In
multivariate calculus, a
differential or
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, ...
is said to be exact or perfect (''exact differential''), as contrasted with an
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 with ...
, if it is equal to the general differential
for some
differentiable function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in it ...
in an
orthogonal coordinate system.
An exact differential is sometimes also called a ''total differential'', or a ''full differential'', or, in the study of
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, it is termed an
exact form 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 ...
.
The integral of an exact differential over any integral path is
path-independent, and this fact is used to identify
state functions 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 of the ...
.
Overview
Definition
Even if we work in three dimensions here, the definitions of exact differentials for other dimensions are structurally similar to the three dimensional definition. In three dimensions, a form of the type
:
is called 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, ...
. This form is called ''exact'' on an open domain
in space if there exists some
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
scalar function
In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number (dimensionless) or a scalar physical quantity ( ...
defined on
such that
:
throughout
, where
are
orthogonal coordinates In mathematics, orthogonal coordinates are defined as a set of ''d'' coordinates q = (''q''1, ''q''2, ..., ''q'd'') in which the coordinate hypersurfaces all meet at right angles (note: superscripts are indices, not exponents). A coordinate su ...
(e.g.,
Cartesian,
cylindrical, or
spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' meas ...
).
[Need to verify if exact differentials in non-orthogonal coordinate systems can also be defined.] In other words, in some open domain of a space, a differential form is an ''exact differential'' if it is equal to the general differential of a differentiable function in an orthogonal coordinate system.
::Note: In this mathematical expression, the subscripts outside the parenthesis indicate which variables are being held constant during differentiation. Due to the definition of the
partial derivative, these subscripts are not required, but they are explicitly shown here as reminders.
Integral path independence
The exact differential for a differentiable scalar function
defined in an open domain
is equal to
, where
is the
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
of
,
represents the
scalar product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
, and
is the general differential displacement vector, if an orthogonal coordinate system is used. If
is of differentiability class
(
continuously differentiable), then
is a
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 ...
for the corresponding potential
by the definition. For three dimensional spaces, expressions such as
and
can be made.
The
gradient theorem
The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. The theorem is ...
states
:
that does not depend on which integral path between the given path endpoints
and
is chosen. So it is concluded that ''the integral of an exact differential is independent of the choice of an integral path between given path endpoints
(path independence).''
For three dimensional spaces, if
defined on an open domain
is of
differentiability class
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
(equivalently
is of
), then this integral path independence can also be proved by using the
vector calculus identity
The following are important identities involving derivatives and integrals in vector calculus.
Operator notation
Gradient
For a function f(x, y, z) in three-dimensional Cartesian coordinate variables, the gradient is the vector field:
\o ...
and the
Stokes' theorem
Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan( ...
.
:
for a simply closed loop
with the smooth oriented surface
in it. If the open domain
is
simply connected open space (roughly speaking, a single piece open space without a hole within it), then any irrotational vector field (defined as a
vector field
which curl is zero, i.e.,
) has the path independence by the Stokes' theorem, so the following statement is made; ''In a simply connected open region, any''
''vector field that has the path-independence property (so it is a conservative vector field.) must also be irrotational and vise versa.'' The equality of the path independence and conservative vector fields is shown
here
Here is an adverb that means "in, on, or at this place". It may also refer to:
Software
* Here Technologies, a mapping company
* Here WeGo (formerly Here Maps), a mobile app and map website by Here Technologies, Here
Television
* Here TV (form ...
.
Thermodynamic state function
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 of the ...
, when
is exact, the function
is a
state function
In the thermodynamics of equilibrium, a state function, function of state, or point function for a thermodynamic system is a mathematical function relating several state variables or state quantities (that describe equilibrium states of a system ...
of the system: a
mathematical function
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the functi ...
which depends solely on the current
equilibrium state
Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In thermod ...
, not on the path taken to reach that state.
Internal energy
The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...
,
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 thermodynam ...
,
Enthalpy
Enthalpy , a property of a thermodynamic system, is the sum of the system's internal energy and the product of its pressure and volume. It is a state function used in many measurements in chemical, biological, and physical systems at a constant ...
,
Helmholtz free energy
In thermodynamics, the Helmholtz free energy (or Helmholtz energy) is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature (isothermal
In thermodynamics, an isotherma ...
, and
Gibbs free energy
In thermodynamics, the Gibbs free energy (or Gibbs energy; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work that may be performed by a thermodynamically closed system at constant temperature and pr ...
are
state function
In the thermodynamics of equilibrium, a state function, function of state, or point function for a thermodynamic system is a mathematical function relating several state variables or state quantities (that describe equilibrium states of a system ...
s. Generally, neither
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 tr ...
nor
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 ...
is a state function. (Note:
is commonly used to represent heat in physics. It should not be confused with the use earlier in this article as the parameter of an exact differential.)
One dimension
In one dimension, a differential form
:
is exact
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
has an
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 ...
(but not necessarily one in terms of elementary functions). If
has an antiderivative and let
be an antiderivative of
so
, then
obviously satisfies the condition for exactness. If
does ''not'' have an antiderivative, then we cannot write
with
for a differentiable function
so
is inexact.
Two and three dimensions
By
symmetry of second derivatives
In mathematics, the symmetry of second derivatives (also called the equality of mixed partials) refers to the possibility of interchanging the order of taking partial derivatives of a function
:f\left(x_1,\, x_2,\, \ldots,\, x_n\right)
of ''n'' ...
, for any "well-behaved" (non-
pathological
Pathology is the study of the causal, causes and effects of disease or injury. The word ''pathology'' also refers to the study of disease in general, incorporating a wide range of biology research fields and medical practices. However, when us ...
) function
, we have
:
Hence, in a
simply-connected region ''R'' of the ''xy''-plane, a differential form
:
is an exact differential if and only if the equation
:
holds. If it is an exact differential so
and
, then
is a differentiable (smoothly continuous) function along
and
, so
. If
holds, then
and
are differentiable (again, smoothly continuous) functions along
and
respectively, and
is only the case.
For three dimensions, in a simply-connected region ''R'' of the ''xyz''-coordinate system, by a similar reason, a differential
:
is an exact differential if and only if between the functions ''A'', ''B'' and ''C'' there exist the relations
:
;
;
These conditions are equivalent to the following sentence: If ''G'' is the graph of this vector valued function then for all tangent vectors ''X'',''Y'' of the ''surface'' ''G'' then ''s''(''X'', ''Y'') = 0 with ''s'' the
symplectic form In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form.
A symplectic bilinear form is a mapping that is
; Bilinear: Linear in each argument ...
.
These conditions, which are easy to generalize, arise from the independence of the order of differentiations in the calculation of the second derivatives. So, in order for a differential ''dQ'', that is a function of four variables, to be an exact differential, there are six conditions (the
combination
In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are th ...
) to satisfy.
Partial differential relations
If a
differentiable function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in it ...
is
one-to-one (injective) for each independent variable, e.g.,
is one-to-one for
at a fixed
while it is not necessarily one-to-one for
, then the following
total differential
In calculus, the differential represents the principal part of the change in a function ''y'' = ''f''(''x'') with respect to changes in the independent variable. The differential ''dy'' is defined by
:dy = f'(x)\,dx,
where f'(x) is the ...
s exist because each independent variable is a differentiable function for the other variables, e.g.,
.
:
:
Substituting the first equation into the second and rearranging, we obtain
:
:
:
Since
and
are independent variables,
and
may be chosen without restriction. For this last equation to generally hold, the bracketed terms must be equal to zero.
The left bracket equal to zero leads to the reciprocity relation while the right bracket equal to zero goes to the cyclic relation as shown below.
Reciprocity relation
Setting the first term in brackets equal to zero yields
:
A slight rearrangement gives a reciprocity relation,
:
There are two more
permutations
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
of the foregoing derivation that give a total of three reciprocity relations between
,
and
.
Cyclic relation
The cyclic relation is also known as the cyclic rule or the
Triple product rule. Setting the second term in brackets equal to zero yields
:
Using a reciprocity relation for
on this equation and reordering gives a cyclic relation (the
triple product rule),
:
If, ''instead'', reciprocity relations for
and
are used with subsequent rearrangement, a
standard form for implicit differentiation is obtained:
:
Some useful equations derived from exact differentials in two dimensions
(See also
Bridgman's thermodynamic equations
In thermodynamics, Bridgman's thermodynamic equations are a basic set of thermodynamic equations, derived using a method of generating multiple thermodynamic identities involving a number of thermodynamic quantities. The equations are named after ...
for the use of exact differentials in the theory of
thermodynamic equations
Thermodynamics is expressed by a mathematical framework of ''thermodynamic equations'' which relate various thermodynamic quantities and physical properties measured in a laboratory or production process. Thermodynamics is based on a fundamental ...
)
Suppose we have five state functions
, and
. Suppose that the state space is two-dimensional and any of the five quantities are differentiable. Then by the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
but also by the chain rule:
and
so that (by substituting (2) and (3) into (1)):
which implies that (by comparing (4) with (1)):
Letting
in (5) gives:
Letting
in (5) gives:
Letting
and
in (7) gives:
using (
gives the
triple product rule:
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 ...
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 mathe ...
*
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 with ...
*
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 calcul ...
for solving non-exact differential equations by making them exact
*
Exact differential equation
In mathematics, an exact differential equation or total differential equation is a certain kind of ordinary differential equation which is widely used in physics and engineering.
Definition
Given a simply connected and open subset ''D'' of R2 a ...
References
*Perrot, P. (1998). ''A to Z of Thermodynamics.'' New York: Oxford University Press.
*
External links
Inexact Differential– from Wolfram MathWorld
– University of Arizona
– University of Texas
– from Wolfram MathWorld
{{DEFAULTSORT:Exact Differential
Thermodynamics
Multivariable calculus