ผลต่างระหว่างรุ่นของ "ผู้ใช้:Keeplearn/กระบะทราย2ExteriorAlgebra"
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บรรทัด 10:
In a precise sense, given by what is known as a [[universal property|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 [[tensor]]s. The definition of the exterior algebra makes sense for spaces not just of geometric vectors, but of other vector-like objects such as [[vector field]]s or [[function (mathematics)|functions]]. In full generality, the exterior algebra can be defined for [[module (mathematics)|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 [[integral|integrated]] over surfaces and higher dimensional [[manifold]]s in a way that generalizes the [[line integral]]s 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 form]]s on ''V'', and the pairing between the exterior algebra and its dual is given by the [[interior product]].
==ตัวอย่างจูงใจ==
<!--The purpose of this section is to motivate the skewness of the exterior product on vectors in ''V''-->
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