I just started learning differential geometry, and my knowledge on exterior algebra is basically nonexistent. While reading about the Invariant Stokes theorem, I encountered this lemma:
If F = (f1, f2, f3) is a vector field in ℝ3, then div(F) = ∇ · F = ∗d(∗F♭)
The document where I found this does not provide a definition of *, nor of ♭. I'm having a hard time digesting all this notation... What do these symbols mean? Is * related to the wedge product? If so, how? What does this mean in terms of differential forms?
Thank you for your time! (Please assume very little previous knowledge from me)
Source
$*$ is the Hodge star.
$♭$ is the musical isomorphism.
In this particular case, this amounts to
\begin{align*} (F_1,F_2,F_3)&\stackrel{♭}{\rightarrow}F_1dx_1+F_2dx_2+F_3dx_3 \\ &\stackrel{*}{\rightarrow}F_1dx_2dx_3-F_2dx_1dx_3+F_3dx_1dx_2 \\ &\stackrel{d}{\rightarrow}\partial_1F_1dx_1dx_2dx_3+\partial_2F_2dx_1dx_2dx_3+\partial_3F_3dx_1dx_2dx_3 \\ &\stackrel{*} {\rightarrow}\partial_1F_1+\partial_2F_2+\partial_3F_3, \end{align*} as you can check in the definitions given in the links above.