If $f$ and $g$ are $C^{∞}$ functions and $X$ and $Y$ are $C^{∞}$ vector fields on a manifold $M$, show that $$[fX,gY]= fg[X,Y]+f(Xg)Y−g(Yf)X.$$
This is a proposition in a book. But I cannot prove this.
2026-03-27 02:37:39.1774579059
Show that $[fX,gY]= fg[X,Y]+f(Xg)Y−g(Yf)X$.
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By definition, for any two vector fields $X,Y$, we have $$[X,Y]=XY-YX.$$ Note also that for any two vector fields $X,Y$ and a smooth function $g$, we have $$X(gY)=(Xg)Y+gXY.$$ Using these, we have $$[fX,gY]=fX(gY)-gY(fX)= f(Xg)Y+fgXY-g(Yf)X-gfYX$$ $$ =f(Xg)Y-g(Yf)X+fg(XY-YX)=f(Xg)Y-g(Yf)X+fg[X,Y],$$ as required.