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Differential k-Form

A differential $k$-form is a Tensor of Rank $k$ which is antisymmetric under exchange of any pair of indices. The number of algebraically independent components in $n$-D is ${n\choose p}$, where this is a Binomial Coefficient. In particular, a 1-form (often simply called a ``differential'') is a quantity

\omega^1 = b_1\,dx_1+b_2\,dx_2,
\end{displaymath} (1)

where $b_1=b_1(x_1,x_2)$ and $b_2=b_2(x_1,x_2)$ are the components of a Covariant Tensor. Changing variables from ${\bf x}$ to ${\bf y}$ gives
\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} (2)

{\bar b}_j \equiv \sum_{i=1}^2 b_i {\partial x_i\over \partial y_j},
\end{displaymath} (3)

which is the covariant transformation law. 2-forms can be constructed from the Wedge Product of 1-forms. Let
\theta_1\equiv b_1\,dx_1+b_2\,dx_2
\end{displaymath} (4)

\theta_2\equiv c_1\,dx_1+c_2\,dx_2,
\end{displaymath} (5)

then $\theta_1 \wedge \theta_2$ is a 2-form denoted $\omega^2$. Changing variables $x_1(y_1, y_2)$ to $x_2(y_1,y_2)$ gives
dx_1 ={\partial x_1\over \partial y_1}dy_1+{\partial x_1\over \partial y_2}dy_2
\end{displaymath} (6)

dx_2 = {\partial x_2\over \partial y_1}dy_1+{\partial x_2\over \partial y_2}dy_2,
\end{displaymath} (7)

$\displaystyle dx_1\wedge dx_2$ $\textstyle =$ $\displaystyle \left({{\partial x_1\over \partial y_1}{\partial x_2\over \partia... x_1\over\partial y_2}{\partial x_2\over \partial y_1}}\right)dy_1\wedge dy_2$  
  $\textstyle =$ $\displaystyle {\partial (x_1,x_2)\over \partial (y_1,y_2)} dy_1\wedge dy_2.$ (8)

Similarly, a 4-form can be constructed from Wedge Products of two 2-forms or four 1-forms
\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} (9)

See also Angle Bracket, Bra, Exterior Derivative, Ket, One-Form, Symplectic Form, Wedge Product


Weintraub, S. H. Differential Forms: A Complement to Vector Calculus. San Diego, CA: Academic Press, 1996.

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© 1996-9 Eric W. Weisstein