Reference for studying polar coordinate

Question

There is a theorem about justification of polar-coordinate in Folland-Real analysis p.78.

I find it somewhat terse (Maybe it's just me)..

I guess this kind of transform is possible even when $S^{n-1}$is defined to be $\{x\in\mathbb{R}^n : ||x||=1\}$ where $||\cdot||$ is an arbitrary norm, not just 2-norm.

This is not the only reason I find it terse.

Is there a webpage or a text treating the polar-coordinate with more explanations neatly?

Answer 1

Well, yes, the notion of polar coordinates in $\Bbb R^2$ can be reformulated in the following form:

Every nonzero vector $v$ can be uniquely written in the form $v=\lambda\cdot e_\varphi$ where $\lambda>0$ is a scalar, and $e_\varphi\in S^1\ =\{x\,:\,\|x\|_2=1\}$.

Here, the unit vectors $e_\varphi$ are in a one-to-one correspondence to the angles $\varphi\,\in [0,2\pi)$, by measuring the angle from the positive ray of the $x$-axis towards the positive ray of the $y$-axis.

Observing that $\lambda$ will be $\|v\|_2$, in this form this has a straightforward generalization to any normed space $(X,\|\_\|)$:

Every nonzero vector $v\in X$ can be uniquely written in the form $v=\lambda\cdot v_1$ where $\lambda>0$ is a scalar, and $v_1\in \{x\,:\,\|x\|=1\}$.

And, of course, we must have $\lambda=\|x\|$ again.