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The theorem is easily generalized to [[multivariate calculus|multivariate function]]s, replacing intervals with the more general concept of connected open sets, that is, [[Domain (mathematics)#Real and complex analysis|___domain]]s, and the original function with some {{math|F(<var>'''x'''</var>) : '''R'''<sup>n</sup>→'''R'''}}, with the constraints of continuity and nullity of its integral over any subdomain {{math|<var>D</var>⊂<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'''<sub>0</sub></var> ∈ <var>Ω</var>}} such that {{math|F(<var>'''x'''<sub>0</sub></var>)≠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>\int\limits_{V_f} \rho(\vec x, t) 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 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 ] 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>{{\partial \rho} \over {\partial t}} + \nabla \cdot (\rho \vec v) = 0</math>
[[Category:Mathematics]]
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