If a differentiable function  is one-to-one (injective) for each independent variable, e.g.,  is one-to-one for  at a fixed  while it is not necessarily one-to-one for , then the following total differentials exist because each independent variable is a differentiable function for the other variables, e.g., .
Substituting the first equation into the second and rearranging, we obtain
Since  and  are independent variables,  and  may be chosen without restriction. For this last equation to generally hold, the bracketed terms must be equal to zero. The left bracket equal to zero leads to the reciprocity relation while the right bracket equal to zero goes to the cyclic relation as shown below.