Integration Over Spheres in R^d

[Mathmajor2010]

Homework Statement

I'm a bit confused at a single step in a proof. Let $\phi \in L^1(\mathbb{R}) \cap C(\mathbb{R}^d)$ be a function such that for any $\omega \in \mathbb{R}^d$, $\phi(\omega) = \psi(||\omega)||$. That is, the function depends solely on the norm of the vector input, so it is constant on spheres I suppose.

Let $S = \{ x \in \mathbb{R}^d : ||x|| = 1\}$. Then, we have
$$\int_{\mathbb{R}^d} \phi(\omega) e^{-ix^T \omega} d\omega = \int_0^{\infty} t^{d-1} dt \int_{S} \phi(t ||\omega||) e^{-ix^t \omega} dS(\omega)$$

I'm not sure exactly what they did to jump from the first integral to the second. I understand the "idea" is that since the function is constant on spheres, simply integrate on the sphere of radius t and then integrate as t goes over all positive numbers, but I'm not sure how they got that $t^{d-1}$ . It's been a while since I've had a multivariable calculus class, so I'm not sure what I'm missing. I assume this is some coordinate transformation and the function of t comes out of a Jacobian of some sort, but I'm not exactly sure. Any suggestions in the right direction would be great. Thanks!

 

[hunt_mat]

I don't quite understand as the sphere S you have defined is d-1 dimensional.

As an aside the $t^{d-1}$ comes from the Jacobian