germ of smooth functions
If is a point on a smooth manifold , then a germ of smooth functions near is represented by a pair where is an open neighbourhood of , and is a smooth function . Two such pairs and are considered equivalent if there is a third open neighbourhood of , contained in both and , such that . To be precise, a germ of smooth functions near is an equivalence class of such pairs.
In more fancy language: the set of germs at is the stalk at of the sheaf of smooth functions on . It is clearly an -algebra.
Germs are useful for defining the tangent space in a coordinate-free manner: it is simply the space of all -linear maps satisfying Leibniz’ rule . (Such a map is called an -linear derivation of with values in .)
|Title||germ of smooth functions|
|Date of creation||2013-03-22 13:05:08|
|Last modified on||2013-03-22 13:05:08|
|Last modified by||rspuzio (6075)|