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14. Homogeneous and Isotropic Tensors

DOI: 10.1016/s0076-5392(08)60291-x
  • Mathematics


Publisher Summary This chapter describes homogeneous and isotropic tensors. An isotropic tensor in the Euclidean metric space is defined as a tensor such that its components, in any rectangular system, are unaltered in value by orthogonal transformations of coordinates. The homogeneous tensor poses no mathematical problem because the required condition—for example, the condition of homogeneity, can be satisfied by the direct assumption that the components of the tensor are constant in any rectangular system. A material medium—for example, a fluid or a solid—is said to be homogeneous if its properties are independent of position and isotropic if they are independent of direction. These properties have their precise mathematical expression in the so-called homogeneous and isotropic tensors. By a homogeneous tensor in a Euclidean metric space is understood a tensor whose components are constants in any one of the preferred or rectangular coordinate systems commonly used as reference frames for the medium.

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