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I'm trying to re-derive a result in a paper that I'm struggling with. Here is the problem:

I wish to calculate [tex] (\nabla \otimes \nabla) h [/tex] where [itex] \nabla [/itex] is defined as [tex] \nabla = \frac{\partial}{\partial r} \hat{\mathbf{r}}+ \frac{1}{r} \frac{\partial}{\partial \psi} \hat{\boldsymbol{\psi}} [/tex] and [itex]h[/itex] is a scalar field.

I got something like this:

[tex] (\nabla \otimes \nabla) = \frac{\partial^2}{\partial r^2}\hat{\mathbf{r}} \otimes \hat{\mathbf{r}} + \frac{\partial}{\partial r} \left( \frac{1}{r} \frac{\partial}{\partial \psi} \right)\hat{\mathbf{r}} \otimes \hat{\boldsymbol{\psi}} + \frac{1}{r} \frac{\partial^2}{\partial \psi \partial r}\hat{\boldsymbol{\psi}} \otimes \hat{\mathbf{r}} + \frac{1}{r^2} \frac{\partial^2}{\partial \psi^2} \hat{\boldsymbol{\psi}} \otimes \hat{\boldsymbol{\psi}} [/tex]

but I have a feeling this is wrong.

After that I wish to calculate [tex] \mathbf{r} \cdot (\nabla \otimes \nabla) [/tex] where [itex] \hat{\mathbf{r}} [/itex] is a position vector in cylindrical coordinates, so [itex] \mathbf{r} = r \hat{\mathbf{r}}[/itex]. However, I'm now struggling with the fact that [itex]\mathbf{r}[/itex] can be written as either [itex]\mathbf{r} \otimes \mathbf{1}[/itex] or [itex] \mathbf{1} \otimes \mathbf{r} [/itex] and I'm not sure which one to choose.

Any insight would be much appreciated.

I wish to calculate [tex] (\nabla \otimes \nabla) h [/tex] where [itex] \nabla [/itex] is defined as [tex] \nabla = \frac{\partial}{\partial r} \hat{\mathbf{r}}+ \frac{1}{r} \frac{\partial}{\partial \psi} \hat{\boldsymbol{\psi}} [/tex] and [itex]h[/itex] is a scalar field.

I got something like this:

[tex] (\nabla \otimes \nabla) = \frac{\partial^2}{\partial r^2}\hat{\mathbf{r}} \otimes \hat{\mathbf{r}} + \frac{\partial}{\partial r} \left( \frac{1}{r} \frac{\partial}{\partial \psi} \right)\hat{\mathbf{r}} \otimes \hat{\boldsymbol{\psi}} + \frac{1}{r} \frac{\partial^2}{\partial \psi \partial r}\hat{\boldsymbol{\psi}} \otimes \hat{\mathbf{r}} + \frac{1}{r^2} \frac{\partial^2}{\partial \psi^2} \hat{\boldsymbol{\psi}} \otimes \hat{\boldsymbol{\psi}} [/tex]

but I have a feeling this is wrong.

After that I wish to calculate [tex] \mathbf{r} \cdot (\nabla \otimes \nabla) [/tex] where [itex] \hat{\mathbf{r}} [/itex] is a position vector in cylindrical coordinates, so [itex] \mathbf{r} = r \hat{\mathbf{r}}[/itex]. However, I'm now struggling with the fact that [itex]\mathbf{r}[/itex] can be written as either [itex]\mathbf{r} \otimes \mathbf{1}[/itex] or [itex] \mathbf{1} \otimes \mathbf{r} [/itex] and I'm not sure which one to choose.

Any insight would be much appreciated.

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