Take the group of all Translations in $R_{1}$.

The way to think about this is:

- the translation group operates on the argument of a function $f(x)$
- the generator is an operator that operates on $f$ itself

So let's take the exponential map:
and we notice that this is exactly the Taylor series of $f(x)$ around the identity element of the translation group, which is 0! Therefore, if $f(x)$ behaves nicely enough, within some radius of convergence around the origin we have for finite $x_{0}$:

$e_{x_{0}∂x∂}f(x)=(1+x_{0}∂x∂ +x_{0}∂x_{2}∂_{2} +…)f(x)$

$e_{x_{0}∂x∂}f(x)=f(x+x_{0})$

This example shows clearly how the exponential map applied to a (differential) operator can generate finite (non-infinitesimal) Translations!

- Translation group | 19, 225, 1
- Translation | 13, 238, 2
- Galilean transformation | 0, 610, 5
- Poincaré group | 192, 1k, 13
- Important Lie groups | 0, 3k, 45
- Lie group | 278, 5k, 72
- Differential geometry | 12, 5k, 73
- Geometry | 0, 7k, 118
- Mathematics | 17, 28k, 633
- Ciro Santilli's Homepage | 262, 218k, 4k

- Exponential map | 130