Transformation Theorem on n-dimensional Sphere

Question

$ n \in \mathbb{N},b >0$

Define $S_n$ as the n-dimensional Sphere in $\mathbb{R}^n$.

I cannot figure out the appropiate transformation to use the transformation theorem such that the following integral transforms into an integral having the following area of integration

$\int_{||x||_2^2 \leq b} e^{-\frac{1}{2} ||x||_2^2} \lambda_n(dx) \to \int_0^\sqrt{b}\int_{S_{n-1}} .... $

Does anyone have an idea?

Answer 1

For $f \in L^1(\mathbb{R}^n)$ the formula $$\int_{B_R(0)} f(x) \ dx = \int_0^R \int_{\partial B_\rho(0) } f(\xi) \ dS(\xi) \ d \rho = \int_0^R \int_{\partial B_1(0)} f(\xi)\ dS(\xi)\ \rho^{n-1} \ d\rho $$ holds. In your case you can apply this for $R = \sqrt{b}$.