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1 - Some mathematical essentials

Published online by Cambridge University Press:  05 June 2012

William I. Newman
Affiliation:
University of California, Los Angeles
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Summary

Scalars, vectors, and Cartesian tensors

Geometry is a vital ingredient in the description of continuum problems. Our treatment will focus on the mathematically simplest representation for this subject. Although curvilinear coordinates can be more natural, they introduce complications that go beyond the scope of this book. The initial part of our treatment will parallel the Cartesian approach of Mase and Mase (1990) rather than the curvilinear approach of Narasimhan (1993) and Fung (1965). Hence, we will adhere to a Cartesian description of problems and be spared the need to distinguish between covariant and contravariant notation. Moreover, we will generally employ second-rank tensors which are matrices that possess some very special and important (coordinate) transformation properties.

We will distinguish between three classes of objects: namely, scalars, vectors, and tensors. In reality, all quantities may be regarded as tensors of a specific rank. Scalar (nonconstant) quantities, such as density and temperature, are zero rank or order tensors, while vector quantities (which have an associated direction, such as velocity) are first-rank tensors. Second-rank tensors, such as the stress tensor, are a special case of square matrices. We will usually denote vector quantities by bold-face lower-case letters, while second-rank tensors will be denoted by bold-face upper-case letters.

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Publisher: Cambridge University Press
Print publication year: 2012

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  • Some mathematical essentials
  • William I. Newman, University of California, Los Angeles
  • Book: Continuum Mechanics in the Earth Sciences
  • Online publication: 05 June 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9780511980121.002
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  • Some mathematical essentials
  • William I. Newman, University of California, Los Angeles
  • Book: Continuum Mechanics in the Earth Sciences
  • Online publication: 05 June 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9780511980121.002
Available formats
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Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Some mathematical essentials
  • William I. Newman, University of California, Los Angeles
  • Book: Continuum Mechanics in the Earth Sciences
  • Online publication: 05 June 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9780511980121.002
Available formats
×