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 [[interval (mathematics)|interval]] <var>Ω</var> of the [[real line]] that is [[Continuous function|continuous]] in <var>Ω</var>. Let <var>D</var> be an arbitrary subinterval contained in <var>Ω</var>. The theorem states the following implication:

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

: <math>\int\limits_D F(x) \, \mathrm{d}x = 0 ~ \forall D \subset \Omega ~ \Rightarrow ~ F(x) = 0 ~ \forall x \in \Omega</math>

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

A simple proof is as follows: if there were a point <var>x</var>₀ within <var>Ω</var> for which {{math|<var>F</var>(<var>x</var>₀) &ne;≠ 0}}, then the continuity of {{mathmvar|<var>F</var>}} would require the existence of a [[neighborhood (mathematics)|neighborhood]] of <var>x</var>₀ in which the value of {{mathmvar|<var>F</var>}} was nonzero, and in particular of the same sign than in <var>x</var>₀. 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|<var>F</var>}} over <var>N</var> would evaluate to a nonzero value. However, since <var>x</var>₀ is part of the ''open'' set <var>Ω</var>, all neighborhoods of <var>x</var>₀ smaller than the distance of <var>x</var>₀ to the frontier of <var>Ω</var> are included within it, and so the integral of {{mathmvar|<var>F</var>}} over them must evaluate to zero. Having reached the contradiction that {{math|∫_<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 in <var>Ω</var> for which {{math|<var>F</var>(<var>x</var>₀) &ne;≠ 0}}.

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''&rarr; → '''R'''}}, with the constraints of continuity and nullity of its integral over any subdomain {{math|<var>D</var>&sub;<var> ⊂ Ω</var>}}. The proof is completely analogous to the single variable case, and concludes with the impossibility of finding a point {{math|<var>'''x'''</var>₀ &isin;∈ <var>Ω</var>}} such that {{math|<var>F</var>(<var>'''x'''</var>₀) &ne;≠ 0}}.

{{clear}}

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==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\limits_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>\frac{\mathrm{d}}{\mathrm{d}t}\int\limits_{V_f} \rho(\vec x, t) \, \mathrm{d}\Omega = 0</math>

: <math display="block">\int\limits_int_{V_c} \left [ {{\partial \rho} \over {\partial t}} + \nabla \cdot (\rho \vec v) \right ] \, \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>\int\limits_{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>

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

[[Category:Integral calculus]]