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Consider working on a domain $\Omega$ in $ R^N$ and we assume that $r=|x|$ and $ \theta$ is the angle between the $x_N$ axis and the $ R^{N-1}$ plane. I am looking at functions and domains that depend only on $ r$ and $ \theta$. Is there a name for these coordinates and is there a reference for a bunch of computations for this type of thing. I can write the the Laplacian in these coordinates but I can't do much else and just having some references that use this stuff would help me a lot.

For instance I think I know how to write a gradient (by copying spherical coordinates) ...but even writing some integrals seems to confusing me. I realize this is not a research level question. thanks

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$r$ is called the radius, $\theta$ is called the polar angle.

I'm not sure exactly what kind of computations you are looking for. Most computations in these coordinates are special cases of computations in differential geometry in general coordinates.

For example, the gradient of a function $f=f(r,\theta)$ is

$$ \begin{aligned} \nabla f &= g^{ij}\frac{\partial f}{\partial x^i}\,\frac\partial{\partial x^j} \\ &= \frac{\partial f}{\partial r}\,\frac{\partial}{\partial r} + \frac1{r^2}\,\frac{\partial f}{\partial\theta}\,\frac\partial{\partial\theta} \end{aligned}$$

because the metric tensor of Euclidean space $\mathbb{R}^N$, in polar coordinates, is

$$ g = \mathrm{d}r^2 + r^2\,\mathrm{d}\theta^2 + r^2(\sin\theta)^2\,g_{\mathbb{S}^{N-2}} \;.$$

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  • $\begingroup$ 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. $\endgroup$
    – Math604
    Commented Mar 12, 2021 at 5:50
  • $\begingroup$ $\frac\partial{\partial\theta}$ 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 $r^{N-1}(\sin\theta)^{N-2}\,\mathrm{d}r\,\mathrm{d}\theta\,\mathrm{dVol}_{\mathbb{S}^{N-2}}$. $\endgroup$
    – Zhuo Min Harold Lim
    Commented Mar 12, 2021 at 5:58
  • $\begingroup$ 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 $\endgroup$
    – Math604
    Commented Mar 12, 2021 at 17:45
  • $\begingroup$ 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. $\endgroup$
    – Zhuo Min Harold Lim
    Commented Mar 20, 2021 at 14:14

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