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In terms of the standard unit vectors i, j, k of Cartesian 3-space, these specific types of vector-valued functions are given by expressions such as = + + where f(t), g(t) and h(t) are the coordinate functions of the parameter t, and the domain of this vector-valued function is the intersection of the domains of the functions f, g, and h.
One can define the pullback of vector-valued forms by smooth maps just as for ordinary forms. The pullback of an E-valued form on N by a smooth map φ : M → N is an (φ*E)-valued form on M, where φ*E is the pullback bundle of E by φ. The formula is given just as in the ordinary case. For any E-valued p-form ω on N the pullback φ*ω is ...
A vector may also result from the evaluation, at a particular instant, of a continuous vector-valued function (e.g., the pendulum equation). In the natural sciences, the term "vector quantity" also encompasses vector fields defined over a two-or three-dimensional region of space, such as wind velocity over Earth's surface.
In mathematics and physics, the concept of a vector is an important fundamental and encompasses a variety of distinct but related notions. Wikimedia Commons has media related to Vectors . Subcategories
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 following are important identities in vector algebra.Identities that only involve the magnitude of a vector ‖ ‖ and the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension, while identities that use the cross product (vector product) A×B only apply in three dimensions, since the cross product is only defined there.
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For a symmetric matrix A, the vector vec(A) contains more information than is strictly necessary, since the matrix is completely determined by the symmetry together with the lower triangular portion, that is, the n(n + 1)/2 entries on and below the main diagonal.