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Real advantage in considering germs of smooth functions

Nicer Description of Germs of Continuous FunctionsHow special are the polynomials amongst the smooth functions?Germs of Smooth Functions vs Germs of Analytic Functions?Is there a relationship between germs and Taylor coefficients?Definition of smooth manifold using sheaves.$(z_1,dotsc,z_n)$ is a prime ideal of $mathcalO_mathbbC^n,0$: why?Two functions that have the same germ have the same derivationDifferent definitions of derivation at a pointSmoothness is localEquivalence of definitions for ring of germs $C_p^infty(mathbb R^n)$

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I went back to read some manifold theory recently and I realized that I can't justify to myself the reason to consider germs of smooth functions over simply smooth functions other than formalism, because we know that a point-derivation of smooth functions depends only on the germ of a given smooth function, that is, its local behaviour around a point. But for germs we can always consider a representative defined on the whole manifold via bump functions, and the algebraic structure of the space of germs $mathcalC^infty_p (mathcalM) $ is nothing more special than the ring structure of $mathcalC^infty(mathcalM)$, so I have this question: what do I earn by considering germs of functions, really?

smooth-manifolds smooth-functions germs

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asked Mar 16 at 21:55

B. ChinaskiB. Chinaski

462211

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$begingroup$

I went back to read some manifold theory recently and I realized that I can't justify to myself the reason to consider germs of smooth functions over simply smooth functions other than formalism, because we know that a point-derivation of smooth functions depends only on the germ of a given smooth function, that is, its local behaviour around a point. But for germs we can always consider a representative defined on the whole manifold via bump functions, and the algebraic structure of the space of germs $mathcalC^infty_p (mathcalM) $ is nothing more special than the ring structure of $mathcalC^infty(mathcalM)$, so I have this question: what do I earn by considering germs of functions, really?

smooth-manifolds smooth-functions germs

share|cite|improve this question

asked Mar 16 at 21:55

B. ChinaskiB. Chinaski

462211

$endgroup$

add a comment | 

$begingroup$

I went back to read some manifold theory recently and I realized that I can't justify to myself the reason to consider germs of smooth functions over simply smooth functions other than formalism, because we know that a point-derivation of smooth functions depends only on the germ of a given smooth function, that is, its local behaviour around a point. But for germs we can always consider a representative defined on the whole manifold via bump functions, and the algebraic structure of the space of germs $mathcalC^infty_p (mathcalM) $ is nothing more special than the ring structure of $mathcalC^infty(mathcalM)$, so I have this question: what do I earn by considering germs of functions, really?

smooth-manifolds smooth-functions germs

share|cite|improve this question

asked Mar 16 at 21:55

B. ChinaskiB. Chinaski

462211

$endgroup$

share|cite|improve this question

asked Mar 16 at 21:55

B. ChinaskiB. Chinaski

462211

share|cite|improve this question

asked Mar 16 at 21:55

B. ChinaskiB. Chinaski

462211

asked Mar 16 at 21:55

B. ChinaskiB. Chinaski

462211

asked Mar 16 at 21:55

B. ChinaskiB. Chinaski

462211