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• Exterior_algebra abstract "In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs. The exterior product of two vectors u and v, denoted by u ∧ v, is called a bivector and lives in a space called the exterior square, a geometrical vector space that differs from the original space of vectors. The magnitude of u ∧ v can be interpreted as the area of the parallelogram with sides u and v, which in three dimensions can also be computed using the cross product of the two vectors. Also like the cross product, the exterior product is anticommutative, meaning that u ∧ v = −(v ∧ u) for all vectors u and v. One way to visualize a bivector is as a family of parallelograms all lying in the same plane, having the same area, and with the same orientation of their boundaries—a choice of clockwise or counterclockwise.When regarded in this manner the exterior product of two vectors is called a 2-blade. More generally, the exterior product of any number k of vectors can be defined and is sometimes called a k-blade. It lives in a geometrical space known as the k-th exterior power. The magnitude of the resulting k-blade is the volume of the k-dimensional parallelotope whose sides are the given vectors, just as the magnitude of the scalar triple product of vectors in three dimensions gives the volume of the parallelepiped spanned by those vectors.The exterior algebra, or Grassmann algebra after Hermann Grassmann, is the algebraic system whose product is the exterior product. The exterior algebra provides an algebraic setting in which to answer geometric questions. For instance, whereas blades have a concrete geometrical interpretation, objects in the exterior algebra can be manipulated according to a set of unambiguous rules. The exterior algebra contains objects that are not just k-blades, but sums of k-blades; such a sum is called a k-vector. The k-blades, because they are simple products of vectors, are called the simple elements of the algebra. The rank of any k-vector is defined to be the smallest number of simple elements of which it is a sum. The exterior product extends to the full exterior algebra, so that it makes sense to multiply any two elements of the algebra. Equipped with this product, the exterior algebra is an associative algebra, which means that α ∧ (β ∧ γ) = (α ∧ β) ∧ γ for any elements α, β, γ. The k-vectors have degree k, meaning that they are sums of products of k vectors. When elements of different degrees are multiplied, the degrees add like multiplication of polynomials. This means that the exterior algebra is a graded algebra.In a precise sense, given by what is known as a universal construction, the exterior algebra is the largest algebra that supports an alternating product on vectors, and can be easily defined in terms of other known objects such as tensors. The definition of the exterior algebra makes sense for spaces not just of geometric vectors, but of other vector-like objects such as vector fields or functions. In full generality, the exterior algebra can be defined for modules over a commutative ring, and for other structures of interest in abstract algebra. It is one of these more general constructions where the exterior algebra finds one of its most important applications, where it appears as the algebra of differential forms that is fundamental in areas that use differential geometry. Differential forms are mathematical objects that represent infinitesimal areas of infinitesimal parallelograms (and higher-dimensional bodies), and so can be integrated over surfaces and higher dimensional manifolds in a way that generalizes the line integrals from calculus. The exterior algebra also has many algebraic properties that make it a convenient tool in algebra itself. The association of the exterior algebra to a vector space is a type of functor on vector spaces, which means that it is compatible in a certain way with linear transformations of vector spaces. The exterior algebra is one example of a bialgebra, meaning that its dual space also possesses a product, and this dual product is compatible with the exterior product. This dual algebra is precisely the algebra of alternating multilinear forms on V, and the pairing between the exterior algebra and its dual is given by the interior product.".
• Exterior_algebra thumbnail N_vector_positive_png_version.png?width=300.
• Exterior_algebra wikiPageExternalLink burali-forti_-_diff._geom._following_grassmann.pdf.
• Exterior_algebra wikiPageExternalLink burali-forti_-_grassman_and_proj._geom..pdf.
• Exterior_algebra wikiPageExternalLink grassmann_-_mechanics_and_extensions.pdf.
• Exterior_algebra wikiPageExternalLink purl?PPN534901565.
• Exterior_algebra wikiPageExternalLink linalg.
• Exterior_algebra wikiPageExternalLink index.htm.
• Exterior_algebra wikiPageExternalLink 978-3-642-30993-9.
• Exterior_algebra wikiPageID "221537".
• Exterior_algebra wikiPageRevisionID "603117824".
• Exterior_algebra author "Onishchik, A.L.".
• Exterior_algebra caption "Orientation defined by an ordered set of vectors.".
• Exterior_algebra caption "Reversed orientation corresponds to negating the exterior product.".
• Exterior_algebra footer "Geometric interpretation of grade n elements in a real exterior algebra for , 1 , 2 , 3 . The exterior product of n vectors can be visualized as any n-dimensional shape ; with magnitude , and orientation defined by that on its -dimensional boundary and on which side the interior is.".
• Exterior_algebra hasPhotoCollection Exterior_algebra.
• Exterior_algebra id "E/e037080".
• Exterior_algebra image "N vector negative png version.png".
• Exterior_algebra image "N vector positive png version.png".
• Exterior_algebra title "Exterior algebra".
• Exterior_algebra width "220".
• Exterior_algebra subject Category:Algebras.
• Exterior_algebra subject Category:Differential_forms.
• Exterior_algebra subject Category:Multilinear_algebra.
• Exterior_algebra type Abstraction100002137.
• Exterior_algebra type Algebra106012726.
• Exterior_algebra type Algebras.
• Exterior_algebra type Cognition100023271.
• Exterior_algebra type Content105809192.
• Exterior_algebra type DifferentialForms.
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• Exterior_algebra type Mathematics106000644.
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• Exterior_algebra type PureMathematics106003682.
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• Exterior_algebra type Word106286395.
• Exterior_algebra comment "In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs. The exterior product of two vectors u and v, denoted by u ∧ v, is called a bivector and lives in a space called the exterior square, a geometrical vector space that differs from the original space of vectors.".
• Exterior_algebra label "Algebra esterna".
• Exterior_algebra label "Algèbre extérieure".
• Exterior_algebra label "Exterior algebra".
• Exterior_algebra label "Graßmann-Algebra".
• Exterior_algebra label "Внешняя алгебра".
• Exterior_algebra label "外代数".
• Exterior_algebra label "外積代数".
• Exterior_algebra sameAs Graßmann-Algebra.
• Exterior_algebra sameAs Algèbre_extérieure.
• Exterior_algebra sameAs Algebra_esterna.
• Exterior_algebra sameAs 外積代数.
• Exterior_algebra sameAs 외대수.
• Exterior_algebra sameAs m.01gbjw.
• Exterior_algebra sameAs Q1196652.
• Exterior_algebra sameAs Q1196652.
• Exterior_algebra sameAs Exterior_algebra.
• Exterior_algebra wasDerivedFrom Exterior_algebra?oldid=603117824.
• Exterior_algebra depiction N_vector_positive_png_version.png.
• Exterior_algebra isPrimaryTopicOf Exterior_algebra.