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



name of coordinates and reference (elliptic pde)
$r$ 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


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


