Physical meaning (and name) of dyadic of second order vector derivative $nabla^{(2)}$ and vector
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I have a vector $mathbf{v}=begin{pmatrix} v_1 & v_2 & v_3 end{pmatrix}^top$ and the matrix $G$ . $$G = begin{pmatrix} partial_1^2v_1 & partial_2^2v_1 & partial_3^2v_1 \ partial_1^2v_2 & partial_2^2v_2 & partial_3^2v_2 \ partial_1^2v_3 & partial_2^2v_3 & partial_3^2v_3end{pmatrix}$$ $G$ is not the same as the gradient of $mathbf{v}$ $$nablamathbf{v}=begin{pmatrix} partial_1v_1 & partial_2v_1 & partial_3v_1 \ partial_1v_2 & partial_2v_2 & partial_3v_2 \ partial_1v_3 & partial_2v_3 & partial_3v_3end{pmatrix}$$ nor is it the Laplacian of $mathbf{v}$ . $$nabla^2mathbf{v}=Deltamathbf{v}=begin{pmatrix} partial_1^2v_1 + partial_2^2v_1 + partial_3^2v_1\ partial_1^2v_2 + partial_2^2v_2 + partial_3^2v_2\partial_1^2v_3 + partial_2^2v_3 + partial_3^2v...