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I'm my current studies in Noether's theorem, the two that I liked the most are joshphysics answer to this Phys SE. post, and the derivation in chapter $4$ of An Elementary Introduction to Classical Fields by R. Aldrovandi and J. G. Pereira. Johphysics answer is nice because he introduces a notion of variation of a field $\delta \phi$ as derivative of a flow instead of the usual treatment of "infinitesimals". He derives Noether's theorem only using what is called "functional variations", wich does not change the integration domain of an action functional $S[\phi]$, and affects only the functional form of the fields and its lagrangian $\mathcal{L}(\phi(x), \partial_\mu \phi(x))$.

Aldrovandi, however, even using the notion of variations as "infinitesimas", make clear statements about what is going on. Specifically in equations $4.16 - 4.18$, he defines the coordinate variation as well as total and functional variation of fields $\phi(x)$. And states in the paragraph before page 98 the following:

Let us go back to the action functional $(4.15)$. It does not depend on x. Its is a functional of the elds, depending on the integration domain. Variations $(4.16)$ and $(4.17)$ will have eects of two kinds: changes in the integration volume and in the Lagrangian density. We shall indicate this by writing $$ \delta A[\phi] = \int [\delta(d^4x)\mathcal{L} + d^4x \delta \mathcal{L}]$$

In Aldrovandi, the term $\delta\mathcal{L}$ is used as a total variation, wich is different from the definition of $\delta \mathcal{L}$ used by johphysics.

The result of what is called Noether's first theorem is given by the equation

$$\partial_\mu J^\mu = 0,$$ which is the same for both derivations, but one uses only variation of functional form of fields, and the other uses contributions due to coordinate change, how is that possible?

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