Suppose is a vector field which is defined and with continuous partial derivatives for all points, then
Note that if is not defined and differentiable everywhere, it is possible to have a nonconservative field with .
Proof in 2D
Both direction of this theorem can be proved by the Green’s theorem.
Part 1:
This is a consequence of the Green’s theorem. If is conservative, then
for all simple closed curve . Then Green’s theorem says
The only way for the area integral of to be 0 over all regions is if .
Part 2:
For the converse, if everywhere, then by Green’s theorem,
for all simple closed curve .