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I The unit dyadic.A general expression for?

  1. Oct 31, 2017 #1

    rocdoc

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    Gold Member

    Does anyone know of a general expression for the unit dyadic, given in terms of the language of general curvilinear coordinate systems?Perhaps an expression for its components?Perhaps an expression, just appropriate for use with orthogonal coordinate systems?$$~$$
    In cartesian coordinates the unit dyadic$$\underline{\underline{I}}$$is given by
    $$\underline{\underline{I}}=\mathbf{i}\mathbf{i}+\mathbf{j}\mathbf{j}+\mathbf{k}\mathbf{k}$$
    The unit dyadic has the property that the two dot products between it and a vector, act so as to multiply the vector by a unit scalar,as below$$\underline{\underline{I}}\cdot\mathbf{v}=\mathbf{v}\cdot\underline{\underline{I}}= 1\mathbf{v}= \mathbf{v}$$
     
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  3. Oct 31, 2017 #2

    clem

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    Science Advisor

    It would be [itex][{\bf I}]=\sum_i{\bf\hat n}_i{\bf\hat n}_i[/itex] in ny orthogonal coordinate system.
     
  4. Nov 1, 2017 #3
    The expression for the identity tensor (aka the Idemfactor, aka the metric tensor) is given in terms of generalized coordinate basis vectors and Einstein summation convention by:

    $$\mathbf{I}=g_{ij}\mathbf{a^i}\mathbf{a^j}$$
    where $$g_{ij}=\mathbf{a_i}\centerdot \mathbf{a_j}$$with ##\mathbf{a_i}## representing the i'th coordinate basis vector, and ##\mathbf{a^i}## representing the i'th reciprocal basis vector.
     
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