MHB Problem: How to prove the vector field identity $[v,fw]=(L_vf)w+f[v,w]$?

[Chris L T521]

Here's this week's problem!

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Problem: Let $v$ and $w$ be smooth vector fields on a smooth manifold $M$ and let $f$ be a smooth function. Prove that\[[v,fw]=(L_vf)w+f[v,w].\]

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[Chris L T521]

No one answered this week's problem. You can find my solution below.

[sp]For $v,w$ smooth vector fields on $M$ and $f$ a smooth function, we have
\[\begin{aligned}{[v,fw]} &= (v(fw)^i-fwv^i)\frac{\partial}{\partial x^i}\\ &= \left(\sum v_j\frac{\partial}{\partial x^j}(fw)^i-fwv^i\right)\frac{\partial}{\partial x^i}\\ &= \left(\sum v_j\left(\frac{\partial f}{\partial x^j}w^i + f\frac{\partial w^i}{\partial x^j}\right)-fwv^i\right)\frac{\partial}{\partial x^i}\\ &= \left(L_v fw^i +f\sum v_j\frac{\partial w^i}{\partial x^j}-fwv^i\right)\frac{\partial}{\partial x^i}\\ &=L_v f w^i\frac{\partial}{\partial x^i}+(fvw^i-fwv^i)\frac{\partial}{\partial x^i}\\ &= (L_v f)w+f[v,w].\end{aligned}\][/sp]