Localization theorem: Difference between revisions

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{{unreferenced|date=June 2021}}

In [[mathematics]], particularly in [[integral calculus]], the '''localization theorem''' allows, under certain conditions, to infer the nullity of a [[function (mathematics)|function]] given only information about its [[continuity (mathematics)|continuity]] and the value of its integral.

Let {{math|<var>F</var>(<var>x</var>)}} be a [[real-valued function]] defined on some open [[Domaininterval (mathematics)#Real and complex analysis|___domaininterval]] <var>Ω</var> of the [[real line]] that is [[Continuous function|continuous]] in <var>Ω</var>. Let <var>D</var> be an arbitrary ___domainsubinterval contained in <var>Ω</var>. The theorem states the following implication:

<math display="block">\int\limits_Dint_D F(x) dx\, \mathrm{d}x = 0 ~ \forall D \insubset \Omega ~ \Rightarrow ~ F(x) = 0 ~ \forall x \in \Omega</math>

[[File:Localization Theorem.svg|float|right]]

Let {{math|F(<var>x</var>)}} be a real-valued function defined on some [[Domain (mathematics)#Real and complex analysis|___domain]] <var>Ω</var> of the real line that is [[Continuous function|continuous]] in <var>Ω</var>. Let <var>D</var> be an arbitrary ___domain contained in <var>Ω</var>. The theorem states the following implication:

A simple proof is as follows: if there were a point <var>x</var>₀</var> within <var>Ω</var> for which {{math|<var>F</var>(<var>x</var>₀</var>)&ne; ≠ 0}}, then the continuity of {{mathmvar|F}} would require the existence of a [[neighborhood (mathematics)|neighborhood]] of <var>x</var>₀</var> in which the value of {{mathmvar|F}} was nonzero, and in particular of the same sign than in <var>x<sub/var>0</sub>0</varsub>. Since such a neighborhood <var>N</var>, which can be taken to be arbitrarily small, must however be of a nonzero width on the real line, the integral of {{mathmvar|F}} over <var>N</var> would evaluate to a nonzero value. However, since <var>x</var>₀</var> is part of the ''open'' set <var>Ω</var>, all neighborhoods of <var>x</var>₀</var> smaller than the distance of <var>x</var>₀</var> to the frontier of <var>Ω</var> are included within it, and so the integral of {{mathmvar|F}} over them must evaluate to zero. Having reached the contradiction that {{math|&int;∫_<var>N</var> <var>F</var>(<var>x</var>) <var>dx</var>}} must be both zero and nonzero, the initial hypothesis must be wrong, and thus there is no <var>x</var>₀</var> in <var>Ω</var> for which {{math|<var>F</var>(<var>x</var>₀</var>)&ne; ≠ 0}}.

<math>\int\limits_D F(x) dx = 0 ~ \forall D \in \Omega ~ \Rightarrow ~ F(x) = 0 ~ \forall x \in \Omega</math>

The theorem is easily generalized to [[multivariate calculus|multivariate function]]s, replacing intervals with the more general concept of connected [[open set]]s, that is, [[Domain (mathematical analysis)|domains]], and the original function with some {{math|<var>F</var>(<var>'''x'''</var>) : '''R'''^''n'' → '''R'''}}, with the constraints of continuity and nullity of its integral over any subdomain {{math|<var>D</var> ⊂ Ω}}. The proof is completely analogous to the single variable case, and concludes with the impossibility of finding a point {{math|<var>'''x'''</var>₀ ∈ Ω}} such that {{math|<var>F</var>(<var>'''x'''</var>₀) ≠ 0}}.

{{clear}}

A simple proof is as follows: if there were a point <var>x₀</var> within <var>Ω</var> for which {{math|F(<var>x₀</var>)&ne;0}}, then the continuity of {{math|F}} would require the existence of a [[neighborhood (mathematics)|neighborhood]] of <var>x₀</var> in which the value of {{math|F}} was nonzero, and in particular of the same sign than in <var>x₀</var>. Since such a neighborhood <var>N</var>, which can be taken to be arbitrarily small, must however be of a nonzero width on the real line, the integral of {{math|F}} over <var>N</var> would evaluate to a nonzero value. However, since <var>x₀</var> is part of the ''open'' set <var>Ω</var>, all neighborhoods of <var>x₀</var> smaller than the distance of <var>x₀</var> to the frontier of <var>Ω</var> are included within it, and so the integral of {{math|F}} over them must evaluate to zero. Having reached the contradiction that {{math|&int;_<var>N</var>F(<var>x</var>)dx}} must be both zero and nonzero, the initial hypothesis must be wrong, and thus there is no <var>x₀</var> in <var>Ω</var> for which {{math|F(<var>x₀</var>)&ne;0}}.

==Example==

An example of the use of this theorem in physics is the law of [[conservation of mass]] for fluids, which states that the mass of any fluid volume must not change:

<math display="block">\frac{\mathrm{d}}{\mathrm{d}t} \int_{V_f} \rho(\vec x, t) \, \mathrm{d}\Omega = 0</math>

Applying the [[Reynolds transport theorem]], one can change the reference to an arbitrary (non-fluid) [[control volume]] <var>V_c</var>. Further assuming that the [[density|density function]] is continuous (i.e. that our fluid is monophasic and thermodynamically metastable) and that <var>V_c</var> is not moving relative to the chosen system of reference, the equation becomes:

<math display="block">\int_{V_c} \left [ {{\partial \rho} \over {\partial t}} + \nabla \cdot (\rho \vec v) \right ] \, \mathrm{d}\Omega = 0</math>

As the equation holds for ''any'' such control volume, the localization theorem applies, rendering the common [[partial differential equation]] for the conservation of mass in monophase fluids:

<math display="block">{\partial \rho \over \partial t} + \nabla \cdot (\rho \vec v) = 0</math>

[[Category:Integral calculus]]