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14: Appendix C- Isotropic Tensors

  • Page ID
    18041
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    Isotropic tensors play a fundamental role in tensor algebra as well as in fluid mechanics. We can easily test whether or not a given tensor is isotropic, but is there a way to identify all isotropic tensors?

    Because an isotropic tensor is invariant under under every rotation, it must in particular be invariant under very slight rotations. In the limit of an infinitesimal rotation, the rotation matrix is just the identity plus a small added matrix. This allows us to simplify the algebra needed to derive requirements for isotropy.


    14: Appendix C- Isotropic Tensors is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Bill Smyth via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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