Timeline for name of coordinates and reference (elliptic pde)
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Mar 20, 2021 at 14:14 | comment | added | Zhuo Min Harold Lim | I'm not entirely sure what you're asking here. But such computations (e.g. computation of volume form in local coordinates) are completely standard in Riemannian geometry, you might want to check with a relevant book. | |
| Mar 15, 2021 at 23:02 | vote | accept | Math604 | ||
| Mar 12, 2021 at 17:45 | comment | added | Math604 | another quick question. Is the volume have a $\sin(\theta)$ or? Note the $\theta$ i am using i think is not the standard one... its not between x_N axis and x but rather x and R^{N-1} plane | |
| Mar 12, 2021 at 5:58 | comment | added | Zhuo Min Harold Lim | $\frac\partial{\partial\theta}$ has length $r$ (because $g_{\theta\theta}=r^2$), so the ``unit vector" is $\hat{\theta}=\frac1r\,\frac\partial{\partial\theta}$, and your formula is correct. The volume form is given by the standard formula $\sqrt{\det g}\mathrm{d}x^1\wedge\cdots\wedge\mathrm{d}x^N$. In our case it's $r^{N-1}(\sin\theta)^{N-2}\,\mathrm{d}r\,\mathrm{d}\theta\,\mathrm{dVol}_{\mathbb{S}^{N-2}}$. | |
| Mar 12, 2021 at 5:50 | comment | added | Math604 | thanks for the comment. For the gradient i don't understand the last line. Is there a way to write it with some unit vectors. I figured that $ \nabla f= f_r \hat{r} + \frac{ f_\theta}{r} \hat{\theta}$ where the `hat' are some unit vectors... so that formula is wrong? also suppose i wanted to write out some integrals over $ \Omega$. How do i write the volume elements. | |
| Mar 12, 2021 at 4:50 | review | First posts | |||
| Mar 12, 2021 at 6:31 | |||||
| Mar 12, 2021 at 4:50 | history | answered | Zhuo Min Harold Lim | CC BY-SA 4.0 |