Definition of dx: What is its Domain & Formalization?

[brunoschiavo]

Homework Statement

http://imgur.com/goozE9f

Homework Equations

$(dx_i)_p i= 1,2,3$

3. The Attempt at a Solution [/B]
I'm reading Manfredo and Do Carmo's Differential Forms and Applications. This is the very first page

Would you mind explaining me what is meant by dx, as highlighted in the picture? I guess it is "differential", like in Calculus textbooks, but what kind of mathematical object is it? A set, a line, a point? What is its domain? How is it formalized?

Instances of

compose the space's basis. Are they arbitrary, or should they be selected in some way?PS: Is there any way to post the image here without uploading it to your server?

 

[andrewkirk]

Welcome to physicsforums Bruno.

That text explains it in rather a confusing fashion.

Using the notation of the text, $(dx_i)_p$ is a linear function from the tangent space $\mathbb{R}_p^3$ to $\mathbb{R}$. Its value is given by its action on basis vectors of that tangent space:

$(dx_i)_p \big((\vec{e}_j)_p\big)=\delta_{ij}$ where $\delta_{ij}$ is the Kronecker Delta, which is 0 unless $i=j$, in which case it is 1.

Note that, under this notation, $(dx_i)_p$ is defined with respect to a basis $\{(\vec{e}_1)_p,(\vec{e}_2)_p,(\vec{e}_3)_p\}$ for the tangent space. Each differential form $(dx_i)_p$ corresponds to a particular basis vector $(\vec{e}_1)_p$ and is called its 'dual'.

To post an image, you can just copy the image on your computer and paste it into the editing area with ctrl-V.