|
In the study of geometric algebras, a blade is a generalization of the concept of scalars and vectors to include ''simple'' bivectors, trivectors, etc. Specifically, a -blade is any object that can be expressed as the exterior product (informally ''wedge product'') of vectors, and is of ''grade'' . In detail: *A 0-blade is a scalar. *A 1-blade is a vector. Every vector is simple. *A 2-blade is a ''simple'' bivector. Linear combinations of 2-blades also are bivectors, but need not be simple, and are hence not necessarily 2-blades. A 2-blade may be expressed as the wedge product of two vectors and : *: *A 3-blade is a simple trivector, that is, it may expressed as the wedge product of three vectors , , and : *: *In a space of dimension , a blade of grade is called a ''pseudovector''. *The highest grade element in a space is called a ''pseudoscalar'', and in a space of dimension is an -blade. *In a space of dimension , there are dimensions of freedom in choosing a -blade, of which one dimension is an overall scaling multiplier.〔For Grassmannians (including the result about dimension) a good book is: . The proof of the dimensionality is actually straightforward. Take vectors and wedge them together and perform elementary column operations on these (factoring the pivots out) until the top block are elementary basis vectors of . The wedge product is then parametrized by the product of the pivots and the lower block.〕 For an -dimensional space, there are blades of all grades from 0 to inclusive. A vector subspace of finite dimension may be represented by the -blade formed as a wedge product of all the elements of a basis for that subspace. ==Examples== For example, in 2-dimensional space scalars are described as 0-blades, vectors are 1-blades, and area elements are 2-blades known as pseudoscalars, in that they are one-dimensional objects distinct from regular scalars. In three-dimensional space, 0-blades are again scalars and 1-blades are three-dimensional vectors, but in three-dimensions, areas have an orientation, so while 2-blades are area elements, they are oriented. 3-blades (trivectors) represent volume elements and in three-dimensional space, these are scalar-like—i.e., 3-blades in three-dimensions form a one-dimensional vector space. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Blade (geometry)」の詳細全文を読む スポンサード リンク
|