Critical notes about Noether's theorem (Part I)

In this brief critical notes, we state and prove the classic Noether's theorem, in a language and in the mood of modern Differential Geometry. This modern approach, on one hand, wants to stimulate a dialogue between the classic Noether's theorem version users and the modern dierential geometry developers, on the other hand, it opens the way for some extensions of the theorem, with possible physical insights about the matter.