The exact differential for a differentiable scalar function  defined in an open domain  is equal to , where  is the gradient of ,  represents the scalar product, and  is the general differential displacement vector, if an orthogonal coordinate system is used. If  is of differentiability class  (continuously differentiable), then  is a conservative vector field for the corresponding potential  by the definition. For three dimensional spaces, expressions such as  and  can be made.
The gradient theorem states
that does not depend on which integral path between the given path endpoints  and  is chosen. So it is concluded that the integral of an exact differential is independent of the choice of an integral path between given path endpoints (path independence).
For three dimensional spaces, if  defined on an open domain  is of differentiability class  (equivalently  is of ), then this integral path independence can also be proved by using the vector calculus identity  and the Stokes' theorem.
for a simply closed loop  with the smooth oriented surface  in it. If the open domain  is simply connected open space (roughly speaking, a single piece open space without a hole within it), then any irrotational vector field (defined as a  vector field  which curl is zero, i.e., ) has the path independence by the Stokes theorem, so the following statement is made; In a simply connected open region, any  vector field that has the path-independence property (so it is a conservative vector field.) must also be irrotational and vise versa.' The equality of the path independence and conservative vector fields is shown here.