Gradient as vector vs differential one-form

[martinbn]

That was the whole point of the post, that the definition that is often presented to students in multivariable calculus is not coordinate free.

Just to emphasize the obvious, it is not coordinate free, but it is a perfectly valid and correct definition.

 

[vanhees71]

And @jbergman explicitly said that this only works in $\mathbb{R}^n$ and does not work in the absence of coordinates.And that part of @jbergman's post, as he explicitly said, was restricted to this case. He then went on, in the latter part of his post, to give a better formulation that is not dependent on any particular choice of coordinates. So he already gave the clarification that you are trying to give here. You're just repeating what he said in different words.

It works only with Cartesian bases, not for general bases, even not in $\mathbb{R}^n$.

 

[PeterDonis]

It works only with Cartesian bases

Yes, in other words, with a particular choice of coordinates. @jbergman even gave an example of how it does not work in polar coordinates. Did you actually read what he wrote?

 

[vanhees71]

Maybe I misunderstood the notation. I think we all agree finally.