Exact differential
In multivariate calculus, a differential or differential form is said to be exact or perfect, as contrasted with an inexact differential, if it is equal to the general differential for some differentiable function 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, 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 thermodynamics.
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 typeis called a differential form. This form is called exact on an open domain in space if there exists some differentiable scalar function defined on such that
throughout, where are orthogonal coordinates. 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, 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 of, represents the scalar product, and is the general differential displacement vector, if an orthogonal coordinate system is used. If is of differentiability class , then is a conservative vector field 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.
For three dimensional spaces, if defined on an open domain is of differentiability class , then this integral path independence can also be proved by using the vector calculus identity and Stokes' theorem.
for a simply closed loop with the smooth oriented surface in it. If the open domain is simply connected open space, then any irrotational vector field 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 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, when is exact, the function is a state function 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, Entropy, Enthalpy, Helmholtz free energy, and Gibbs free energy are state functions. Generally, neither work nor heat is a state function.One dimension
In one dimension, a differential formis exact if and only if has an antiderivative. 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" function, we haveHence, in a simply-connected region R of the xy-plane, where are independent, 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 function along and, so. If holds, then and are differentiable 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 = 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.
Partial differential relations
If a differentiable function is one-to-one 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 differentials 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 yieldsA 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.
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 yieldsUsing a reciprocity relation for on this equation and reordering gives a cyclic relation,
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
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 rulebut also by the chain rule:
and
so that and into ):
which implies that with ):
Letting in gives:
Letting in gives:
Letting and in gives:
using ( gives the triple product rule: