Derivation

Calculus Level 3

A derivation is a R \mathbb R -linear map D : C ∞ ( R n ) → C ∞ ( R n ) D:C^\infty(\mathbb R^n)\to C^\infty(\mathbb R^n) that satisfies Leibniz's law: D ( f g ) = D ( f ) g + f D ( g ) . D(fg)=D(f)g+fD(g). Must the composition of two derivations (i.e. D 1 ∘ D 2 D_1\circ D_2 ) be a derivation?

Yes No

Brian Lie by Brian Lie , 26, China

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1 solution

Otto Bretscher
Nov 16, 2018

As a simple counterexample, with n = 1 n=1 , consider the fact that the first derivative is a derivation, but the second derivative is not since ( f g ) ′ ′ = f ′ ′ g + f g ′ ′ + 2 f ′ g ′ (fg)''=f''g+fg''+2f'g' .

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