Let be a vector field defined everywhere on the domain .
(a) Suppose that has a potential such that for . Show that
for any smooth path from a to in . Show further that necessarily on .
(b) State a condition for which ensures that implies is pathindependent.
(c) Compute the line integral for the vector field
where denotes the anti-clockwise path around the unit circle in the -plane. Compute and comment on your result in the light of (b).