1. Properties of Gradient Fields
Theorem 5.8 (page 354 in Lovrić)

2. Properties of a Gradient Vector Field---Part I
Let F be a C1 vector field defined on an open, connected set U  2 or (3) . The following statements are equivalent:
F is a gradient vector field; F=f.
For any oriented, simple closed curve c,
F is path-independent: for any two oriented, simple curves c1 and c2 having the same initial an terminal points.

3. Properties of a Gradient Vector Field---Part II
Let F be a C1 vector field defined on an open, connected set U  2 or (3) .
F=f.
Integral of F around any oriented, simple closed curve c is 0.
F is path-independent.
Each of (a), (b), and (c) imply
Scalar curl of F is 0 (or curl F = 0 if U  3) .
If, in addition, U is simply connected all are equivalent.