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  • ...arentheses, <math>(x\,\,y\,\,z\,\,...)</math>. The set of vectors over a [[field]] is called a [[vector space]]. ...ally inherent that vectors have the same properties as those elements in a field.
    11 KB (1,875 words) - 14:35, 7 January 2025
  • ...ts of a [[set]] (of [[vector]]s) and two operations, vector addition and [[scalar]] multiplication, which obey the following rules: * Scalar multiplication is associative, so if <math>r, s \in F</math> and <math>{\ma
    3 KB (561 words) - 23:47, 20 March 2009
  • ...avier-Stokes equations must be found with the help of computers. This is a field of sciences by its own called computational fluid dynamics. ...osity, and <math>f_i</math> be the <math>i</math>th component of the force field, the equations are:
    3 KB (553 words) - 21:08, 2 May 2022
  • ...M,+)</math> together with an operation <math>R\times M\to M</math> (called scalar multiplication) written as <math>r\cdot x</math> or <math>rx</math>, which If <math>R</math> is a [[field]] then <math>M</math> is a vector space over <math>R</math>.
    883 bytes (156 words) - 19:11, 23 January 2017
  • ...than itself and the zero module, and it is not itself the zero module and scalar products are not all equal to 0. ring of <math>R</math> that is a [[field]]. This is not the case when <math>R</math> is not
    1 KB (253 words) - 10:02, 29 September 2012
  • Given a [[field extension]] <math>F\subseteq K</math>, if <math>\alpha\in K</math> is [[alg ...ay <math>g(x)</math>, of minimum degree. Now multiplying a polynomial by a scalar does not change it's roots, so we can find some nonzero <math>a\in F</math>
    2 KB (361 words) - 22:02, 1 March 2018
  • Suppose <math>A</math> is a <math>n \times n</math> [[matrix]] (over a field <math>K</math>). Then the characteristic polynomial of <math>A</math> is de ...satisfies the relation <math>A\bold{v} = \lambda\bold{v}</math>, for some scalar <math>\lambda \in K</math>. In other words, applying a linear operator to a
    19 KB (3,412 words) - 13:57, 21 September 2022
  • ...<math>n \times n</math> matrices <math>A</math> and <math>B</math> and any scalar <math>c</math>.
    759 bytes (120 words) - 18:27, 2 March 2010
  • The gradient applies when <math>f</math> is a [[scalar]]-valued function of many variables. For example, the gradient of temperatu ...whose values are in the same dimensions as its arguments, the [[Vector#Dot(Scalar) Product|dot product]] of <math>\nabla</math> with <math>\mathbf{f}</math>
    4 KB (734 words) - 19:51, 4 May 2022
  • ...fields]]. In modern mathematics, vector analysis is often taken to be sub-field of [[Differential geometry|differential geometry]]. In terms of university === Scalar fields ===
    9 KB (1,488 words) - 23:07, 19 January 2025