An antisymmetric operation on Differential Forms (also called the Exterior Derivative)
![\begin{displaymath}
dx_i\wedge dx_j = -dx_j\wedge dx_i,
\end{displaymath}](w_418.gif) |
(1) |
which Implies
The wedge product is Associative
![\begin{displaymath}
(s\wedge t)\wedge u=s\wedge(t\wedge u),
\end{displaymath}](w_428.gif) |
(6) |
and Bilinear
![\begin{displaymath}
(\alpha _1s_1+\alpha _2s_2)\wedge t=\alpha _1(s_1\wedge t)+\alpha _2(s_2\wedge t)
\end{displaymath}](w_429.gif) |
(7) |
![\begin{displaymath}
s\wedge(\alpha _1t_1+\alpha _2t_2)=\alpha _1(s\wedge t_1)+\alpha _2(s\wedge t_2),
\end{displaymath}](w_430.gif) |
(8) |
but not (in general) Commutative
![\begin{displaymath}
s\wedge t=(-1)^{pq} (t\wedge s),
\end{displaymath}](w_431.gif) |
(9) |
where
is a
-form and
is a
-form. For a 0-form
and 1-form
,
![\begin{displaymath}
(s\wedge t)_\mu = st_\mu.
\end{displaymath}](w_432.gif) |
(10) |
For a 1-form
and 1-form
,
![\begin{displaymath}
(s\wedge t)_{\mu\nu} = {\textstyle{1\over 2}}(s_\mu t_\nu-s_\nu t_\mu).
\end{displaymath}](w_433.gif) |
(11) |
The wedge product is the ``correct'' type of product to use in computing a Volume Element
![\begin{displaymath}
dV=dx_1\wedge \ldots\wedge dx_n.
\end{displaymath}](w_434.gif) |
(12) |
See also Differential Form, Exterior Derivative, Inner Product, Volume Element
© 1996-9 Eric W. Weisstein
1999-05-26