- #1

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I'm trying to get my head around the Hodge dual and how it exactly works. In the book "Gauge Fields, Knots and Gravity" by John Baez and Javier P. Muniain they define:

\begin{equation}

\omega \wedge * \mu = \langle \omega , \mu \rangle \mathrm{vol}

\end{equation}

for two p-forms. This implies that:

\begin{equation}

\omega \wedge * \mu = * \mu \wedge \omega

\end{equation}

Therefore, if we consider a vector space with basis dx, dy, dz, and

\begin{equation}

\omega = \omega_x \mathrm{d}x

\end{equation}

\begin{equation}

*\mu = \mu_y \mathrm{d} y

\end{equation}

Then the definition by Baez and Muniain yields:

\begin{equation}

\omega_x \mu_y \mathrm{d}x \wedge \mathrm{d} y = \mu_y \omega_x \mathrm{d} y \wedge \mathrm{d}x

\end{equation}

However, if I would try to calculate the above equation using 'anti-commuting' property of the wedge product, then I get:

\begin{equation}

\omega \wedge * \mu = \omega_x \mu_y \mathrm{d}x \wedge \mathrm{d} y = - \omega_x \mu_y \mathrm{d}y \wedge \mathrm{d} x \neq * \mu \wedge \omega

\end{equation}

So clearly, I am going wrong somewhere, but I can't see where I'm going wrong.