confusion regarding tangent vector basis

[naive swan]

If $T_p\mathbb{R}^n \cong D_p\mathbb{R}^n$, is it sufficient for equating the tangent space with the derivation space?

[minor flame]

majesty

[naive swan]

i.e for the rest of the book Tu just explicitly makes the partial derivative basis of the derivation space the basis of the tangent space, instead of using the arbitrary $(e_1,\dots,e_n)$ basis

is this justified only given that isomorphism is proven?

[dawn pagoda]

well by definition a tangent vector is based on a derivative of another vector or graph

$(e_1,\dots,e_n)$

[minor flame]

ashton_the_ghostly_nerd

[naive swan]

i mean like

wouldnt $$\sum v^i \left. \frac{\partial}{\partial x^i} \right|_p$$ just be a representation of $v \in T_p\mathbb{R}^n$ or something

[minor flame]

majesty

[naive swan]

like the isomorphism allows for that but i dont get when it is justified to directly say $$\sum_i v^i \left. \frac{\partial}{\partial x^i} \right|_p \in T_p\mathbb{R}^n$$

[minor flame]

majesty

[naive swan]

since the partial derivative basis is the basis of the derivation space

not yk

the tangent space

even though theyre isomorphic

[dawn pagoda]

well if they're isomorphic doesn't a=b and b=c lead to the conclusion a=c?

[naive swan]

but isomorphic jus means bijective tho (the map used for the proof is linear so a bijection should be sufficient)

like direct equality (=) isnt the same as isomorphism

[dawn pagoda]

man i shoulda taken a logic class

so we're basically running into the issue of "we know the conclusion is right but the reasoning is unclear or faulty"

[naive swan]

kinda

[dawn pagoda]

is this for a class?

[naive swan]

nah im self learning sumt

[dawn pagoda]

ah i see

what's the topic this is building up to?

bc this may be something that will make more sense if the big picture is made clear

[naive swan]

well, diffgeo obv

im trying to do the maths up until doing integration on manifolds

probably

then id consider doing general relativity with this idk

[dawn pagoda]

ooh

now you really have my attention

[dawn pagoda]

well yeah but i guess i mean more "what conclusions are made that rely on this being true"

more datapoints in the train of thought could help us find a better reasoning for this step in the thought process

[naive swan]

yeah id prolly do GR

black holes and shi feels cool

[dawn pagoda]

oh yeah

there's this thing I'd like to prove or at least investigate mathematically using general and special relativity