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

$$\omega^1 = b_1\,dx_1+b_2\,dx_2,$$ (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

$$\omega^1=\sum_{i=1}^2 b_i\,dx_i =\sum_{i=1}^2\sum_{j=1}^2 b_i{\partial x_i\over \partial y_j}=\sum_{j=1}^2{\bar b}_j\,dy_j,$$ (2)

where

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

which is the covariant transformation law. 2-forms can be constructed from the Wedge Product of 1-forms. Let

$$\theta_1\equiv b_1\,dx_1+b_2\,dx_2$$ (4)

$$\theta_2\equiv c_1\,dx_1+c_2\,dx_2,$$ (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

$$dx_1 ={\partial x_1\over \partial y_1}dy_1+{\partial x_1\over \partial y_2}dy_2$$ (6)

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

so

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

$$= {\partial (x_1,x_2)\over \partial (y_1,y_2)} dy_1\wedge dy_2.$$ (8)

Similarly, a 4-form can be constructed from Wedge Products of two 2-forms or four 1-forms

$$\omega^4={\omega_1}^2\wedge{\omega_2}^2 = ({\omega_1}^1\wedge{\omega_2}^1)\wedge ({\omega_3}^1\wedge{\omega_4}^1).$$ (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.