Hi all, I have been struggling (really) with this and hope someone can help me out.(adsbygoogle = window.adsbygoogle || []).push({});

I would just like to compute the gradient of a tensor in cylindrical coordinates.

I thought I got the right way to calculate and successfully computed several terms and check against the results given by wikipedia (see attached images).

However, there are some terms I computer are different from what's given in wikipedia.

For example the following term:

[tex] \nabla S = ... +\frac{1}{r}\left[ \frac{\partial S_{\theta z}}{\partial \theta} + S_{rz}\right] e_{\theta} \otimes e_{z} \otimes e_{\theta}+... [/tex]

The formula I use is the following

[tex] \nabla S = \left[ \frac{\partial S_{ij}}{\partial \xi^k} - \Gamma^l_{ki} S_{lj}-\Gamma^l_{kj}S_{il}\right] g^i \otimes g^j \otimes g^k

[/tex]

Denoting [tex]1:r, \ 2:\theta, 3:z[/tex], I know that

[tex]

S_{11} = S_{rr}, \ S_{12} = r S_{r \theta}, \ S_{13} = S_{rz}

[/tex]

[tex]

S_{21} = r S_{\theta r}, \ S_{22} = r^2 S_{\theta \theta}, \ S_{23} = r S_{\theta z}

[/tex]

[tex]

S_{31} = S_{z r}, \ S_{32} = r S_{z \theta}, \ S_{33} = S_{z z}

[/tex]

And

[tex]

g^1 = e_r, \ g^2 =\frac{1}{r} e_{\theta}, \ g^3 = e_{z}

[/tex]

And the non-zeros Christoffel symbols are:

[tex]

\Gamma^2_{12} = \frac{1}{r}, \ \Gamma^1_{22} = -r

[/tex]

Then, using the definition above, I naively think that the term I mentioned above in the very beginning will be computed as

[tex]

\begin{align}

(\nabla S)_{232} &= \left( \frac{\partial S_{23}}{\partial \xi^2}-\Gamma^1_{23} S_{13}-\Gamma^2_{23} S_{23}-\Gamma^3_{23} S_{33}-\Gamma^1_{22} S_{21}-\Gamma^2_{22} S_{22}-\Gamma^3_{22} S_{23} \right) g^2 \otimes g^3 \otimes g^2

\\

&= \left[ \frac{r S_{\theta z}}{\partial \theta} +r (r S_{\theta r}) \right] \frac{1}{r^2} e_{\theta} \otimes e_{z} \otimes e_{\theta} \\

&= \frac{1}{r} \left( \frac{\partial S_{\theta z}}{\partial \theta} + r S_{\theta r} \right) e_\theta \otimes e_z \otimes e_\theta

\end{align}

[/tex]

which is clearly different from what wikipedia says. I don't understand how [tex]S_{rz}[/tex] could possibly remain because it is multiplied by a zero Chirstoffel symbol....

I have been using this approach to successfully calculate many other terms, but it worked. I do not understand why it doesn't work in these terms.

If I turn to another formula for the gradient of a tensor

[tex]

\nabla S = \left( \frac{\partial S^{ij}}{\partial z^k} + S^{lj}\Gamma^i_{lk}+S^{il}\Gamma^j_{lk} \right) g_i \otimes g_j \otimes g^k

[/tex]

it seems that this might work because after expansion,

[tex]S_{rz}[/tex]

is multiplied by [tex]\Gamma^2_{12}[/tex], which is non-zero.

I am very confused about this. What the problem with my first approach?

Can someone help me out? Thanks.

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# Gradient of a tensor in cylindrical coordinates

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