Mapping with change of variable

[jason12345]

For an infinitesimal mapping with u = 1,2,3,4:

[tex]x ^u \rightarrow x^u + \xi^u(x) [/tex]

Now suppose we introduce a new set of variables:

[tex]x^{'u} = x^{'u}(x)[/tex]

I would have thought the infinitesimal mapping in terms of the new variables should be written as:

[tex] \xi^{'u}(x^{'}) = \frac{\partial \xi^{u}(x)}{\partial x^p} x^{'p} (x)[/tex]

However, it is written as:

[tex] \xi^{'u}(x^{'}) = \frac{\partial x^{'u}}{\partial x^u} \xi^{u} (x)[/tex]

Does this look correct to you?

 

[tiny-tim]

Hi jason12345!

The infinitesimal transformation is linear, and is essentially a matrix:

y = (I + Z)x
y' = (I + Z')x'

The coordinates of Z transform as Z' = (Jacobian)Z