An exact differential is a differential equation , e.g. of two variables, of the form , where . For example, the total differential of , where is a constant, is . Comparing this with the general form, 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

For the first differential, we have . For the second differential, if we integrate with respect to , we have

Since and are independent, and we have , where is only a constant with respect to . Therefore, we can rewrite as

If we integrate the differential with respect to , we have

Clearly, , for all and . Therefore, the second differential does not integrate directly 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.