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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Gradient of a tensor in cylindrical coordinates

Loading...

Similar Threads for Gradient tensor cylindrical |
---|

I Intuitive explanation for Riemann tensor definition |

I Riemann tensor components |

I Calculating the Ricci tensor on the surface of a 3D sphere |

I Riemann curvature tensor derivation |

I Gradient in the rate of time vs acceleration |

**Physics Forums | Science Articles, Homework Help, Discussion**