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A two-vector or bivector [1] is a tensor of type () and it is the dual of a two-form, meaning that it is a linear functional which maps two-forms to the real numbers (or more generally, to scalars). The tensor product of a pair of vectors is a two-vector. Then, any two-form can be expressed as a linear combination of tensor products of pairs of ...
A vector pointing from point A to point B. In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector [1] or spatial vector [2]) is a geometric object that has magnitude (or length) and direction.
In the natural sciences, a vector quantity (also known as a vector physical quantity, physical vector, or simply vector) is a vector-valued physical quantity. [9] [10] It is typically formulated as the product of a unit of measurement and a vector numerical value (), often a Euclidean vector with magnitude and direction.
The transpose (indicated by T) of any row vector is a column vector, and the transpose of any column vector is a row vector: […] = [] and [] = […]. The set of all row vectors with n entries in a given field (such as the real numbers ) forms an n -dimensional vector space ; similarly, the set of all column vectors with m entries forms an m ...
For a finite planar surface of scalar area S and unit normal n̂, the vector area S is defined as the unit normal scaled by the area: = ^. For an orientable surface S composed of a set S i of flat facet areas, the vector area of the surface is given by = ^ where n̂ i is the unit normal vector to the area S i.
Compared to the graph y = f(x), the graph y = f(x − a) has been translated horizontally by a, while the graph y = f(x) + b has been translated vertically by b. The graph of a real function f , the set of points ( x , f ( x ) ) {\displaystyle (x,f(x))} , is often pictured in the real coordinate plane with x as the horizontal coordinate ...
In mathematics and physics, vector notation is a commonly used notation for representing vectors, [1] [2] which may be Euclidean vectors, or more generally, members of a vector space. For denoting a vector, the common typographic convention is lower case, upright boldface type, as in v .
Given a differentiable manifold, a vector field on is an assignment of a tangent vector to each point in . [2] More precisely, a vector field F {\displaystyle F} is a mapping from M {\displaystyle M} into the tangent bundle T M {\displaystyle TM} so that p ∘ F {\displaystyle p\circ F} is the identity mapping where p {\displaystyle p} denotes ...