If A is a p-vector and B is a (n-p)-vector, then the hodge dual, *A, is defined by:(adsbygoogle = window.adsbygoogle || []).push({});

[TEX] A\ \wedge\ B = (*A,B)E \ \ \forall B\in \Lambda ^{(n-p)} [/TEX], where E=[TEX]e_1 \wedge\ ... \ \wedge e_n [/TEX]

I am having trouble in deriving the tensor components of the dual (n-p)-vector - *A.

Specifically, I am getting stuck when I write down the components of the (n,0)-tensor on both sides and then comparing the coefficients - because, the LHS involves the antisymmetrization, [TEX] A^{[i_1 ... i_p}B^{j_1 ... j_{n-p} ] } [/TEX].

I got stuck with the LHS even when I took B to be just a simple (n-p)-vector of basis vectors. Because, when I do that, I get the following (n,0)-tensor on LHS...

[TEX]\frac{1}{n!} \sum_\sigma\ (-1)^\sigma\ A^{i_{\sigma (1)} ... i_{\sigma (p)}}\ \epsilon ^ {j_{\sigma (1)} ... j_{\sigma (n-p)}}\ \ e_{i_1}\otimes ... \otimes e_{i_p}\otimes e_{j_1} \otimes ... e_{j_{n-p}} [/TEX]

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Tensor components of a Hodge dual

**Physics Forums - The Fusion of Science and Community**