Английская Википедия:Blade (geometry)

Материал из Онлайн справочника
Перейти к навигацииПерейти к поиску

Шаблон:Short description In the study of geometric algebras, a Шаблон:Math-blade or a simple Шаблон:Math-vector is a generalization of the concept of scalars and vectors to include simple bivectors, trivectors, etc. Specifically, a Шаблон:Math-blade is a [[Multivector|Шаблон:Math-vector]] that can be expressed as the exterior product (informally wedge product) of 1-vectors, and is of grade Шаблон:Math.

In detail:[1]

A vector subspace of finite dimension Шаблон:Math may be represented by the Шаблон:Math-blade formed as a wedge product of all the elements of a basis for that subspace.[6] Indeed, a Шаблон:Math-blade is naturally equivalent to a Шаблон:Math-subspace endowed with a volume form (an alternating Шаблон:Math-multilinear scalar-valued function) normalized to take unit value on the Шаблон:Math-blade.

Examples

In two-dimensional space, scalars are described as 0-blades, vectors are 1-blades, and area elements are 2-blades in this context known as pseudoscalars, in that they are elements of a one-dimensional space distinct from regular scalars.

In three-dimensional space, 0-blades are again scalars and 1-blades are three-dimensional vectors, while 2-blades are oriented area elements. In this case 3-blades are called pseudoscalars and represent three-dimensional volume elements, which form a one-dimensional vector space similar to scalars. Unlike scalars, 3-blades transform according to the Jacobian determinant of a change-of-coordinate function.

See also

Notes

Шаблон:Reflist

References

External links

  1. Шаблон:Cite book
  2. Шаблон:Cite book
  3. Шаблон:Cite book
  4. Шаблон:Cite book
  5. For Grassmannians (including the result about dimension) a good book is: Шаблон:Citation. The proof of the dimensionality is actually straightforward. Take the exterior product of Шаблон:Math vectors <math>v_1\wedge\cdots\wedge v_k</math> and perform elementary column operations on these (factoring the pivots out) until the top Шаблон:Math block are elementary basis vectors of <math>\mathbb{F}^k</math>. The wedge product is then parametrized by the product of the pivots and the lower Шаблон:Math block. Compare also with the dimension of a Grassmannian, Шаблон:Math, in which the scalar multiplier is eliminated.
  6. Шаблон:Cite book