See Exact differential for a list of mathematical relationships.

The statement of the first law, using exact differentials is now:

In other words, for a holonomic process function, may be defined such that is an exact differential.

Those that can are referred to as exact differentials.

Only the internal energy is an exact differential.

The symbol for exact differentials is the lowercase letter d.

The quantity is an exact differential, while is not.

Suppose that the state space is two dimensional and any of the five quantities are exact differentials.

Exact differential (has another derivation of the triple product rule)

For an idealized continuous process, this means that incremental changes in such variables are exact differentials.

By way of notation, we will specify the use of d to denote an exact differential.

This is especially useful in thermodynamics where temperature becomes the integrating factor that makes entropy an exact differential.

So, in order for a differential 'dQ', that is a function of four variables to be an exact differential, there are six conditions to satisfy.

(See also Bridgman's thermodynamic equations for the use of exact differentials in the theory of thermodynamic equations)

It is contrasted with the concept of the exact differential in calculus, which can be expressed as the gradient of another function and is therefore path independent.

For example, thermodynamic work is a holonomic process function since the integrating factor (where p is pressure) will yield exact differential of the volume state function .

An exact differential is sometimes also called a 'total differential', or a 'full differential', or, in the study of differential geometry, it is termed an exact form.

However, when divided by the absolute temperature and when the exchange occurs at reversible conditions (therefore the subscript), it produces an exact differential: the entropy S is also a state function.

If the initial and final states are the same, then the integral of an inexact differential may or may not be zero, but the integral of an exact differential will always be zero.

Since the entropy is an exact differential, using the chain rule, the change in entropy when going from a reference state 0 to some other state with entropy S may be written as where:

It follows from the properties of an exact differential (see equation 7 in the exact differential article) and from the energy/entropy equation of state that, for a closed system:

The integral of an inexact differential depends upon the particular path taken through the space of thermodynamic parameters while the integral of an exact differential depends only upon the initial and final states.

The letter d indicates an exact differential, expressing that internal energy U is a property of the state of the system; they depend only on the original state and the final state, and not upon the path taken.

Therefore, the sum of exchanged heat and work is an exact differential (dU), but since they are equivalent and the lack of one can be compensated by the presence of the other, singularly they are inexact differentials.

For a homogeneous system, with a well-defined temperature and pressure, the expression for dU can be written in terms of exact differentials, if the work that the system does is equal to its pressure times the infinitesimal increase in its volume.