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After proving Fubini-Tonelli theorem a formula on polar coordinates in $\mathbb R^n$ is given in my class as follows. Let $f$ be a real-valued integrable function on $\mathbb R^n$ and $S^{n-1}$ be the surface of $n$-dimensional unit ball with respect to Euclidean norm. Then define a new function $\tilde f(r, \omega) :\mathbb R^+ \times S^{n-1} \to \mathbb R$ by $$\tilde f(r, \omega) = f(r\omega).$$ In addition, let $E$ be a measurable subset of $\mathbb R^n$ and $r>0$. Define $E_r \subset S^{n-1}$ as $$E_r := \{\omega\in S^{n-1}:r\omega \in E\}.$$ Then finally the polar coordinate formula in $\mathbb R^n$ is as follows.

$$ \int_E f(x) dx = \int_0^\infty \left( \int_{E_r} \tilde f(r, \omega)d\omega \right) r^{n-1} dr. $$

I have the following questions about this formula.

  1. How is this formula related to the two-dimensional case? Recall in two dimensional case, we have $x=r\cos\theta$ and $y=r\sin\theta$ and $$\iint f(x, y) dxdy = \iint f(r\cos\theta, r\sin\theta) rdrd\theta.$$ Hence, the general formula is similar but still different, especially the ingegrad $\tilde f(r, \omega)$. How do I intepret this, please?
  2. How can $\mathbb R^n$ be written as $\mathbb R^+ \times S^{n-1} $? Comparing with the two dimensional situation, $r$ seems to be the same radius. However, the angle $\theta$ in $2$-d case is now replaced by $\omega \in S^{n-1}$. How can this be treated as angle like argument, please?
  3. Why and how do we define $\tilde f(r, \omega) = f(r\omega)$?
  4. How to interpret the definition of $E_r := \{\omega\in S^{n-1}:r\omega \in E\}$?

Could anyone explain these to me, please? More detailed answers are appreciated since I know very little here. Thank you!

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  1. When $n=2$, the polar coordinates formula you write is exactly the formula above. Because the elements of $S^1$ are of the form $\omega=(\cos\theta,\sin\theta)$, $0\leq\theta<2\pi$.

  2. Given any $x\in\mathbb R^n$, we can write $x=\|x\|\,\frac{x}{\|x\|}$. Since $\omega=\left\|\frac x{\|x\|}\right\|=1$, we have $\omega\in S^{n-1}$.

  3. It is just notation. You can write $f(r\omega)$ wherever it says $\bar f(r,\omega)$.

  4. $E_r$ is the shadow onto $S^{n-1}$ of the "slice" of $E$ by the sphere of radius $r$.

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