Exact differential

Exact differential

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A mathematical
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

  differential is said to be exact, as contrasted with an inexact differential
Inexact differential
An inexact differential or imperfect differential is a specific type of differential used in thermodynamics to express the path dependence of a particular differential. It is contrasted with the concept of the exact differential in calculus, which can be expressed as the gradient of another...

, if it is of the form dQ, for some differentiable function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 Q.

The form A(xyzdx + B(xyzdy + C(xyzdz is called a differential form. A differential form is exact on a domain D in space if
A dx + B dy + C dz = df for some scalar function f throughout D. This is equivalent to saying that the field is conservative.

Overview


For one dimension, a differential


is always exact.

For two dimensions, in order that a differential


be an exact differential in a simply-connected region R of the xy-plane, it is necessary and sufficient that between A and B there exists the relation:


For three dimensions, a differential


is an exact differential in a simply-connected region R of the xyz-coordinate system if between the functions A, B and C there exist the relations:
  ;     ;  
Note: The subscripts outside the parenthesis indicate which variables are being held constant during differentiation. Due to the definition of the partial derivative
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...

, these subscripts are not required, but they are included as a reminder.


These conditions are equivalent to the following one: If G is the graph of this vector valued function then for all tangent vectors X,Y of the surface G then s(XY) = 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 to satisfy.

In summary, when a differential dQ is exact:
  • the function Q exists;
  • independent of the path followed.


In thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

, when dQ is exact, the function Q is a state function of the system. The thermodynamic functions U
Internal energy
In thermodynamics, the internal energy is the total energy contained by a thermodynamic system. It is the energy needed to create the system, but excludes the energy to displace the system's surroundings, any energy associated with a move as a whole, or due to external force fields. Internal...

, S
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...

, H
Enthalpy
Enthalpy is a measure of the total energy of a thermodynamic system. It includes the internal energy, which is the energy required to create a system, and the amount of energy required to make room for it by displacing its environment and establishing its volume and pressure.Enthalpy is a...

, A
Helmholtz free energy
In thermodynamics, the Helmholtz free energy is a thermodynamic potential that measures the “useful” work obtainable from a closed thermodynamic system at a constant temperature and volume...

and G
Gibbs free energy
In thermodynamics, the Gibbs free energy is a thermodynamic potential that measures the "useful" or process-initiating work obtainable from a thermodynamic system at a constant temperature and pressure...

are state function
State function
In thermodynamics, a state function, function of state, state quantity, or state variable is a property of a system that depends only on the current state of the system, not on the way in which the system acquired that state . A state function describes the equilibrium state of a system...

s. Generally, neither work
Work (thermodynamics)
In thermodynamics, work performed by a system is the energy transferred to another system that is measured by the external generalized mechanical constraints on the system. As such, thermodynamic work is a generalization of the concept of mechanical work in mechanics. Thermodynamic work encompasses...

 nor heat
Heat
In physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...

 is a state function. An exact differential is sometimes also called a 'total differential', or a 'full differential', or, in the study of differential geometry, it is termed an exact form.

Partial differential relations


For three variables, , and bound by some differentiable function , the following total differentials exist


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 hold in general, the bracketed terms must be equal to zero.

Reciprocity relation


Setting the first term in brackets equal to zero yields


A slight rearrangement gives a reciprocity relation,


There are two more permutations of the foregoing derivation that give a total of three reciprocity relations between , and . Reciprocity relations show that the inverse of a partial derivative is equal to its reciprocal.

Cyclic relation


The cyclic relation is also known as the cyclic rule or the Triple product rule
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...

. 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
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...

),


If, instead, a reciprocity relation for is 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
Bridgman's thermodynamic equations
In thermodynamics, Bridgman's thermodynamic equations are a basic set of thermodynamic equations, derived using a method of generating a large number of thermodynamic identities involving a number of thermodynamic quantities. The equations are named after the American physicist Percy Williams...

 for the use of exact differentials in the theory of thermodynamic equations
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...

)

Suppose we have five state functions , and . Suppose that the state space is two dimensional and any of the five quantities are exact differentials. Then by the chain rule
Chain rule
In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...





but also by the chain rule:



and



so that:




which implies that:



Letting gives:



Letting gives:



Letting , gives:



using ( gives the triple product rule
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...

:


See also

  • Closed and exact differential forms
    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 that is the exterior derivative of another differential form β...

     for a higher-level treatment
  • Differential (mathematics)
    Differential (mathematics)
    In mathematics, the term differential has several meanings.-Basic notions:* In calculus, the differential represents a change in the linearization of a function....

  • Inexact differential
    Inexact differential
    An inexact differential or imperfect differential is a specific type of differential used in thermodynamics to express the path dependence of a particular differential. It is contrasted with the concept of the exact differential in calculus, which can be expressed as the gradient of another...

  • Integrating factor
    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 calculus, in this case often multiplying through by an...

     for solving non-exact differential equations by making them exact
  • Exact differential equation
    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:...


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