An inexact differential or imperfect differential is a
differential whose integral is path dependent. It is most often used in
thermodynamics
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws o ...
to express changes in path dependent quantities such as heat and work, but is defined more generally within
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
as a type of
differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
. In contrast, an integral of an
exact differential
In multivariate calculus, a differential or differential form is said to be exact or perfect (''exact differential''), as contrasted with an inexact differential, if it is equal to the general differential dQ for some differentiable function&n ...
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
In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is al ...
and
work
Work may refer to:
* Work (human activity), intentional activity people perform to support themselves, others, or the community
** Manual labour, physical work done by humans
** House work, housework, or homemaking
** Working animal, an animal ...
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
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
which is not an
exact differential
In multivariate calculus, a differential or differential form is said to be exact or perfect (''exact differential''), as contrasted with an inexact differential, if it is equal to the general differential dQ for some differentiable function&n ...
, 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
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or ...
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" 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 form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s 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
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...
. 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
The first law of thermodynamics is a formulation of the law of conservation of energy, adapted for thermodynamic processes. It distinguishes in principle two forms of energy transfer, heat and thermodynamic work for a system of a constant amo ...
,
where
is the energy,
is the differential change in heat and
is the differential change in work. Based on the constants of the thermodynamic system, we are able to parameterize the average energy in several different ways. E.g., in the first stage of the
Carnot cycle
A Carnot cycle is an ideal thermodynamic cycle proposed by French physicist Sadi Carnot in 1824 and expanded upon by others in the 1830s and 1840s. By Carnot's theorem, it provides an upper limit on the efficiency of any classical thermodynam ...
(which is isentropic) a gas is heated by a reservoir, giving us an isothermal expansion of that gas. During this stage, the volume is constant while some differential amount of heat
enters the gas. 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
and
must be inexact differentials.
Internal energy 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 and .
Since we can go from state to state either by providing heat or work , such a change of state does not uniquely identify the amount of work done to the system or heat transferred, but only the change in internal energy .
Heat and work
A fire requires heat, fuel, and an oxidizing agent. The energy required to overcome the
activation energy
In chemistry and physics, activation energy is the minimum amount of energy that must be provided for compounds to result in a chemical reaction. The activation energy (''E''a) of a reaction is measured in joules per mole (J/mol), kilojoules pe ...
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.
Integrating factors
It is sometimes possible to convert an inexact differential into an exact one by means of an
integrating factor
In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calculu ...
.
The most common example of this in thermodynamics is the definition of
entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...
:
In this case, is an inexact differential, because its effect on the state of the system can be compensated by .
However, when divided by the absolute
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
''and'' when the exchange occurs at reversible conditions (therefore the
rev subscript), it produces an exact differential: the entropy is also a state function.
Example
Consider the inexact differential form,
This must be inexact by considering going to the point . If we first increase and then increase , then that corresponds to first integrating over and then over . Integrating over first contributes
and then integrating over 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 , yielding
. And so
is an exact differential.
See also
*
Closed and exact differential forms In mathematics, especially vector calculus and differential topology, a closed form is a differential form ''α'' whose exterior derivative is zero (), and an exact form is a differential form, ''α'', that is the exterior derivative of another diff ...
for a higher-level treatment
*
Differential (mathematics)
In mathematics, differential refers to several related notions derived from the early days of calculus, put on a rigorous footing, such as infinitesimal differences and the derivatives of functions.
The term is used in various branches of mathema ...
*
Exact differential
In multivariate calculus, a differential or differential form is said to be exact or perfect (''exact differential''), as contrasted with an inexact differential, if it is equal to the general differential dQ for some differentiable function&n ...
*
Exact differential equation
*
Integrating factor
In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calculu ...
for solving non-exact differential equations by making them exact
*
Conservative vector field
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of any path between two points does not ...
References
External links
Inexact Differential– from Wolfram MathWorld
– University of Arizona
– University of Texas
– from Wolfram MathWorld
{{DEFAULTSORT:Inexact Differential
Thermodynamics
Multivariable calculus