↑ arvau

Wikipedia / Exact differential

In multivariate calculus, a differential or differential form is said to be exact or perfect (exact differential), as contrasted with an inexact differential for some differentiable function in an orthogonal coordinate system is a multivariable function whose variables are independent, as they are always expected to be when treated in multivariable calculus).

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

is called a differential form. This form is called exact in space if there exists some differentiable scalar function are orthogonal coordinates (e.g., Cartesian, cylindrical, or spherical 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

is the gradient represents the scalar product (continuously differentiable is a conservative vector field can be made.

The gradient theorem states

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).

is of differentiability class), then this integral path independence can also be proved by using the vector calculus identity and Stokes' theorem.

is simply connected open space) 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 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 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 has an antiderivative (but not necessarily one in terms of elementary functions does not is inexact.

Two and three dimensions

By symmetry of second derivatives, for any "well-behaved" (non-pathological, we have

Hence, in a simply-connected region R of the xy are independent, a differential form

is an exact differential if and only if the equation

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

  1. ;;

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) to satisfy.

Partial differential relations

If a differentiable function is one-to-one (injective), then the following total differentials.

Substituting the first equation into the second and rearranging, we obtain

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.

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

on this equation and reordering gives a cyclic relation (the triple product rule),

If, instead 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)

. Suppose that the state space is two-dimensional and any of the five quantities are differentiable. Then by the chain rule

but also by the chain rule:

and

so that (by substituting (2) and (3) into (1)):

which implies that (by comparing (4) with (1)):

in (5) gives:

in (5) gives:

in (7) gives:

gives the triple product rule:

See also

References

  • Perrot, P. (1998). A to Z of Thermodynamics. New York: Oxford University Press.