In
multivariate calculus
Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving multiple variables ('' mult ...
, 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, 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 ...
in an
orthogonal coordinate system
In mathematics, orthogonal coordinates are defined as a set of coordinates \mathbf q = (q^1, q^2, \dots, q^d) in which the coordinate hypersurfaces all meet at right angles (note that superscripts are indices, not exponents). A coordinate sur ...
(hence
is a multivariable function
whose variables are independent, as they are always expected to be when treated in
multivariable calculus
Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving multiple variables ('' mult ...
).
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 Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
, it is termed an
exact form.
The integral of an exact differential over any integral path is
path-independent, and this fact is used to identify
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 syste ...
s in
thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
.
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 ...
scalar function defined on
such that
throughout
, where
are
orthogonal coordinates
In mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. ...
(e.g.,
Cartesian,
cylindrical
A cylinder () has traditionally been a Solid geometry, three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a Prism (geometry), prism with a circle as its base.
A cylinder may ...
, or
spherical coordinates
In mathematics, a spherical coordinate system specifies a given point in three-dimensional space by using a distance and two angles as its three coordinates. These are
* the radial distance along the line connecting the point to a fixed point ...
). 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.
The subscripts outside the parenthesis in the above mathematical expression indicate which variables are being held constant during differentiation. Due to the definition of the
partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). P ...
, 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 f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...
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 for other symmetric bilinear forms, for example in a pseudo-Euclidean space. Not to be confused wit ...
, and
is the general differential displacement vector, if an orthogonal coordinate system is used. If
is of differentiability class
(
continuously differentiable
In mathematics, a differentiable function of one Real number, real variable is a Function (mathematics), function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differ ...
), 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 path between two points does not chan ...
for the corresponding potential
by the definition. For three dimensional spaces, expressions such as
and
can be made.
The
gradient theorem 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 (''differentiability class)'' it has over its domain.
A function of class C^k is a function of smoothness at least ; t ...
(equivalently
is of
), then this integral path independence can also be proved by using the
vector calculus identity and
Stokes' theorem
Stokes' theorem, also known as the Kelvin–Stokes theorem after Lord Kelvin and George Stokes, the fundamental theorem for curls, or simply the curl theorem, is a theorem in vector calculus on \R^3. Given a vector field, the theorem relates th ...
.
:
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 vice versa.'' The equality of the path independence and conservative vector fields is shown
here.
Thermodynamic state function
In
thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
, 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 syste ...
of the system: a
mathematical function
In mathematics, a function from a set (mathematics), set to a set assigns to each element of exactly one element of .; the words ''map'', ''mapping'', ''transformation'', ''correspondence'', and ''operator'' are sometimes used synonymously. ...
which depends solely on the current
equilibrium state
Thermodynamic equilibrium is a notion of thermodynamics with axiomatic status referring to an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable ...
, not on the path taken to reach that state.
Internal energy
The internal energy of a thermodynamic system is the energy of the system as a state function, measured as the quantity of energy necessary to bring the system from its standard internal state to its present internal state of interest, accoun ...
,
Entropy
Entropy is a scientific concept, most commonly associated with states of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the micros ...
,
Enthalpy
Enthalpy () is the sum of a thermodynamic system's internal energy and the product of its pressure and volume. It is a state function in thermodynamics used in many measurements in chemical, biological, and physical systems at a constant extern ...
,
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). The change in the Helmholtz ene ...
, and
Gibbs free energy
In thermodynamics, the Gibbs free energy (or Gibbs energy as the recommended name; symbol is a thermodynamic potential that can be used to calculate the maximum amount of Work (thermodynamics), work, other than Work (thermodynamics)#Pressure–v ...
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 syste ...
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 ani ...
nor
heat
In thermodynamics, heat is energy in transfer between a thermodynamic system and its surroundings by such mechanisms as thermal conduction, electromagnetic radiation, and friction, which are microscopic in nature, involving sub-atomic, ato ...
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" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
has an
antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a continuous function is a differentiable function whose derivative is equal to the original function . This can be stated ...
(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, for any "well-behaved" (non-
pathological) function
, we have
:
Hence, in a
simply-connected region ''R'' of the ''xy''-plane, where
are independent,
[If the pair of independent variables is a (locally reversible) function of dependent variables , all that is needed for the following theorem to hold, is to replace the partial derivatives with respect to or to , by the partial derivatives with respect to and to involving their Jacobian components. That is: is an exact differential, if and only if: ] 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 \mathbb) equipped with a symplectic bilinear form.
A symplectic bilinear form is a mapping \omega : V \times V \to F that is
; Bilinear: ...
.
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 ...
) 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 ...
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 derivative of with resp ...
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 (mathematics), set can mean one of two different things:
* an arrangement of its members in a sequence or linear order, or
* the act or process of changing the linear order of an ordered set.
An example ...
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 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 Function composition, 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 ...
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 dif ...
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
*
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 non-exact ordinary differential equations, but is also used within multivari ...
for solving non-exact differential equations by making them exact
*
Exact differential equation
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