A differential
-form is a Tensor of Rank
which is antisymmetric under exchange of any
pair of indices. The number of algebraically independent components in
-D is
, where this is a
Binomial Coefficient. In particular, a 1-form (often simply called a ``differential'') is a quantity
![\begin{displaymath}
\omega^1 = b_1\,dx_1+b_2\,dx_2,
\end{displaymath}](d1_841.gif) |
(1) |
where
and
are the components of a Covariant Tensor. Changing variables from
to
gives
![\begin{displaymath}
\omega^1=\sum_{i=1}^2 b_i\,dx_i
=\sum_{i=1}^2\sum_{j=1}^2 b_i{\partial x_i\over \partial y_j}=\sum_{j=1}^2{\bar b}_j\,dy_j,
\end{displaymath}](d1_846.gif) |
(2) |
where
![\begin{displaymath}
{\bar b}_j \equiv \sum_{i=1}^2 b_i {\partial x_i\over \partial y_j},
\end{displaymath}](d1_847.gif) |
(3) |
which is the covariant transformation law. 2-forms can be constructed from the Wedge Product of 1-forms. Let
![\begin{displaymath}
\theta_1\equiv b_1\,dx_1+b_2\,dx_2
\end{displaymath}](d1_848.gif) |
(4) |
![\begin{displaymath}
\theta_2\equiv c_1\,dx_1+c_2\,dx_2,
\end{displaymath}](d1_849.gif) |
(5) |
then
is a 2-form denoted
. Changing variables
to
gives
![\begin{displaymath}
dx_1 ={\partial x_1\over \partial y_1}dy_1+{\partial x_1\over \partial y_2}dy_2
\end{displaymath}](d1_854.gif) |
(6) |
![\begin{displaymath}
dx_2 = {\partial x_2\over \partial y_1}dy_1+{\partial x_2\over \partial y_2}dy_2,
\end{displaymath}](d1_855.gif) |
(7) |
so
Similarly, a 4-form can be constructed from Wedge Products of two 2-forms or four 1-forms
![\begin{displaymath}
\omega^4={\omega_1}^2\wedge{\omega_2}^2 = ({\omega_1}^1\wedge{\omega_2}^1)\wedge ({\omega_3}^1\wedge{\omega_4}^1).
\end{displaymath}](d1_859.gif) |
(9) |
See also Angle Bracket, Bra, Exterior Derivative, Ket, One-Form, Symplectic Form,
Wedge Product
References
Weintraub, S. H. Differential Forms: A Complement to Vector Calculus. San Diego, CA:
Academic Press, 1996.
© 1996-9 Eric W. Weisstein
1999-05-24