Physical meaning (and name) of dyadic of second order vector derivative $nabla^{(2)}$ and vector
$begingroup$
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 $mathbf{v}$. $$nabla^2mathbf{v}=Deltamathbf{v}=begin{pmatrix} partial_1^2v_1 + partial_2^2v_1 + partial_3^2v_1\ partial_1^2v_2 + partial_2^2v_2 + partial_3^2v_2\partial_1^2v_3 + partial_2^2v_3 + partial_3^2v_3end{pmatrix}$$
However, $G$ is equivalent to $G^*{^top}$ where $G^*$ is the dyadic between a second order vector derivative $nabla^{(2)}$ and the vector $mathbf{v}$. $$G^*=nabla^{(2)}otimesmathbf{v}=nabla^{(2)}mathbf{v}^top=begin{pmatrix}partial_1^2\partial_2^2\partial_3^2end{pmatrix}begin{pmatrix}v_1 & v_2 & v_3end{pmatrix}=begin{pmatrix}partial_1^2v_1 & partial_1^2v_2 & partial_1^2v_3\partial_2^2v_1 & partial_2^2v_2 & partial_2^2v_3\partial_3^2v_1 & partial_3^2v_2 & partial_3^2v_3end{pmatrix}=G^top$$
Do either $G$ or $G^*$ as I have defined them have a physical interpretation or particular name besides the long and drawn out "dyadic of second order vector derivative and a vector"? I also am unsure if $nabla^{(2)}$ has a name or not so I have used my own notation.
vectors tensor-products
asked Dec 20 '18 at 16:57
WnGatRC456WnGatRC456
10811
$endgroup$
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$begingroup$
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 $mathbf{v}$. $$nabla^2mathbf{v}=Deltamathbf{v}=begin{pmatrix} partial_1^2v_1 + partial_2^2v_1 + partial_3^2v_1\ partial_1^2v_2 + partial_2^2v_2 + partial_3^2v_2\partial_1^2v_3 + partial_2^2v_3 + partial_3^2v_3end{pmatrix}$$
However, $G$ is equivalent to $G^*{^top}$ where $G^*$ is the dyadic between a second order vector derivative $nabla^{(2)}$ and the vector $mathbf{v}$. $$G^*=nabla^{(2)}otimesmathbf{v}=nabla^{(2)}mathbf{v}^top=begin{pmatrix}partial_1^2\partial_2^2\partial_3^2end{pmatrix}begin{pmatrix}v_1 & v_2 & v_3end{pmatrix}=begin{pmatrix}partial_1^2v_1 & partial_1^2v_2 & partial_1^2v_3\partial_2^2v_1 & partial_2^2v_2 & partial_2^2v_3\partial_3^2v_1 & partial_3^2v_2 & partial_3^2v_3end{pmatrix}=G^top$$
Do either $G$ or $G^*$ as I have defined them have a physical interpretation or particular name besides the long and drawn out "dyadic of second order vector derivative and a vector"? I also am unsure if $nabla^{(2)}$ has a name or not so I have used my own notation.
vectors tensor-products
asked Dec 20 '18 at 16:57
WnGatRC456WnGatRC456
10811
$endgroup$
add a comment |
$begingroup$
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 $mathbf{v}$. $$nabla^2mathbf{v}=Deltamathbf{v}=begin{pmatrix} partial_1^2v_1 + partial_2^2v_1 + partial_3^2v_1\ partial_1^2v_2 + partial_2^2v_2 + partial_3^2v_2\partial_1^2v_3 + partial_2^2v_3 + partial_3^2v_3end{pmatrix}$$
However, $G$ is equivalent to $G^*{^top}$ where $G^*$ is the dyadic between a second order vector derivative $nabla^{(2)}$ and the vector $mathbf{v}$. $$G^*=nabla^{(2)}otimesmathbf{v}=nabla^{(2)}mathbf{v}^top=begin{pmatrix}partial_1^2\partial_2^2\partial_3^2end{pmatrix}begin{pmatrix}v_1 & v_2 & v_3end{pmatrix}=begin{pmatrix}partial_1^2v_1 & partial_1^2v_2 & partial_1^2v_3\partial_2^2v_1 & partial_2^2v_2 & partial_2^2v_3\partial_3^2v_1 & partial_3^2v_2 & partial_3^2v_3end{pmatrix}=G^top$$
Do either $G$ or $G^*$ as I have defined them have a physical interpretation or particular name besides the long and drawn out "dyadic of second order vector derivative and a vector"? I also am unsure if $nabla^{(2)}$ has a name or not so I have used my own notation.
vectors tensor-products
asked Dec 20 '18 at 16:57
WnGatRC456WnGatRC456
10811
$endgroup$
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 $mathbf{v}$. $$nabla^2mathbf{v}=Deltamathbf{v}=begin{pmatrix} partial_1^2v_1 + partial_2^2v_1 + partial_3^2v_1\ partial_1^2v_2 + partial_2^2v_2 + partial_3^2v_2\partial_1^2v_3 + partial_2^2v_3 + partial_3^2v_3end{pmatrix}$$
However, $G$ is equivalent to $G^*{^top}$ where $G^*$ is the dyadic between a second order vector derivative $nabla^{(2)}$ and the vector $mathbf{v}$. $$G^*=nabla^{(2)}otimesmathbf{v}=nabla^{(2)}mathbf{v}^top=begin{pmatrix}partial_1^2\partial_2^2\partial_3^2end{pmatrix}begin{pmatrix}v_1 & v_2 & v_3end{pmatrix}=begin{pmatrix}partial_1^2v_1 & partial_1^2v_2 & partial_1^2v_3\partial_2^2v_1 & partial_2^2v_2 & partial_2^2v_3\partial_3^2v_1 & partial_3^2v_2 & partial_3^2v_3end{pmatrix}=G^top$$
Do either $G$ or $G^*$ as I have defined them have a physical interpretation or particular name besides the long and drawn out "dyadic of second order vector derivative and a vector"? I also am unsure if $nabla^{(2)}$ has a name or not so I have used my own notation.
vectors tensor-products
vectors tensor-products
asked Dec 20 '18 at 16:57
WnGatRC456WnGatRC456
10811
asked Dec 20 '18 at 16:57
WnGatRC456WnGatRC456
10811
asked Dec 20 '18 at 16:57
WnGatRC456WnGatRC456
10811
asked Dec 20 '18 at 16:57
WnGatRC456WnGatRC456
10811
asked Dec 20 '18 at 16:57
WnGatRC456WnGatRC456
10811
10811
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