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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 several variables, rather th ...
, 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 application ...
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 wit ...
, if it is equal to the general differential dQ 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 ...
 Q in an
orthogonal coordinate system 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 ...
. 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 mult ...
, 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 functions 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 ...
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 ...
.


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 :A(x,y,z) \,dx + B(x,y,z) \,dy + C(x,y,z) \,dz 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 application ...
. This form is called ''exact'' on an open domain D \subset \mathbb^3 in space if there exists some differentiable scalar function Q = Q(x,y,z) defined on D such that : dQ \equiv \left ( \frac \right )_ \, dx + \left ( \frac \right )_ \, dy + \left ( \frac \right )_ \, dz, dQ = A \, dx + B \, dy + C \, dz throughout D, where x,y,z 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 A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an in ...
, 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 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). Pa ...
, these subscripts are not required, but they are explicitly shown here as reminders.


Integral path independence

The exact differential for a differentiable scalar function Q defined in an open domain D \subset \mathbb^n is equal to dQ = \nabla Q \cdot d \mathbf, where \nabla Q 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 Q, \cdot 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 d \mathbf is the general differential displacement vector, if an orthogonal coordinate system is used. If Q is of differentiability class C^1 (
continuously 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 ...
), then \nabla Q is a conservative vector field for the corresponding potential Q by the definition. For three dimensional spaces, expressions such as d \mathbf = (dx, dy, dz) and \nabla Q = (\frac, \frac,\frac) 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 :\int _^ dQ = \int _^\nabla Q (\mathbf )\cdot d \mathbf = Q \left(f \right) - Q \left(i \right) that does not depend on which integral path between the given path endpoints i and f 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 \nabla Q defined on an open domain D \subset \mathbb^3 is of
differentiability class In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...
C^1 (equivalently Q is of C^2), then this integral path independence can also be proved by using the vector calculus identity \nabla \times ( \nabla Q ) = \mathbf 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( ...
. :\oint _\nabla Q \cdot d \mathbf = \iint _(\nabla \times \nabla Q)\cdot d \mathbf = 0 for a simply closed loop \partial \Sigma with the smooth oriented surface \Sigma in it. If the open domain D 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 C^1 vector field \mathbf which curl is zero, i.e., \nabla \times \mathbf = \mathbf) has the path independence by the Stokes' theorem, so the following statement is made; ''In a simply connected open region, any'' C^1 ''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 Television * Here TV (formerly "here!"), a ...
.


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 ...
, when dQ is exact, the function Q 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 which depends solely on the current equilibrium state, 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 ...
U,
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 ...
S,
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 ...
H,
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 en ...
A, 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 an ...
G 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 t ...
W 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 ...
Q is a state function. (Note: Q 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 :A(x) \, dx 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 bic ...
A 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 symbolica ...
(but not necessarily one in terms of elementary functions). If A has an antiderivative and let Q be an antiderivative of A so \frac = A, then A(x) \, dx obviously satisfies the condition for exactness. If A does ''not'' have an antiderivative, then we cannot write dQ = \fracdx with A = \frac for a differentiable function Q so A(x) \, dx 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 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 used in th ...
) function Q, we have : \frac = \frac. Hence, in a
simply-connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spa ...
region ''R'' of the ''xy''-plane, a differential form :A(x, y)\,dx + B(x, y)\,dy is an exact differential if and only if the equation :\left( \frac \right)_x = \left( \frac \right)_y holds. If it is an exact differential so A=\frac and B=\frac, then Q is a differentiable (smoothly continuous) function along x and y, so \left( \frac \right)_x = \frac = \frac = \left( \frac \right)_y. If \left( \frac \right)_x = \left( \frac \right)_y holds, then A and B are differentiable (again, smoothly continuous) functions along y and x respectively, and \left( \frac \right)_x = \frac = \frac = \left( \frac \right)_y is only the case. For three dimensions, in a simply-connected region ''R'' of the ''xyz''-coordinate system, by a similar reason, a differential :dQ = A(x, y, z) \, dx + B(x, y, z) \, dy + C(x, y, z) \, dz is an exact differential if and only if between the functions ''A'', ''B'' and ''C'' there exist the relations :\left( \frac \right)_ \!\!\!= \left( \frac \right)_\left( \frac \right)_ \!\!\!= \left( \frac \right)_\left( \frac \right)_ \!\!\!= \left( \frac \right)_. 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. 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 C(4,2)=6) 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 ...
z(x,y) is one-to-one (injective) for each independent variable, e.g., z(x,y) is one-to-one for x at a fixed y while it is not necessarily one-to-one for (x,y), then the following total differentials exist because each independent variable is a differentiable function for the other variables, e.g., x(y,z). :d x = _z \, d y + _y \,dz :d z = _y \, d x + _x \,dy. Substituting the first equation into the second and rearranging, we obtain :d z = _y \left _z d y + _y dz \right + _x dy, :d z = \left _y _z + _x \right d y + _y _y dz, :\left 1 - _y _y \right dz = \left _y _z + _x \right d y. Since y and z are independent variables, d y and d z 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 :_y _y = 1. A slight rearrangement gives a reciprocity relation, :_y = \frac. 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 p ...
of the foregoing derivation that give a total of three reciprocity relations between x, y and z.


Cyclic relation

The cyclic relation is also known as the cyclic rule or the
Triple product rule The triple product rule, known variously as the cyclic chain rule, cyclic relation, cyclical rule or Euler's chain rule, is a formula which relates partial derivatives of three interdependent variables. The rule finds application in thermodyna ...
. Setting the second term in brackets equal to zero yields :_y _z = - _x. Using a reciprocity relation for \tfrac on this equation and reordering gives a cyclic relation (the
triple product rule The triple product rule, known variously as the cyclic chain rule, cyclic relation, cyclical rule or Euler's chain rule, is a formula which relates partial derivatives of three interdependent variables. The rule finds application in thermodyna ...
), :_z _x _y = -1. If, ''instead'', reciprocity relations for \tfrac and \tfrac are used with subsequent rearrangement, a standard form for implicit differentiation is obtained: :_z = - \frac .


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) Suppose we have five state functions z,x,y,u, and v. 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 v=y in (5) gives: Letting u=y in (5) gives: Letting u=y and v=z in (7) gives: using (\partial a/\partial b)_c = 1/(\partial b/\partial a)_c gives the
triple product rule The triple product rule, known variously as the cyclic chain rule, cyclic relation, cyclical rule or Euler's chain rule, is a formula which relates partial derivatives of three interdependent variables. The rule finds application in thermodyna ...
:


See also

* Closed and exact differential forms 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 wit ...
*
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


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