How can we verify the result that
$$[f\textbf{X},\textbf{Y}]= f[\textbf{X}, \textbf{Y}] - \textbf{Y} \left( f \right) \bf{X}$$
Where:
f- Some function
$\textbf{X}, \textbf{Y}$ - Some vector fields
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Can you explain your notation? I guess $fX$ denotes the function of directional differentials of $f$ according to $X$. But how do you mean $Y(f)X$? – Berci Mar 19 '20 at 18:04
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Ah by that I mean the vector field $\textbf{Y}$ acting on $f$. – Kirito Mar 19 '20 at 19:04