An exact differential is a differential equation , for instance of two variables, of the form , where .

Consider the function , where is a constant. Its total differential is . Comparing with the general form of an exact differential, we have and . Since , is an exact differential. The equality of the mixed partials implies that the change in is independent of the path taken.

Conversely, an inexact differential is a differential equation of the form , where . The change in , in this case, is dependent on the path taken.

###### Question

Show that is an inexact differential.

###### Answer

We have and . So, .

Another difference between an exact differential and an inexact differential is that an exact differential integrates directly to give the function , whereas an inexact differential does not.

###### Question

Using the integral criterion, show that is an exact differential, while is an inexact differential.

###### Answer

The integrated form of the first differential is evidently . The detailed analysis involves two steps. First, integrating with respect to , treating as a constant, yields

The function accounts for terms in involving or constants, which differentiate to zero when differentiating with respect to . Secondly, integrating with respect to , with as a constant, gives

Comparing eq17a and eq17b, we find , where . In other words, integrates directly from .

For the second differential, integrating with respect to yields , and integrating with respect to gives . Clearly, , for all and , indicating that the second differential does not directly integrate to give a function.

The fact that an exact differential integrates directly to give the function but an inexact differential does not, implies that and for a differentiable function must be of the appropriate forms of and respectively. In other words, the total differential of a differentiable function must be an exact differential.