An inexact differential or imperfect differential is a differential whose integral is path dependent. It is most often used in thermodynamics to express changes in path dependent quantities such as heat and work, but is defined more generally within mathematics as a type of differential form. In contrast, an integral of an exact differential is always path independent since the integral acts to invert the differential operator. Consequently, a quantity with an inexact differential cannot be expressed as a function of only the variables within the differential. I.e., its value cannot be inferred just by looking at the initial and final states of a given system. Inexact differentials are primarily used in calculations involving heat and work because they are path functions, not state functions.
Definition
An inexact differential is a differential for which the integral over some two paths with the same end points is different. Specifically, there exist integrable paths such that , and
In this case, we denote the integrals as and respectively to make explicit the path dependence of the change of the quantity we are considering as .
More generally, an inexact differential is a differential form which is not an exact differential, i.e., for all functions ,
The fundamental theorem of calculus for line integrals requires path independence in order to express the values of a given vector field in terms of the partial derivatives of another function that is the multivariate analogue of the antiderivative. This is because there can be no unique representation of an antiderivative for inexact differentials since their variation is inconsistent along different paths. This stipulation of path independence is a necessary addendum to the fundamental theorem of calculus because in one-dimensional calculus there is only one path in between two points defined by a function.
Notation
Thermodynamics
Instead of the differential symbol , the symbol is used, a convention which originated in the 19th century work of German mathematician Carl Gottfried Neumann, indicating that (heat) and (work) are path-dependent, while (internal energy) is not.
Statistical Mechanics
Within statistical mechanics, inexact differentials are often denoted with a bar through the differential operator, đ. In LaTeX the command "\rlap{ \textrm{ d}}{ \bar{ \phantom{ w}}}" is an approximation or simply "\dj" for a dyet character, which needs the T1 encoding.
Mathematics
Within mathematics, inexact differentials are usually just referred more generally to as differential forms which are often written just as .
Examples
Total distance
When you walk from a point to a point along a line (without changing directions) your net displacement and total distance covered are both equal to the length of said line . If you then return to point (without changing directions) then your net displacement is zero while your total distance covered is . This example captures the essential idea behind the inexact differential in one dimension. Note that if we allowed ourselves to change directions, then we could take a step forward and then backward at any point in time in going from to and in-so-doing increase the overall distance covered to an arbitrarily large number while keeping the net displacement constant, hence the saying two steps forward one step back.
Reworking the above with differentials and taking to be along the -axis, the net distance differential is , an exact differential with antiderivative . On the other hand, the total distance differential is , which does not have an antiderivative. The path taken is where there exists a time such that is strictly increasing before and strictly decreasing afterward. Then is positive before and negative afterward, yielding the integrals,
exactly the results we expected from the verbal argument before.
First law of thermodynamics
Inexact differentials show up explicitly in the first law of thermodynamics,