## Differential k-Form

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

 (1)

where and are the components of a Covariant Tensor. Changing variables from to gives
 (2)

where
 (3)

which is the covariant transformation law. 2-forms can be constructed from the Wedge Product of 1-forms. Let
 (4)

 (5)

then is a 2-form denoted . Changing variables to gives
 (6)

 (7)

so
 (8)

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