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Could some one post the formula (or a reference) on laplacian of a rank two tensor in curvilinear coordinate? I have spent hours looking for it with no success. in particular, I found two contradicting formulas one in mathworld and the other one from internet.

Mathworld suggests the following formula enter image description here

and I also found the following one somewhere else

enter image description here

The second one seems more reasonable to me as it has terms with two christoffel symbols (I think such terms would appear in the laplacian).

I really appreciate it if someone could help. I think it could be helpful to others too.

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    $\begingroup$ The mathworld formula is notoriously wrong. $\endgroup$ Commented Aug 24, 2024 at 6:18
  • $\begingroup$ @KurtG. Thanks! Any chance you know a reference that writes down all these terms? $\endgroup$ Commented Aug 24, 2024 at 11:03
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    $\begingroup$ I don't. Given the mathworld fiasco every such reference should be checked anyway. Cumbersome but doable. $\endgroup$ Commented Aug 24, 2024 at 11:35
  • $\begingroup$ @KurtG. Could you guide me how to do it? In your answer, how can I calculate the first term for example? It would be very helpful if you could show me how to calculate one of the terms $\endgroup$ Commented Aug 24, 2024 at 13:55
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    $\begingroup$ You will learn more when you try it yourself. The first term $A_{\mu\nu\,;\,\lambda,\kappa}$ is the only one where we still need to do a bit of work. This is the $\kappa$-th partial derivative of (A). $\endgroup$ Commented Aug 24, 2024 at 17:52

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