Homework Help: Definition of dx

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1. Sep 23, 2015

brunoschiavo

1. The problem statement, all variables and given/known data
http://imgur.com/goozE9f
2. Relevant equations
$(dx_i)_p i= 1,2,3$

3. The attempt at a solution

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?

2. Sep 23, 2015

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'.

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