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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>\int\limits_D F(x) \,
[[File:Localization Theorem.svg|float|right]]
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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>\frac{\mathrm{d}}{\mathrm{d}t}\int\limits_{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<sub>c</sub></var>. Further assuming that the [[density|density function]] is continuous (i.e. that our fluid is monophasic and thermodinamically metastable) and that <var>V<sub>c</sub></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:
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