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Vectors, Tensors and Their Representation

  • Ralf Greve
  • Heinz Blatter
Chapter
Part of the Advances in Geophysical and Environmental Mechanics and Mathematics book series (AGEM)

Abstract

In mathematics, a vector is defined as an element of a vector space, and a vector space is a commutative (Abelian) group with a scalar multiplication. This is an abstract definition which has many possible realisations (numbers, functions, geometric objects and so on). For our purposes, it is sufficient to consider one of them, namely the geometric object of an arrow in the three-dimensional, Euclidian, physical space ε. Therefore, in our sense a vector a ∈ ε is an arrow which is characterised by a length and a direction. Physical quantities which can be described by such vectors are, for instance, velocity, acceleration, momentum and force. By contrast, scalars are simple numbers and characterise physical quantities without a direction, like mass, density, temperature etc.

Keywords

Cross Product Scalar Multiplication Geometric Object General Tensor Index Notation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Ralf Greve
    • 1
  • Heinz Blatter
    • 2
  1. 1.Inst. Low Temperature ScienceHokkaido UniversitySapporoJapan
  2. 2.Inst. Atmospheric & Climate ScienceETH ZürichZürichSwitzerland

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