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

Latest revision as of 18:42, 24 July 2009

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

Let $X,Y$ be smooth vector fields on a differential manifold $M$ and $f$ be in $C^\infty(M)$ . We then have:

$\! f[X,Y] = [fX,Y] + (Yf)X$

$\! f[X,Y] = [X,fY] - (Xf)Y$

We prove this by showing that for any $g \in C^\infty(M)$ , both sides evaluate to the same thing. Let's do this. Simplifying the right side yields:

$f(X(Yg) - Y(Xg)) - f(X(Yg)) + Y((fX)g) = Y(f(Xg)) - f(Y(Xg))$

Applying the Leibniz rule for $Y$ on the product of functions $f$ and $Xg$ , this simplifies to:

$(Yf)(Xg) + Y(Xg)(f) - f(Y(Xg)) = (Yf)(Xg)$

which is precisely equal to the left side.

We prove this as well by taking any test function $g \in C^\infty(M)$ . Simplifying the right side yields:

$[X,fY]g - (Xf)(Yg) = X(f(Yg)) -f(YXg) - (Xf)(Yg) = (Xf)(Yg) + f(XYg) - f(YXg) - (Xf)(Yg) = f[X,Y]g$

which is the same as the left side.

@@ Line 5: / Line 5: @@
 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==