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# Wedge Product

The wedge product is the product in an exterior algebra. If and are differential k-forms of degrees and , respectively, then

 (1)

It is not (in general) commutative, but it is associative,

 (2)

and bilinear

 (3)
 (4)

(Spivak 1999, p. 203), where and are constants. The exterior algebra is generated by elements of degree one, and so the wedge product can be defined using a basis for :

 (5)

when the indices are distinct, and the product is zero otherwise.

While the formula holds when has degree one, it does not hold in general. For example, consider :

 (6) (7) (8)

If have degree one, then they are linearly independent iff .

The wedge product is the "correct" type of product to use in computing a volume element

 (9)

The wedge product can therefore be used to calculate determinants and volumes of parallelepipeds. For example, write where are the columns of . Then

 (10)

and is the volume of the parallelepiped spanned by .

Cohomology, Cup Product, Determinant, Differential k-Form, Exterior Algebra, Exterior Derivative, Exterior Power, Inner Product, Module Tensor Product, Vector Space, Volume, Volume Element

This entry contributed by Todd Rowland

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## References

Spivak, M. A Comprehensive Introduction to Differential Geometry, Vol. 1, 2nd ed. Berkeley, CA: Publish or Perish Press, 1979a.Spivak, M. A Comprehensive Introduction to Differential Geometry, Vol. 2, 2nd ed. Berkeley, CA: Publish or Perish Press, 1990a.Spivak, M. A Comprehensive Introduction to Differential Geometry, Vol. 3, 2nd ed. Berkeley, CA: Publish or Perish Press, 1990b.Spivak, M. A Comprehensive Introduction to Differential Geometry, Vol. 4, 2nd ed. Berkeley, CA: Publish or Perish Press, 1979b.Spivak, M. A Comprehensive Introduction to Differential Geometry, Vol. 5, 2nd ed. Berkeley, CA: Publish or Perish Press, 1979c.

Wedge Product

## Cite this as:

Rowland, Todd. "Wedge Product." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/WedgeProduct.html