I am trying to show that the different ways of defining the Hodge star operator are equivalent.
I started with the following definition from the following lecture notes "An application of discrete differential geometry to the spectral element method" by G. Oud.
Then one can show that
Where the index sets H and K are defined as $H:=\{1,...,p\}$ and $K:=\{p+1,...,n\}$. I am fine with that. Then it says that from that one can derive the formula:
My Questions:
- Can equation 1.35 be shown by using linearity and the fact that $\star e^{H}=g(e^{K},e^{K})e^{K}$?
- How is the other direction shown, i.e, equation 1.35 implies equation 1.31? My guess is that I need to use $\star\star$, but then I only get the answer up to $(-1)^{k(n-k)}$.
- Is there a way to show equation 1.35 implies equation 1.31 without using $\star\star$?
Thank you very much in advance for your help!