Uniqueness theorem

In mathematics, a uniqueness theorem, also called a unicity theorem, is a theorem asserting the uniqueness of an object satisfying certain conditions, or the equivalence of all objects satisfying the said conditions. ^[1] Examples of uniqueness theorems include:

• Cauchy's rigidity theorem and Alexandrov's uniqueness theorem for three-dimensional polyhedra.
• Black hole uniqueness theorem
• Cauchy–Kowalevski theorem is the main local existence and uniqueness theorem for analytic partial differential equations associated with Cauchy initial value problems.
• Cauchy–Kowalevski–Kashiwara theorem is a wide generalization of the Cauchy–Kowalevski theorem for systems of linear partial differential equations with analytic coefficients.
• Division theorem, the uniqueness of quotient and remainder under Euclidean division.
• Fundamental theorem of arithmetic, the uniqueness of prime factorization.
• Holmgren's uniqueness theorem for linear partial differential equations with real analytic coefficients.
• Picard–Lindelöf theorem, the uniqueness of solutions to first-order differential equations.
• Thompson uniqueness theorem in finite group theory.
• Uniqueness theorem for Poisson's equation. ^[2]
• Electromagnetism uniqueness theorem for the solution of Maxwell's equation.
• Uniqueness case in finite group theory.

The word unique is sometimes replaced by essentially unique , whenever one wants to stress that the uniqueness is only referred to the underlying structure, whereas the form may vary in all ways that do not affect the mathematical content. ^[1]

A uniqueness theorem (or its proof) is, at least within the mathematics of differential equations, often combined with an existence theorem (or its proof) to a combined existence and uniqueness theorem (e.g., existence and uniqueness of solution to first-order differential equations with boundary condition). ^[3]

See also

• Existence theorem
• Rigidity (mathematics)
• Uniqueness quantification

References

1. Weisstein, Eric W. "Uniqueness Theorem". mathworld.wolfram.com. Retrieved 2019-11-29.
2. "The uniqueness theorem". farside.ph.utexas.edu. Retrieved 2019-11-29.
3. "Existence and Uniqueness". www.sosmath.com. Retrieved 2019-11-29.