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Removing Category:Management calculus per Wikipedia:Categories for discussion/Log/2023 February 3#Category:Management calculus

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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]] Ω of the [[real line]] that is [[Continuous function|continuous]] in Ω. Let <var>D</var> be an arbitrary subinterval contained in Ω. The theorem states the following implication:
<math display="block">\int_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 Ω for which {{math|<var>F</var>(<var>x</var>0) ≠ 0}}, then the continuity of {{mvar|F}} would require the existence of a [[neighborhood (mathematics)|neighborhood]] of <var>x</var>0 in which the value of {{mvar|F}} 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 {{mvar|F}} over <var>N</var> would evaluate to a nonzero value. However, since <var>x</var>0 is part of the ''open'' set Ω, all neighborhoods of <var>x</var>0 smaller than the distance of <var>x</var>0 to the frontier of Ω are included within it, and so the integral of {{mvar|F}} 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 Ω for which {{math|<var>F</var>(<var>x</var>0) ≠ 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'' → '''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}}

==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>Vc</var>. Further assuming that the [[density|density function]] is continuous (i.e. that our fluid is monophasic and thermodynamically metastable) and that <var>Vc</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]]

Localization theorem - Revision history

imported>JJMC89 bot III: Removing :Category:Management calculus per Wikipedia:Categories for discussion/Log/2023 February 3#Category:Management calculus