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I saw that in differential topology, we are using the terms $dx,dy..$ . What exactly is that?

For example an example of differential form on $\mathbb{R}^2$ would be $d(X,Y)= x dx + (y^2-x) dy$. My doubt is, what is $dx,dy$ and what is the codomain of this differential form?

permutation_matrix
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  • Possible repeat of this post: https://math.stackexchange.com/questions/2858098/what-is-a-differential-form/2858132 – klein4 May 26 '21 at 17:00
  • can you explain what do they mean by $dx,dy$? I know the definition of a differential form – permutation_matrix May 26 '21 at 17:28
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    $\newcommand{\R}{\mathbb{R}}$ The simplest answer is the following: You should already know that if $f: \R^2 \rightarrow \R$ is a smooth function, then its differential $df$ is a differential $1$-form. If you use $(x,y)$ as coordinates on $\mathbb{R}^2$, then this defines a smooth function $x: \mathbb{R}^2 \rightarrow \R$, where $x(a,b) = a$. Then $dx$ is simply the differential of the function $x$. – Deane May 26 '21 at 18:40

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