Derivation 2

Calculus Level 2

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 commutator of two derivations (i.e. D 1 ∘ D 2 − D 2 ∘ D 1 D_1\circ D_2-D_2\circ D_1 ) be a derivation?

No Yes

Brian Lie by Brian Lie , 26, China

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Otto Bretscher
Nov 16, 2018

One can verify this with a straightforward computation: D 1 ( D 2 ( f g ) ) − D 2 ( D 1 ( f g ) ) = ( D 1 D 2 f ) g + f ( D 1 D 2 g ) − ( D 2 D 1 f ) g − f ( D 2 D 1 g ) D_1(D_2(fg))-D_2(D_1(fg))=(D_1D_2f)g+f(D_1D_2g)-(D_2D_1f)g-f(D_2D_1g) , with the terms ( D 1 f ) ( D 2 g ) (D_1f)(D_2g) and ( D 1 g ) ( D 2 f ) (D_1g)(D_2f) canceling out. A simple but interesting observation.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...