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!

  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 $r$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.