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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 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 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 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 Q 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 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 applications ...
. This form is called ''exact'' on an open domain D \subset \mathbb^3 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 Q = Q(x, y, z) defined on D such that dQ \equiv \left(\frac\right)_ \, dx + \left(\frac\right)_ \, dy + \left(\frac\right)_ \, dz, \quad dQ = A \, dx + B \, dy + C \, dz throughout D, where x, y, z 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 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 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 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 for other symmetric bilinear forms, for example in a pseudo-Euclidean space. Not to be confused wit ...
, 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 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 \nabla Q 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 Q by the definition. For three dimensional spaces, expressions such as d \mathbf = (dx, dy, dz) and \nabla Q = \left(\frac, \frac,\frac\right) can be made. The gradient theorem 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 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 ...
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
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 ...
. :\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 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 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 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 ...
U,
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 ...
S,
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 ...
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 ene ...
A, 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 ...
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 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 ...
W 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 ...
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" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
A 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 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, for any "well-behaved" (non- pathological) function Q, we have : \frac = \frac. Hence, in a simply-connected region ''R'' of the ''xy''-plane, where x,y are independent,If the pair of independent variables (x,y) is a (locally reversible) function of dependent variables (u,v), all that is needed for the following theorem to hold, is to replace the partial derivatives with respect to x or to y, by the partial derivatives with respect to u and to v involving their Jacobian components. That is: A(u, v)du + B(u, v)dv, is an exact differential, if and only if: \frac\frac + \frac\frac = \frac\frac + \frac\frac. 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 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 ...
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 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., 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 (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 x, y and z.


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 :_y _z = - _x. Using a reciprocity relation for \tfrac on this equation and reordering gives a cyclic relation (the triple product rule), :_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 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 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 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 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:


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