If x is a point on a smooth manifold M , then a germ of smooth functions near x is represented by a pair ( U , f ) where U β M is an open neighbourhood of x , and f is a smooth function U β β . Two such pairs ( U , f ) and ( V , g ) are considered equivalent if there is a third open neighbourhood W of x , contained in both U and V , such that f | W = g | W . To be precise, a germ of smooth functions near x is an equivalence class of such pairs.
In more fancy language: the set πͺ x of germs at x is the stalk at x of the sheaf πͺ of smooth functions on M . It is clearly an β -algebra.
Germs are useful for defining the tangent space T x β’ M in a coordinate-free manner: it is simply the space of all β -linear maps X : πͺ x β β satisfying Leibnizβ rule X β’ ( f β’ g ) = X β’ ( f ) β’ g + f β’ X β’ ( g ) . (Such a map is called an β -linear derivation of πͺ x with values in β .)
Title germ of smooth functions Canonical name GermOfSmoothFunctions Date of creation 2013-03-22 13:05:08 Last modified on 2013-03-22 13:05:08 Owner rspuzio (6075) Last modified by rspuzio (6075) Numerical id 4 Author rspuzio (6075) Entry type Definition Classification msc 53B99RetroSearch is an open source project built by @garambo | Open a GitHub Issue
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