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Wedge Product

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

alpha ^ beta=(-1)^(pq)beta ^ alpha.
(1)

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

(alpha ^ beta) ^ u=alpha ^ (beta ^ u),
(2)

and bilinear

(c_1alpha_1+c_2alpha_2) ^ beta=c_1(alpha_1 ^ beta)+c_2(alpha_2 ^ beta)
(3)
alpha ^ (c_1beta_1+c_2beta_2)=c_1(alpha ^ beta_1)+c_2(alpha ^ beta_2)
(4)

(Spivak 1999, p. 203), where c_1 and c_2 are constants. The exterior algebra is generated by elements of degree one, and so the wedge product can be defined using a basis e_i for V:

(e_(i_1) ^ ... ^ e_(i_p)) ^ (e_(j_1) ^ ... ^ e_(j_q))=e_(i_1) ^ ... ^ e_(i_p) ^ e_(j_1) ^ ... ^ e_(j_q)
(5)

when the indices i_1,...,i_p,j_1,...,j_q are distinct, and the product is zero otherwise.

While the formula alpha ^ alpha=0 holds when alpha has degree one, it does not hold in general. For example, consider alpha=e_1 ^ e_2+e_3 ^ e_4:

alpha ^ alpha=(e_1 ^ e_2) ^ (e_1 ^ e_2)+(e_1 ^ e_2) ^ (e_3 ^ e_4)+(e_3 ^ e_4) ^ (e_1 ^ e_2)+(e_3 ^ e_4) ^ (e_3 ^ e_4)
(6)
=0+e_1 ^ e_2 ^ e_3 ^ e_4+e_3 ^ e_4 ^ e_1 ^ e_2+0
(7)
=2e_1 ^ e_2 ^ e_3 ^ e_4.
(8)

If alpha_1,...,alpha_k have degree one, then they are linearly independent iff alpha_1 ^ ... ^ alpha_k!=0.

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

dV=dx_1 ^ ... ^ dx_n.
(9)

The wedge product can therefore be used to calculate determinants and volumes of parallelepipeds. For example, write detA=det(c_1,...,c_n) where c_i are the columns of A. Then

c_1 ^ ... ^ c_n=det(c_1,...,c_n)e_1 ^ ... ^ e_n
(10)

and |det(c_1,...,c_n)| is the volume of the parallelepiped spanned by c_1,...,c_n.

See also

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. 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.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. 1, 3rd ed. Houston, TX: Publish or Perish, 1999.

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Wedge Product

Cite this as:

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

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