text: By symmetry of second derivatives, for any "well-behaved" (non-pathological) function , we have
  Hence, in a simply-connected region R of the xy-plane, where  are independent, a differential form
  is an exact differential if and only if the equation
  holds. If it is an exact differential so  and , then  is a differentiable (smoothly continuous) function along  and , so . If  holds, then  and  are differentiable (again, smoothly continuous) functions along  and  respectively, and  is only the case.
  For three dimensions, in a simply-connected region R of the xyz-coordinate system, by a similar reason, a differential
  is an exact differential if and only if between the functions A, B and C there exist the relations
  These conditions are equivalent to the following sentence: If G is the graph of this vector valued function then for all tangent vectors X,Y of the surface G then s(X, Y) = 0 with s the symplectic form.
  These conditions, which are easy to generalize, arise from the independence of the order of differentiations in the calculation of the second derivatives. So, in order for a differential dQ, that is a function of four variables, to be an exact differential, there are six conditions (the combination ) to satisfy.