Английская Википедия:Blade (geometry)
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 0-blade is a scalar.
- A 1-blade is a vector. Every vector is simple.
- A 2-blade is a simple bivector. Sums of 2-blades are also bivectors, but not always simple. A 2-blade may be expressed as the wedge product of two vectors Шаблон:Math and Шаблон:Math:
- <math>a \wedge b .</math>
- A 3-blade is a simple trivector, that is, it may be expressed as the wedge product of three vectors Шаблон:Math, Шаблон:Math, and Шаблон:Math:
- <math>a \wedge b \wedge c. </math>
- In a vector space of dimension Шаблон:Math, a blade of grade Шаблон:Math is called a pseudovector[2] or an antivector.[3]
- The highest grade element in a space is called a pseudoscalar, and in a space of dimension Шаблон:Math is an Шаблон:Math-blade.[4]
- In a vector space of dimension Шаблон:Math, there are Шаблон:Math dimensions of freedom in choosing a Шаблон:Math-blade for Шаблон:Math, of which one dimension is an overall scaling multiplier.[5]
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
References
- Шаблон:Cite book
- Шаблон:Cite book
- A Lasenby, J Lasenby & R Wareham (2004) A covariant approach to geometry using geometric algebra Technical Report. University of Cambridge Department of Engineering, Cambridge, UK.
- Шаблон:Cite book
External links
- A Geometric Algebra Primer, especially for computer scientists.
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite book
- ↑ 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.
- ↑ Шаблон:Cite book
