,
then by the definition of F,
and
have continuous first-order partial derivatives whenever ( x, y )
( 0, 0).
Further
.
Show that 
for every simple closed path that does not pass through or enclose the origin.
Solution: If ,
then by the definition of F,
and
have continuous first-order partial derivatives whenever ( x, y )
( 0, 0).
Further
So F is conservative on any simply connected domain D ,
that does not contain the origin.
Thus if C is any simple closed path in D, then
by C.A.H.