A Tensor product of gradients

joshmccraney

Does anyone know where I can find the definition of $\nabla \otimes \nabla f$? I tried googling this but nothing comes up. I know it will change depending on the coordinate system, so does anyone know the general definition OR a table for rectangular, spherical, cylindrical coordinates?

Thanks so much.

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Orodruin

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It does not change with the coordinate system. That is the entire point. However, its components in a particular coordinate system may be different.

It is the tensor you obtain by taking the gradient twice.

joshmccraney

It does not change with the coordinate system. That is the entire point. However, its components in a particular coordinate system may be different.

It is the tensor you obtain by taking the gradient twice.
Okay, so in cylindrical coordinates, for example, $\nabla f = \langle f_r , f_\theta r^{-1}, f_z\rangle$. So does this imply $$\nabla \otimes \nabla f = \begin{bmatrix} f_r^2 & f_r f_\theta r^{-1} & f_r f_z\\ f_\theta r^{-1} f_r & f_\theta^2 r^{-2} & f_\theta r^{-1} f_z\\ f_rf_z & f_z f_\theta r^{-1} & f_z^2 \end{bmatrix}$$

Orodruin

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No, you are missing all of the terms involving Christoffel symbols that you get when taking the gradient of a vector

• joshmccraney

joshmccraney

Ohhhhh yeaaaa, because the unit vectors change with position. Is there a table anywhere with this information? I'd prefer not to derive it all from scratch if I can help it.

Chestermiller

Mentor
All you need to do is look up the equations for the gradient of a vector for your particular coordinate system, since $\nabla f$ is a vector.

"Tensor product of gradients"

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