RElation of partial differential operator and Basis vector

[dpa]

Hi everyone:

How is the following derived? Just for example:

$\Delta$x^$\alpha$e_$\alpha$=$\Delta$x^$\alpha$($\delta$/$\delta$x^$\alpha$)

does it not mean?

e_$\alpha$=$\delta$/$\delta$x^$\alpha$

But How?

 

[Matterwave]

Some really odd notation there...but ok. Basically these derivatives of curves form a vector space (one should check that they do, the only non-trivial property of vector spaces to be checked is closure under addition), and so you can identify your vectors with them. They are derivatives of arbitrary functions. One could also associate the vector field with the equivalence class of curves which have the same tangent at the point you are looking at. It depends on what "correspondence" you want to make.

 

[dx]

The symbol ∂/∂xⁱ stands for the tangent vector of the curve a → (0, ..., a, ...) where a is in the i'th position. This set of tangent vectors associated with a coordinate system is a natural basis for the tangent space when working with those coordinates.

 

[dpa]

So, typically almost all coordinate transformations that I came across in GR or even in introduction to tensors include coordinate transformation including the above transformation. <I did not bother to write it again sorry.>

So, does that mean, all transformation involve transformation from curved spacetime to lorentzian flat space time?
Is it radically different if the transformation involves transformation from one curved coordinate system to another curved spacetime?

Thank You.

 

[Matterwave]

What does this have to do with coordinate transformations? The definition of a vector is coordinate independent.

 

[dpa]

I mean in tangent space,

SCR= MTW Gravitation

So is it not associated with coordinate systems and transformations.