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Wedge Product

An antisymmetric operation on Differential Forms (also called the Exterior Derivative)

dx_i\wedge dx_j = -dx_j\wedge dx_i,
\end{displaymath} (1)

which Implies
$\displaystyle dx_i\wedge dx_i$ $\textstyle =$ $\displaystyle 0$ (2)
$\displaystyle b_i\wedge dx_j$ $\textstyle =$ $\displaystyle dx_j\wedge b_i = b_i\,dx_j$ (3)
$\displaystyle dx_i\wedge(b_i \,dx_j)$ $\textstyle =$ $\displaystyle b_i \,dx_i\wedge dx_j$ (4)
$\displaystyle \theta_1\wedge\theta_2$ $\textstyle =$ $\displaystyle (b_1\,dx_1+b_2\,dx_2)\wedge(c_1\,dx_1+c_2\,dx_2)$  
  $\textstyle =$ $\displaystyle (b_1c_2-b_2c_1)\,dx_1\wedge dx_2$  
  $\textstyle =$ $\displaystyle -\theta_2\wedge\theta_1.$ (5)

The wedge product is Associative
(s\wedge t)\wedge u=s\wedge(t\wedge u),
\end{displaymath} (6)

and Bilinear
(\alpha _1s_1+\alpha _2s_2)\wedge t=\alpha _1(s_1\wedge t)+\alpha _2(s_2\wedge t)
\end{displaymath} (7)

s\wedge(\alpha _1t_1+\alpha _2t_2)=\alpha _1(s\wedge t_1)+\alpha _2(s\wedge t_2),
\end{displaymath} (8)

but not (in general) Commutative
s\wedge t=(-1)^{pq} (t\wedge s),
\end{displaymath} (9)

where $s$ is a $p$-form and $t$ is a $q$-form. For a 0-form $s$ and 1-form $t$,
(s\wedge t)_\mu = st_\mu.
\end{displaymath} (10)

For a 1-form $s$ and 1-form $t$,
(s\wedge t)_{\mu\nu} = {\textstyle{1\over 2}}(s_\mu t_\nu-s_\nu t_\mu).
\end{displaymath} (11)

The wedge product is the ``correct'' type of product to use in computing a Volume Element
dV=dx_1\wedge \ldots\wedge dx_n.
\end{displaymath} (12)

See also Differential Form, Exterior Derivative, Inner Product, Volume Element

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© 1996-9 Eric W. Weisstein