"Wedge product" redirects here. For the operation on topological spaces, see Wedge sum.
Orientation defined by an ordered set of vectors.
Reversed orientation corresponds to negating the exterior product.
Geometric interpretation of grade n elements in a real exterior algebra for n = 0 (signed point), 1 (directed line segment, or vector), 2 (oriented plane element), 3 (oriented volume). The exterior product of n vectors can be visualized as any n-dimensional shape (e.g. n-parallelotope, n-ellipsoid); with magnitude (hypervolume), and orientation defined by that of its (n − 1)-dimensional boundary and on which side the interior is.[1][2]
In mathematics, the exterior algebra or Grassmann algebra of a vector space is an associative algebra that contains which has a product, called exterior product or wedge product and denoted with , such that for every vector in The exterior algebra is named after Hermann Grassmann,[3] and the names of the product come from the "wedge" symbol and the fact that the product of two elements of are "outside"
The wedge product of vectors is called a blade of degree or -blade. The wedge product was introduced originally as an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues: The magnitude of a 2-blade is the area of the parallelogram defined by and and, more generally, the magnitude of a -blade is the (hyper)volume of the parallelotope defined by the constituent vectors. The alternating property that implies a skew-symmetric property that and more generally any blade flips sign whenever two of its constituent vectors are exchanged, corresponding to a parallelotope of opposite orientation.
The full exterior algebra contains objects that are not themselves blades, but linear combinations of blades; a sum of blades of homogeneous degree is called a k-vector, while a more general sum of blades of arbitrary degree is called a multivector.[4] The linear span of the -blades is called the -th exterior power of The exterior algebra is the direct sum of the -th exterior powers of and this makes the exterior algebra a graded algebra.
The exterior algebra is universal in the sense that every equation that relates elements of in the exterior algebra is also valid in every associative algebra that contains and in which the square of every element of is zero.
The definition of the exterior algebra can be extended for spaces built from vector spaces, such as vector fields and functions whose domain is a vector space. Moreover, the field of scalars may be any field (however for fields of characteristic two, the above condition must be replaced with which is equivalent in other characteristics). More generally, the exterior algebra can be defined for modules over a commutative ring. In particular, the algebra of differential forms in variables is an exterior algebra over the ring of the smooth functions in variables.
^Penrose, R. (2007). The Road to Reality. Vintage books. ISBN 978-0-679-77631-4.
^Wheeler, Misner & Thorne 1973, p. 83
^Grassmann (1844) introduced these as extended algebras (cf. Clifford 1878).
^The term k-vector is not equivalent to and should not be confused with similar terms such as 4-vector, which in a different context could mean an element of a 4-dimensional vector space. A minority of authors use the term -multivector instead of -vector, which avoids this confusion.
In mathematics, the exterioralgebra or Grassmann algebra of a vector space V {\displaystyle V} is an associative algebra that contains V , {\displaystyle...
related exterioralgebra. In 1878, William Kingdon Clifford greatly expanded on Grassmann's work to form what are now usually called Clifford algebras in his...
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differential forms by the interior product. The algebra of differential forms along with the exterior derivative defined on it is preserved by the pullback...
their manipulation is carried out using exterioralgebra. Following Grassmann, developments in multilinear algebra were made by Victor Schlegel in 1872 with...
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enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal...
p-form. That is to say, d is an antiderivation of degree 1 on the exterioralgebra of differential forms (see the graded product rule). The second defining...
multilinear algebra, a multivector, sometimes called Clifford number or multor, is an element of the exterioralgebra Λ(V) of a vector space V. This algebra is...
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supernumber), is an element of the exterioralgebra over the complex numbers. The special case of a 1-dimensional algebra is known as a dual number. Grassmann...
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a duality between the derived category of a symmetric algebra and that of an exterioralgebra. The importance of the notion rests on the suspicion that...
In mathematics, a bivector or 2-vector is a quantity in exterioralgebra or geometric algebra that extends the idea of scalars and vectors. Considering...
mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings...
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bialgebras. Examples of coalgebras include the tensor algebra, the exterioralgebra, Hopf algebras and Lie bialgebras. Unlike the polynomial case above...
matrixPages displaying short descriptions of redirect targets Exterioralgebra – Algebra of exterior/ wedge products Levi-Civita symbol – Antisymmetric permutation...
tensor algebra can be constructed as quotients: these include the exterioralgebra, the symmetric algebra, the Clifford algebra, the Weyl algebra, and the...
{\displaystyle {\rm {d}}} are constant functions. An exterioralgebra or differential graded algebra structure over A {\displaystyle A} means a compatible...
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