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<math>\forall\phi\in\mathcal{C}\, S[\phi]\equiv\int_M d^nx \mathcal{L}(\phi(x),\partial_\mu\phi(x),x)</math>.
Given [[boundary]] conditions, which is basically a specification of the value of φ at the [[boundary]] of M is [[compact]] or some limit on φ as x approaches <math>\infty</math> (this will help in doing [[integration by parts]]), we can denote as N the [[subset]] of <math>\mathcal{C}</math> consisting of functions, φ such that all [[functional derivative]]s of S at φ are zero and φ satisfies the given boundary conditions.
Now, suppose we have an [[infinitesimal]] [[Transformation (mathematics)|transformation]] on <math>\mathcal{C}</math>, given by a [[functional derivative]], δ such that
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