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I am asking for clarification regarding expressions such as in here:

start with the left-multiplication map $\phi(x)=gx$ and compute its differential $d\phi(x, dx)=gdx$ which is the push-forward map of the vector $dx\in T_xG$ to $T_{gx}G$.

or here:

an assignment of a linear map to each point $x\in M$ which maps small deviations $dx \in TM_{x}$ to deviations $dy\in TN_{\varphi(x)}$ in a linear way.

Both quotes seem to be referring to the pushforward ('differential'). However, the notation $dx$ is used and explicitly said to be in the tangent space of the manifold $G$, as opposed to $\frac \partial {\partial x}$ or $\partial x$ (derivation notation). This is confusing in that $dx$ I believe is used for differential forms, not vectors. Vectors are pushed forward, and differential forms pulled back.

The notation above is likely correct, but I am missing or misunderstanding part of the story. It could also be equivalent, or part of some flexibility in expressing the same concept. I don't know.

JAP
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    This appears to just be an error of notation. (Well, it's not strictly wrong, since you can define $dx$ to be a tangent vector if you want to, but this is not normal notation to use.) – Eric Wofsey Nov 24 '22 at 00:33
  • @EricWofsey Kind of flipping the dual and the vector space, being isomorphic, or something like that? I was asking the question to make sure I didn't completely misunderstood the elements in the construct, and with your comment now, it seems more along slight-of-hand, or loose notation. – JAP Nov 24 '22 at 00:47
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    The second example looks like it is meant to evoke the naive Leibniz notation idea of $dx$ and $dy$ as infinitesimals (paralleling the use of $\Delta r$ earlier in the answer), so for that one I wouldn't call it an error so much as an intentional pedagogical choice of an unusual notation. – Eric Wofsey Nov 24 '22 at 00:53

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