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)
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(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:
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 ElementThis 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.Referenced on Wolfram|Alpha
Wedge ProductCite this as:
Rowland, Todd. "Wedge Product." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/WedgeProduct.html