Examples
Although difficult to express mathematically, the inexact differential is very simple conceptually. There are many everyday examples that are much more relevant to inexact differentials in the actual context in which it is used.
The easiest example is the difference between net distance and total distance. For example, in walking from Point A to Point B one covers a net distance B-A that is equal to the total distance. If one then returns to Point A, however, net distance is now 0 while total distance covered is 2*(B-A). This example captures the essential idea behind the inexact differential in one dimension.
Precisely, the differential of net distance is simply the exact one form with corresponding function . It is exact because 1 has antiderivative x everywhere on the real line. On the other hand, the differential of total distance is the inexact one form (i.e. the sign function). It is inexact because sgn(x) has antiderivative |x| which is not differentiable at x =0. Therefore and instead we must look at the path dependence. In our example, in the first leg of the journey, sgn(dx) is 1 since x is increasing. In the second leg, sgn(dx) is -1 since x is decreasing. We can then evaluate the total distance as:
It is known that, with some skill, it is possible to start a fire only using friction and tinder. This is a way to convert mechanical energy (work, W) into an increase of internal energy, ΔU, which finally results into an increase in the local temperature of wood, its gasification and combustion, thereby creating a fire.
It is also possible to start a fire by adding heat using a lighter. This is a way to convert heat (Q) into an increase of internal energy, ΔU, but it has the same result as in the example involving work.
Both friction and heat transfer increase the internal energy of the system, since work and heat are both form of energy transform. 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. In other words, specifying the change in internal energy alone is insufficient to determine the heat evolved or the work done because there is no differentiation between the two forms of energy.
Read more about this topic: Inexact Differential
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