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Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, . [1] The term vector calculus is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration.
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 ...
The cross product operation is an example of a vector rank function because it operates on vectors, not scalars. Matrix multiplication is an example of a 2-rank function, because it operates on 2-dimensional objects (matrices). Collapse operators reduce the dimensionality of an input data array by one or more dimensions. For example, summing ...
There were significant reviews given near the time of original publication. G.J.Whitrow:. Although many books have been published in recent years in which vector and tensor methods are used for solving problems in geometry and mathematical physics, there has been a lack of first-class treatises which explain the methods in full detail and are nevertheless suitable for the undergraduate student.
A free vector is a vector quantity having an undefined support or region of application; it can be freely translated with no consequences; a displacement vector is a prototypical example of free vector. Aside from the notion of units and support, physical vector quantities may also differ from Euclidean vectors in terms of metric.
The simplest example of a vector space over a field F is the field F itself with its addition viewed as vector addition and its multiplication viewed as scalar multiplication. More generally, all n -tuples (sequences of length n ) ( a 1 , a 2 , … , a n ) {\displaystyle (a_{1},a_{2},\dots ,a_{n})} of elements a i of F form a vector space that ...
Scalar multiplication of a vector by a factor of 3 stretches the vector out. A vector may also be multiplied, or re-scaled, by any real number r. In the context of conventional vector algebra, these real numbers are often called scalars (from scale) to distinguish them from vectors.
In other words, a K[x]-module is a K-vector space M combined with a linear map from M to M. Applying the structure theorem for finitely generated modules over a principal ideal domain to this example shows the existence of the rational and Jordan canonical forms. The concept of a Z-module agrees with the notion of an abelian group.