Exact Differetial Theorem

To give a clear representation of a state function, it is often helpful to investigate the two theorems on exact differentials.

Theorem 1:

If

Provided that M and N have continuous derivatives:

Proof: The total derivative of the function φ can be expressed in the form:

Comparison of the above two dφ equations can be expressed as:

Since the right-side of the last two equations are equal to each other, then:

Thus, the representation, M dx + N dy can be written as the differential of φ, i.e., dφ, often called an exact differential.

Theorem 2

A necessary and sufficient condition for M dx + N dy to be independent of the path C joining any two points A and B is that: