The unit dyadic.A general expression for?

• I
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$$\underline{\underline{I}}\cdot\mathbf{v}=\mathbf{v}\cdot\underline{\underline{I}}= 1\mathbf{v}= \mathbf{v}$$

Meir Achuz
It would be $[{\bf I}]=\sum_i{\bf\hat n}_i{\bf\hat n}_i$ in ny orthogonal coordinate system.
$$\mathbf{I}=g_{ij}\mathbf{a^i}\mathbf{a^j}$$
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.