Derivation
From Maths
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Contents
[hide]Definition
If a∈Rn, we say that a map, α:C∞(Rn)→R is a derivation at a if it is [[Linear map|R-linear and satisfies the following[1]:
- Given f,g∈C∞(Rn) we have:
- α(fg)=f(a)α(g)+g(a)α(f)
Questions to answer
- What is fg? Clearly we somehow have ×:C∞(Rn)×C∞(Rn)→C∞(Rn) but what it is?
References
- Jump up ↑ Introduction to Smooth Manifolds - John M. Lee - Second Edition - Springer GTM
OLD PAGE
Warning: the definitions below are very similar
Definition
Derivation of C∞p
A derivation at a point is any R−Linear map: D:C∞p(Rn)→R that satisfies the Leibniz rule - that is D(fg)|p=f(p)Dg|p+g(p)Df|p
Recall that C∞p(Rn) is a set of germs - specifically the set of all germs of smooth functions at a point
Derivation at a point
One doesn't need the concept of germs to define a derivation (at p), it can be done as follows:
D:C∞(Rn)→Rn is a derivation if it is R−Linear and satisfies the Leibniz rule, that is:
D(fg)=f(p)Dg+g(p)Df
Warnings
These notions are VERY similar (and are infact isomorphic (both isomorphic to the Tangent space)) - but one must still be careful.