# Difference between revisions of "Corollary of Leibniz rule for Lie bracket"

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Let <math>X,Y</math> be smooth [[vector field]]s on a [[differential manifold]] <math>M</math> and <math>f</math> be in <math>C^\infty(M)</math>. We then have: | Let <math>X,Y</math> be smooth [[vector field]]s on a [[differential manifold]] <math>M</math> and <math>f</math> be in <math>C^\infty(M)</math>. We then have: | ||

− | <math>f[X,Y] = [fX,Y] + (Yf)X</math> | + | <math>\! f[X,Y] = [fX,Y] + (Yf)X</math> |

− | <math>f[X,Y] = [X,fY] - (Xf)Y</math> | + | <math>\! f[X,Y] = [X,fY] - (Xf)Y</math> |

==Applications== | ==Applications== |

## Latest revision as of 18:42, 24 July 2009

## Statement

This is an identity that uses the Leibniz rule to measure the failure of the Lie bracket operation from being -linear.

Let be smooth vector fields on a differential manifold and be in . We then have:

## Applications

## Proof

### First identity

We prove this by showing that for any , both sides evaluate to the same thing. Let's do this. Simplifying the right side yields:

Applying the Leibniz rule for on the product of functions and , this simplifies to:

which is precisely equal to the left side.

### Second identity

We prove this as well by taking any test function . Simplifying the right side yields:

which is the same as the left side.