Localization theorem

{{unreferenced|date=June 2021}} In mathematics, particularly in integral calculus, the '''localization theorem''' allows, under certain conditions, to infer the nullity of a function given only information about its 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 Ω of the real line that is 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>

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 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 functions, replacing intervals with the more general concept of connected open sets, that is, 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}}.

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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_{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 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