Inexact differential

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.

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.

Instead of the differential symbol d, the symbol δ is used, a convention which originated in the 19th century work of German mathematician Carl Gottfried Neumann, indicating that Q (heat) and W (work) are path dependent, while U (internal energy) is not.

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.

Within mathematics, inexact differentials are usually just referred more generally to as differential forms which are often written just as .

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. 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.

A fire requires heat, fuel, and an oxidizing agent. The energy required to overcome the activation energy barrier for combustion is transferred as heat into the system, resulting in changes to the system's internal energy. In a process, the energy input to start a fire may comprise both work and heat, such as when one rubs tinder (work) and experiences friction (heat) to start a fire. The ensuing combustion is highly exothermic, which releases heat. The overall change in internal energy does not reveal the mode of energy transfer and quantifies only the net work and heat. The difference between initial and final states of the system's internal energy does not account for the extent of the energy interactions transpired. Therefore, internal energy is a state function (i. e. exact differential), while heat and work are path functions (i. e. inexact differentials) because integration must account for the path taken.

Consider the inexact differential form, This must be inexact by considering going to the point (1,1). If we first increase y and then increase x, then that corresponds to first integrating over y and then over x. Integrating over y first contributes and then integrating over x contributes Thus, along the first path we get a value of 2. However, along the second path we get a value of We can make an exact differential by multiplying it by x, yielding And so is an exact differential.