In mathematics,

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History

The so-called Pfaffian form, supposedly, was derived in circa 1805 by German mathematician Johann Pfaff.

Truncated Pfaffian

The expression:

where W is an amount of work done by the system, Y is an intensive property of the system, and Z is an extensive property of the system, is called a "truncated Pfaffian". [3]

Note

The Pfaffian expression is somehow related to the so-called Maxwell relations as found in the thermodynamic work of James Maxwell. [3]

Etymology

The names Pfaffian form, Pfaffian function, Pfaffian expression, truncated Pfaffian, etc., may have been introduced in Greek mathematician Constantin Caratheodory's 1908 "Studies in the Foundation of Thermodynamics", and what has since been called Caratheodory's theorem; although this needs to be fact-checked.

Another reference seems to state that the names Pfaffian function and Pfaffian form for these types of expressions were introduced in the 1970s by Russian mathematician

Conjugate variables

which is called the "conjugate", whereby, according to the first law, the change in internal energy

The combined law of thermodynamics:

being the simplest example, temperature

References

1. Sychev, Viacheslav V. (1991).

2. Caratheodory, Constantin. (1908).

3. Kestin, Joseph. (1966).

Further reading

● Sychev, Viacheslav. (1981).

● Emanuel, George. (1987).

● Ott, Bevan J. and Boerio-Goates Juliana. (2000).

External links

● Pfaffian function – Wikipedia.

● Pfaffian form – Wolfram MathWorld.

**Pfaffian form**is an expression which takes the form: [1](add)

History

The so-called Pfaffian form, supposedly, was derived in circa 1805 by German mathematician Johann Pfaff.

Truncated Pfaffian

The expression:

where W is an amount of work done by the system, Y is an intensive property of the system, and Z is an extensive property of the system, is called a "truncated Pfaffian". [3]

Note

The Pfaffian expression is somehow related to the so-called Maxwell relations as found in the thermodynamic work of James Maxwell. [3]

Etymology

The names Pfaffian form, Pfaffian function, Pfaffian expression, truncated Pfaffian, etc., may have been introduced in Greek mathematician Constantin Caratheodory's 1908 "Studies in the Foundation of Thermodynamics", and what has since been called Caratheodory's theorem; although this needs to be fact-checked.

Another reference seems to state that the names Pfaffian function and Pfaffian form for these types of expressions were introduced in the 1970s by Russian mathematician

__Askold Khovanskii__and named in honor of German mathematician Johann Pfaff.Conjugate variables

Pfaffian forms is a common formulation in thermodynamics, where the summation pairs, each taking the form of intensive (Xi) and extensive (xi) conjugate variable pair, act as quantifiable energy representations of transformations of the internal energy of the system. The right side of the Gibbs fundamental equation, for example, is a Pfaffian form. In short, the general use of the conjugate pairs perspective is that one can quantify the internal energy of a system as the sum of the conjugate variables. In short, with any extensitySee main: Conjugate variables

*xi*(extensive variable) it is always possible to associate a tension variable*Xi*(intensive variable):which is called the "conjugate", whereby, according to the first law, the change in internal energy

*dU*of a system is given by the summation of the product of the conjugate pairs:The combined law of thermodynamics:

being the simplest example, temperature

*T*and pressure*P*being the intensive variables (dependent on position in the body) and*S*and*V*being the extensive variables (independent on position in the body).References

1. Sychev, Viacheslav V. (1991).

*The Differential Equations of Thermodynamics*(2.2: Pfaffian forms and Total Differentials, pgs. 11-). Taylor & Francis.2. Caratheodory, Constantin. (1908).

*Studies in the Foundation of Thermodynamics*(*Untersuchungen uber die Grundlagen der Thermodynamik*). Bonn; published in:*Math. Ann*., 67: 355-386, 1909.3. Kestin, Joseph. (1966).

*A Course in Thermodynamics*(truncated Pfaffian, pg. 531)*.*London: Blaisdell Publishing Co.Further reading

● Sychev, Viacheslav. (1981).

*The Differential Equations of Thermodynamics*(Pfaffian, 5+ pgs)*.*Taylor & Francis.● Emanuel, George. (1987).

*Advanced Classical Thermodynamics*(3.3: Legendre Transformation, pgs. 25-28). AIAA.● Ott, Bevan J. and Boerio-Goates Juliana. (2000).

*Chemical Thermodynamics – Principles and Applications*(2.2d: Caratheodory and Pfaffian Differentials, pgs. 63-). Academic Press.External links

● Pfaffian function – Wikipedia.

● Pfaffian form – Wolfram MathWorld.