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===One independent variable===
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Consider the simplest case, a system with one independent variable, time. Suppose the dependent variables '''q''' are such that the action integral
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===Field-theoretic derivation===
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Noether's theorem may also be derived for tensor fields <math>\varphi^A</math> where the index ''A'' ranges over the various components of the various tensor fields. These field quantities are functions defined over a four-dimensional space whose points are labeled by coordinates ''x''<sup>μ</sup> where the index ''μ'' ranges over time (''μ'' = 0) and three spatial dimensions (''μ'' = 1, 2, 3). These four coordinates are the independent variables; and the values of the fields at each event are the dependent variables. Under an infinitesimal transformation, the variation in the coordinates is written
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