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## Main Question or Discussion Point

This may not be a differential geometry question (and a posting of this in Linear & Abstract Algebra forum didnt help either) but Hodge dual is used in diff geom in a slightly different form. Hence posting here:

If A is a p-vector, then the hodge dual, [tex]*A[/tex] is a (n-p)-vector and is defined by:

[tex] A\ \wedge\ B = (*A,B)E \ \ ,\ \forall B\in \Lambda ^{(n-p)} [/tex]

[tex]\ where,\ \ E = e_1 \wedge\ ... \ \wedge e_n [/tex]

I am having trouble in deriving the tensor components of the dual (n-p)-vector - [tex]*A[/tex] in an orthonormal basis.

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 proceeded as follows, taking B to be a

[tex] i.e.\ \ Let\ B\ =\ e_{i_{p+1}}\ \wedge\ e_{i_{p+2}}\ \wedge\ ... \wedge\ e_{i_n}

\ =\ \frac{1}{n!} \epsilon^{i_{p+1}, ... ,i_n}\ e_{i_{p+1}} \otimes ... \otimes e_{i_n} [/tex]

where [tex] \epsilon [/tex] (epsilon) is the Levi-Civita symbol and [tex] e_{i_x} [/tex] (subscripted e) are the o.n. basis vectors.

[tex] LHS [/tex]

[tex]=\ A\ \wedge\ B\ [/tex]

[tex]=\ A^{[i_1 ... i_p}B^{j_1 ... j_{n-p} ] }\ e_{i_1}\otimes ... \otimes e_{i_p}\otimes e_{i_{p+1}} \otimes ... \otimes e_{i_n} [/tex]

[tex] =\ \left(\ \frac{1}{n!} \sum_\sigma\ (-1)^\sigma\ A^{i_{\sigma (1)} ... i_{\sigma (p)}}\ \epsilon ^ {i_{\sigma (p+1)} ... i_{\sigma (n)}}\ \right)\ \ e_{i_1}\otimes ... \otimes e_{i_p}\otimes e_{i_{p+1}} \otimes ... \otimes e_{i_n} [/tex]

[tex] RHS [/tex]

[tex] =\ (*A,B)\ E [/tex]

[tex] =\ (*A,B)\ e_1 \wedge\ ... \ \wedge e_n [/tex]

[tex] =\ \left(\ (*A,B)\ \frac{1}{n!}\ \epsilon^{i_1, ... ,i_p,i_{p+1}, ... ,i_n}\ \right) e_{i_1}\otimes ... \otimes e_{i_p}\otimes e_{i_{p+1}} \otimes ... \otimes e_{i_n} [/tex]

[tex] =\ \left(\ \frac{1}{n!}\ \left(\ (n-p)!\ *A^{j_1,...,j_{n-p}} \epsilon_{j_1, ... ,j_{n-p}}\ \right)\ \epsilon^{i_1, ... ,i_p,i_{p+1}, ... ,i_n}\ \right) e_{i_1}\otimes ... \otimes e_{i_p}\otimes e_{i_{p+1}} \otimes ... \otimes e_{i_n} [/tex]

Any pointers on how to proceed further to get the components of [tex]*A^{i_1,...,i_{n-p}}[/tex] ?

If A is a p-vector, then the hodge dual, [tex]*A[/tex] is a (n-p)-vector and is defined by:

[tex] A\ \wedge\ B = (*A,B)E \ \ ,\ \forall B\in \Lambda ^{(n-p)} [/tex]

[tex]\ where,\ \ E = e_1 \wedge\ ... \ \wedge e_n [/tex]

I am having trouble in deriving the tensor components of the dual (n-p)-vector - [tex]*A[/tex] in an orthonormal basis.

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 proceeded as follows, taking B to be a

*simple*(n-p)-vector of orthonormal basis vectors...[tex] i.e.\ \ Let\ B\ =\ e_{i_{p+1}}\ \wedge\ e_{i_{p+2}}\ \wedge\ ... \wedge\ e_{i_n}

\ =\ \frac{1}{n!} \epsilon^{i_{p+1}, ... ,i_n}\ e_{i_{p+1}} \otimes ... \otimes e_{i_n} [/tex]

where [tex] \epsilon [/tex] (epsilon) is the Levi-Civita symbol and [tex] e_{i_x} [/tex] (subscripted e) are the o.n. basis vectors.

[tex] LHS [/tex]

[tex]=\ A\ \wedge\ B\ [/tex]

[tex]=\ A^{[i_1 ... i_p}B^{j_1 ... j_{n-p} ] }\ e_{i_1}\otimes ... \otimes e_{i_p}\otimes e_{i_{p+1}} \otimes ... \otimes e_{i_n} [/tex]

[tex] =\ \left(\ \frac{1}{n!} \sum_\sigma\ (-1)^\sigma\ A^{i_{\sigma (1)} ... i_{\sigma (p)}}\ \epsilon ^ {i_{\sigma (p+1)} ... i_{\sigma (n)}}\ \right)\ \ e_{i_1}\otimes ... \otimes e_{i_p}\otimes e_{i_{p+1}} \otimes ... \otimes e_{i_n} [/tex]

[tex] RHS [/tex]

[tex] =\ (*A,B)\ E [/tex]

[tex] =\ (*A,B)\ e_1 \wedge\ ... \ \wedge e_n [/tex]

[tex] =\ \left(\ (*A,B)\ \frac{1}{n!}\ \epsilon^{i_1, ... ,i_p,i_{p+1}, ... ,i_n}\ \right) e_{i_1}\otimes ... \otimes e_{i_p}\otimes e_{i_{p+1}} \otimes ... \otimes e_{i_n} [/tex]

[tex] =\ \left(\ \frac{1}{n!}\ \left(\ (n-p)!\ *A^{j_1,...,j_{n-p}} \epsilon_{j_1, ... ,j_{n-p}}\ \right)\ \epsilon^{i_1, ... ,i_p,i_{p+1}, ... ,i_n}\ \right) e_{i_1}\otimes ... \otimes e_{i_p}\otimes e_{i_{p+1}} \otimes ... \otimes e_{i_n} [/tex]

Any pointers on how to proceed further to get the components of [tex]*A^{i_1,...,i_{n-p}}[/tex] ?