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Euler's homogeneous function theorem is a characterization of positively homogeneous differentiable functions, which may be considered as the fundamental theorem on homogeneous functions. Examples [ edit ]
This formula is known as an Euler relation, because Euler's theorem on homogeneous functions leads to it. [15] [16] (It was not discovered by Euler in an investigation of thermodynamics, which did not exist in his day.). Substituting into the expressions for the other main potentials we have: = +
Euclid–Euler theorem (number theory) Euler's partition theorem (number theory) Euler's polyhedron theorem ; Euler's quadrilateral theorem ; Euler's rotation theorem ; Euler's theorem (differential geometry) Euler's theorem (number theory) Euler's theorem in geometry (triangle geometry) Euler's theorem on homogeneous functions (multivariate ...
Because all of the natural variables of the internal energy U are extensive quantities, it follows from Euler's homogeneous function theorem that = + Substituting into the expressions for the other main potentials we have the following expressions for the thermodynamic potentials:
In 1736, Leonhard Euler published a proof of Fermat's little theorem [1] (stated by Fermat without proof), which is the restriction of Euler's theorem to the case where n is a prime number. Subsequently, Euler presented other proofs of the theorem, culminating with his paper of 1763, in which he proved a generalization to the case where n is ...
Euler's theorem, on modular exponentiation; Euler's partition theorem relating the product and series representations of the Euler function Π(1 − x n) Goldbach–Euler theorem, stating that sum of 1/(k − 1), where k ranges over positive integers of the form m n for m ≥ 2 and n ≥ 2, equals 1; Gram–Euler theorem
The internal energy is thus a first-order homogenous function. Applying Euler's homogeneous function theorem, one finds the following relation when taking only volume, number of particles, and entropy as extensive variables: = + = Taking the total differential, one finds
The function defined by a homogeneous polynomial is always a homogeneous function. An algebraic form, or simply form, is a function defined by a homogeneous polynomial. [notes 1] A binary form is a form in two variables. A form is also a function defined on a vector space, which may be expressed as a homogeneous function of the coordinates over ...