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.
I.e., its value cannot be inferred just by looking at the initial and final states of a given system.
is a differential for which the integral over some two paths with the same end points is different.
respectively to make explicit the path dependence of the change of the quantity we are considering as
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 an unnecessary 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,[2] indicating that Q (heat) and W (work) are path-dependent, while U (internal energy) is not.
[3] 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.
(without changing directions) your net displacement and total distance covered are both equal to the length of said line
(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
and in-so-doing increase the overall distance covered to an arbitrarily large number while keeping the net displacement constant.
exactly the results we expected from the verbal argument before.
Inexact differentials show up explicitly in the first law of thermodynamics,
Based on the constants of the thermodynamic system, we are able to parameterize the average energy in several different ways.
During the second stage, the gas is allowed to freely expand, outputting some differential amount of work
The third stage is similar to the first stage, except the heat is lost by contact with a cold reservoir, while the fourth cycle is like the second except work is done onto the system by the surroundings to compress the gas.
Because the overall changes in heat and work are different over different parts of the cycle, there is a nonzero net change in the heat and work, indicating that the differentials
Internal energy U is a state function, meaning its change can be inferred just by comparing two different states of the system (independently of its transition path), which we can therefore indicate with U1 and U2.
Since we can go from state U1 to state U2 either by providing heat Q = U2 − U1 or work W = U2 − U1, such a change of state does not uniquely identify the amount of work W done to the system or heat Q transferred, but only the change in internal energy ΔU.
A fire requires heat, fuel, and an oxidizing agent.
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 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.
It is sometimes possible to convert an inexact differential into an exact one by means of an integrating factor.
In this case, δQ is an inexact differential, because its effect on the state of the system can be compensated by δW.
However, when divided by the absolute temperature and when the exchange occurs at reversible conditions (therefore the rev subscript), it produces an exact differential: the entropy S is also a state function.