Inexact Differentials

In a previous post, I discussed the proper techniques to integrate an exact total differential. The major point to be drawn from exact differentials is that their parent function is independent of the path of integration. For example, the work done by gravity is independent of the path taken. It only depends on the end points of the path. This has to do with the fact that the gravitational force can be expressed as the gradient of a scalar. We call this type of force a conservative force field.

In general, many physical processes cannot be represented by conservative fields and therefore, their total differentials are inexact. One can think of the total differential as the a small increment taken on an arbitrary path. A very popular example of an inexact field is the work (and subsequently heat) in thermodynamics.

The work done by or on a system is in general dependent on the path taken. It is a summation of infinitesimal steps along the path. In contrast, the internal energy of the system is independent of the path taken. This has to do with the macrostates of a system. A macrostate of a system is a state where external parameters are specified. These include volume, temperature, pressure, mean total energy.

Then, for the mean energy U, the total differential is simply the difference between two known macrostates (remember, that the energy is specified for a macrostate). In contrast, the work done cannot, in general, be written as the difference between two well defined quantities. You can find more details on this in Prof. Richard Fitzpatrick's online textbook on thermodynamics.

So how do we integrate inexact differentials? Simple. If the path is known then the integration can be carried out along that path!

However, we will now show that if the inexact differential is multiplied by some function of the independent variables, one can construct an exact differential. To show this, I will follow the exposition given by Prof. Richard Fitzpatrick (http://farside.ph.utexas.edu/teaching/sm1/lectures/node36.html).

Consider the inexact differential equation

where I have used the symbol \delta to denote an inexact differential. An immediate consequence is that

Furthermore, the integral of F over a closed path is not equal to zero

To make further headway, let us consider the solution of

or

Dividing by H dx, we get

This equation describes the slope of some set of curves at every point in the x-y plane. These curves can be written as

where c is a constant labeling parameter. Think of this a set of controur lines for \Gamma. Note that Gamma is a function of (x,y), the constant on the RHS merely says that Gamma is constant on a given contour line. We now form the total differential of \Gamma

Now we want to connect the total differential of Gamma to the ratio dy/dx. To achieve this, we divide the previous equation by dx

upon substitution of dy/dx, we get

or

then

where sigma(x,y) is an arbitrary function of the independent variables. Then

Upon substitution into the original inexact differential, we have

therefore

and thus, by multiplying the inexact differential by a proper factor, one arrives at an exact differential. If this factor exists, it is called an integrating factor (its reciprocal in fact is the integrating factor). Such a factor may not exist in higher dimensions however.

In thermodynamics, for a reversible process, the entropy is written as

Note that the total differential of Q is inexact. But when dividing it by the temperature, one arrives to an exact differential. In this case, the temperature is an integrating factor and the total differential of entropy is exact.

Voila!

Cite as:
Saad, T. "Inexact Differentials". Weblog entry from Please Make A Note. http://pleasemakeanote.blogspot.com/2010/07/inexact-differentials.html