= Exponential map
{disambiguate=Lie theory}
{wiki}
Like everything else in <Lie group> theory, you should first look at the <matrix> version of this operation: the <matrix exponential>.
The <exponential map> links small transformations around the origin (infinitely small) back to larger finite transformations, and small transformations around the origin are something we can deal with a <Lie algebra>, so this map links the two worlds.
The idea is that we can decompose a finite transformation into infinitely arbitrarily small around the origin, and proceed just like the <product definition of the exponential function>.
The definition of the exponential map is simply the same as that of the regular exponential function as given at <Taylor expansion definition of the exponential function>, except that the argument $x$ can now be an operator instead of just a number.
Examples:
* <the derivative is the generator of the translation group>