Exact and inexact differentials

Thinking about distance

Consider the subtle difference between two kinds of distances:

  • $D\equiv$ 'distance travelled' from Goshen: The distance from Goshen to Elkhart is 11 miles. So, if I travel from Goshen to Elkhart and back again along the shortest path, $D=22$ miles at the end of my trip.
  • $R\equiv$ 'distance away' from Goshen: If I travel from Goshen to Elkhart and back again, $R=0$ at the end of my trip.

To do: Joe travels along old CR 17, from CR 40 to CR 36. Knowing that evenly numbered county roads are 1 mile from the next one, estimate:

  • The total distance that Joe travels, $\Delta D$.
  • The change in Joes "distance away" from Goshen, $\Delta R$.

Distances and differentials of distance

Using our two kinds of distances again...

  • $\Delta D=\int_{\cal P}\delta D$ depends on the path ${\cal P}$ travelled (or, we could say, on the history of how you get from one point to another),
  • $D$ is not very useful as a coordinate on a map.
  • $\Rightarrow \delta D$ is an inexact differential.
  • That's why I wrote the differential, $\delta D$, with the $\delta$ symbol in front of $D$.

On the other hand...

  • $\Delta R = \int_{\cal P} dR$ is also found by integrating along a particular path,
  • But the total change in "distance from Goshen", $\Delta R$ is the same for any path between two given points.
  • $R$ could be used as part of a coordinate system to locate you on a map.
  • $\Rightarrow dR$ is an exact differential.
  • That's why I wrote the differential, $dR$, with a '$d$' symbol in front of $R$

What is $\Delta R=\int_{\cal P}dR$ for a path, $\cal P$, that ends at the same point where it started?

Combinations of differentials

Let's say that you keep track of the miles you run, and the miles that you walk along a line--say for example, always staying on CR 40--like this:

  • $x_r \equiv$ 'miles run north along CR 40',
  • $x_w\equiv$ 'miles walked north along CR 40'.

To do: A 'journey' might consist of any mix of walking and running.

  • How might you argue that the differential of $x_w$ is exact or inexact? (Consider a variety of journeys...)
  • Is there any algebraic combination of differentials for walking and running that is exact?


At least sometimes it is possible to make an exact differential out of a combination of inexact differentials.

Is a sum of exact differentials exact or inexact? What about a product or a difference?