# Uniqueness theorem

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In mathematics, a **uniqueness theorem**, also called a **unicity theorem**, is a [theorem](/source/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](/source/Cauchy's_theorem_(geometry)) and [Alexandrov's uniqueness theorem](/source/Alexandrov's_uniqueness_theorem) for three-dimensional polyhedra.
- [Black hole uniqueness theorem](/source/Black_hole_uniqueness_theorem)
- [Cauchy–Kowalevski theorem](/source/Cauchy%E2%80%93Kowalevski_theorem) is the main local [existence](/source/Existence_theorem) and uniqueness theorem for [analytic](/source/Analytic_function) [partial differential equations](/source/Partial_differential_equation) associated with [Cauchy initial value problems](/source/Cauchy_problem).
- [Cauchy–Kowalevski–Kashiwara theorem](/source/Cauchy%E2%80%93Kowalevski_theorem#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](/source/Euclidean_division#Statement_of_the_theorem), the uniqueness of quotient and remainder under Euclidean division.
- [Fundamental theorem of arithmetic](/source/Fundamental_theorem_of_arithmetic), the uniqueness of prime factorization.
- [Holmgren's uniqueness theorem](/source/Holmgren's_uniqueness_theorem) for linear partial differential equations with real analytic coefficients.
- [Picard–Lindelöf theorem](/source/Picard%E2%80%93Lindel%C3%B6f_theorem), the uniqueness of solutions to first-order differential equations.
- [Thompson uniqueness theorem](/source/Thompson_uniqueness_theorem) in finite group theory.
- [Uniqueness theorem for Poisson's equation](/source/Uniqueness_theorem_for_Poisson's_equation).[2]
- [Electromagnetism uniqueness theorem](/source/Electromagnetism_uniqueness_theorem) for the solution of Maxwell's equation.
- [Uniqueness case](/source/Uniqueness_case) in finite group theory.

The word *unique* is sometimes replaced by *[essentially unique](/source/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](/source/Existence_theorem)
- [Rigidity (mathematics)](/source/Rigidity_(mathematics))
- [Uniqueness quantification](/source/Uniqueness_quantification)

## References

1. Weisstein, Eric W. ["Uniqueness Theorem"](http://mathworld.wolfram.com/UniquenessTheorem.html). *mathworld.wolfram.com*. Retrieved 2019-11-29.

1. ["The uniqueness theorem"](http://farside.ph.utexas.edu/teaching/em/lectures/node62.html). *farside.ph.utexas.edu*. Retrieved 2019-11-29.

1. ["Existence and Uniqueness"](http://www.sosmath.com/diffeq/first/existence/existence.html). *www.sosmath.com*. Retrieved 2019-11-29.

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