germ of smooth functions

If x is a point on a smooth manifoldMathworldPlanetmath M, then a germ of smooth functions near x is represented by a pair (U,f) where UβŠ†M is an open neighbourhood of x, and f is a smooth functionMathworldPlanetmath U→ℝ. Two such pairs (U,f) and (V,g) are considered equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath 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 classMathworldPlanetmath of such pairs.

In more fancy languagePlanetmathPlanetmath: the set π’ͺx of germs at x is the stalk at x of the sheaf π’ͺ of smooth functions on M. It is clearly an ℝ-algebra.

Germs are useful for defining the tangent spaceMathworldPlanetmathPlanetmath Tx⁒M in a coordinate-free manner: it is simply the space of all ℝ-linear maps X:π’ͺx→ℝ satisfying Leibniz’ rule X⁒(f⁒g)=X⁒(f)⁒g+f⁒X⁒(g). (Such a map is called an ℝ-linear derivation of π’ͺx with values in ℝ.)

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 53B99