How to see that this is uniformly distributed on the unit sphere?

Question

Assume $X_i$ are independent Gaussian $(0,1)$ and

$$ Y:=\left( Y_1 := \frac{X_1}{\sqrt{X_1^2+...+X_n^2}}, ... , Y_n := \frac{X_n}{\sqrt{X_1^2+...+X_n^2}}\right)$$

Then Y is uniformly distributed on the unit sphere. That's what I want to show at least.

Now because $$Y_1^2+...+Y_n^2=1$$ we obviously see that Y is on the unit sphere - but how can I see that it is also uniformly distributed?

Answer 1

Ultimately, this boils down to the fact that $$e^{-x_1^2}\cdot e^{-x_2^2}\cdot\ldots\cdot e^{-x_n^2} =e^{-(x_1^2+x_2^2+\ldots+x_n^2)}$$ is rotational symmetric.