We work in three dimensions, with similar definitions holding in
any other number of dimensions. In three dimensions, a form of
the type is called a differential form. This form is called
exact on a domain in space if there exists some scalar function
defined on such that throughout D. This is equivalent to saying
that the vector field is a conservative vector field, with
corresponding potential . In one dimension, a differential form
is exact as long as has an antiderivative; in this case let be
the antiderivative of . Otherwise, if does not have an
antiderivative, we cannot write and so the differential form is
inexact. Two and three dimensions Hence, it follows that in a
simply-connected region R of the xy-plane, a differential is an
exact differential if and only if the following holds: For three
dimensions, a differential is an exact differential in a simply-
connected region R of the xyz-coordinate system if between the
functions A, B and C there exist the relations: - ; ; - Note:
The subscripts outside