"Is the converse true, i.e., if is such that for given points the integral is independent of the path joining to , is conservative?"
Thus, given a vector-field with the above property, one would like to construct a potential function for it, i.e., try to extend equation to line integrals. Recall that, for a function of one variable, to construct an antiderivative in , we simply defined it to be

In the present situation, given a point , we would like to define
                                                 ---------(40)
where is a curve in joining to any arbitrary point .